Chapter 21 Chapter 21 The Kinetic Theory of The Kinetic Theory of Gases Gases
Dec 18, 2015
Chapter 21Chapter 21
The Kinetic Theory of GasesThe Kinetic Theory of Gases
Molecular Model of an Ideal GasMolecular Model of an Ideal Gas
The model shows that the pressure that a gas The model shows that the pressure that a gas exerts on the walls of its container is a exerts on the walls of its container is a consequence of the collisions of the gas molecules consequence of the collisions of the gas molecules with the wallswith the walls
It is consistent with the macroscopic description It is consistent with the macroscopic description developed earlierdeveloped earlier
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd Look at the motion Look at the motion
of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energy Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
After colliding with the right hand wall After colliding with the right hand wall
the the x x - component of moleculersquos - component of moleculersquos velocity will not change until it hits the velocity will not change until it hits the left-hand wall and its left-hand wall and its x x - velocity will - velocity will again reverses again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy The average force due to the The average force due to the
each molecule on the walleach molecule on the wall
To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molecular Model of an Ideal GasMolecular Model of an Ideal Gas
The model shows that the pressure that a gas The model shows that the pressure that a gas exerts on the walls of its container is a exerts on the walls of its container is a consequence of the collisions of the gas molecules consequence of the collisions of the gas molecules with the wallswith the walls
It is consistent with the macroscopic description It is consistent with the macroscopic description developed earlierdeveloped earlier
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd Look at the motion Look at the motion
of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energy Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
After colliding with the right hand wall After colliding with the right hand wall
the the x x - component of moleculersquos - component of moleculersquos velocity will not change until it hits the velocity will not change until it hits the left-hand wall and its left-hand wall and its x x - velocity will - velocity will again reverses again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy The average force due to the The average force due to the
each molecule on the walleach molecule on the wall
To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd Look at the motion Look at the motion
of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energy Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
After colliding with the right hand wall After colliding with the right hand wall
the the x x - component of moleculersquos - component of moleculersquos velocity will not change until it hits the velocity will not change until it hits the left-hand wall and its left-hand wall and its x x - velocity will - velocity will again reverses again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy The average force due to the The average force due to the
each molecule on the walleach molecule on the wall
To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd Look at the motion Look at the motion
of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energy Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
After colliding with the right hand wall After colliding with the right hand wall
the the x x - component of moleculersquos - component of moleculersquos velocity will not change until it hits the velocity will not change until it hits the left-hand wall and its left-hand wall and its x x - velocity will - velocity will again reverses again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy The average force due to the The average force due to the
each molecule on the walleach molecule on the wall
To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Pressure and Kinetic EnergyPressure and Kinetic Energy
Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd Look at the motion Look at the motion
of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energy Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
After colliding with the right hand wall After colliding with the right hand wall
the the x x - component of moleculersquos - component of moleculersquos velocity will not change until it hits the velocity will not change until it hits the left-hand wall and its left-hand wall and its x x - velocity will - velocity will again reverses again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy The average force due to the The average force due to the
each molecule on the walleach molecule on the wall
To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Pressure and Kinetic EnergyPressure and Kinetic Energy Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
After colliding with the right hand wall After colliding with the right hand wall
the the x x - component of moleculersquos - component of moleculersquos velocity will not change until it hits the velocity will not change until it hits the left-hand wall and its left-hand wall and its x x - velocity will - velocity will again reverses again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy The average force due to the The average force due to the
each molecule on the walleach molecule on the wall
To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Pressure and Kinetic EnergyPressure and Kinetic Energy The average force due to the The average force due to the
each molecule on the walleach molecule on the wall
To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
N
v
V
mN
V
vmP xx
22
Since is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
2xvV
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv 2
yv 2zv
Since the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = 3v = 3vxx22 or or vvxx
22 = v = v223 3 Then the expression for Then the expression for
pressurepressure
2222zyx vvvv
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Pressure and Kinetic EnergyPressure and Kinetic Energy
The relationship can be writtenThe relationship can be written
This tells us that pressure is proportional to the This tells us that pressure is proportional to the number of molecules per unit volume (number of molecules per unit volume (NNVV) and ) and to the average translational kinetic energy of the to the average translational kinetic energy of the moleculesmolecules
___22 1
3 2
NP mv
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Pressure and Kinetic EnergyPressure and Kinetic Energy
This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
A A 200-mol 200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L 500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
TnkmvV
NP B
2
2
1
3
2
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molecular Interpretation of Molecular Interpretation of TemperatureTemperature
Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2mv k T
___2
B
1 1
2 2xmv k T
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
A Microscopic Description of A Microscopic Description of TemperatureTemperature
Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Total Kinetic EnergyTotal Kinetic Energy The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy
of each moleculeof each molecule
If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N mv Nk T nRT
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Root Mean Square SpeedRoot Mean Square Speed The root mean square (The root mean square (rmsrms) speed is the square root of ) speed is the square root of
the average of the squares of the speedsthe average of the squares of the speeds Square average take the square rootSquare average take the square root
Solving for Solving for vvrmsrms we findwe find
MM is the molar mass and is the molar mass and MM = = mNmNAA
Brms
3 3k T RTv
m M
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Some Example Some Example vvrmsrms ValuesValues
At a given At a given temperature lighter temperature lighter molecules move molecules move faster on the faster on the average than average than heavier moleculesheavier molecules
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molar Specific HeatMolar Specific Heat
Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
The heat is different for the The heat is different for the different pathsdifferent paths
The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molar Specific HeatMolar Specific Heat
We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molar Specific HeatMolar Specific Heat Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processes for constant-volume processes QQ = = nCnCPP ΔΔTT for constant-pressure processes for constant-pressure processes
QQ (constant pressure) must account for both the (constant pressure) must account for both the increase in internal energy and the transfer of increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Ideal Monatomic GasIdeal Monatomic Gas
A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gas There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Ideal Monatomic GasIdeal Monatomic Gas
Therefore Therefore
ΔΔEEintint is a function of is a function of TT only only At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3int
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Monatomic GasesMonatomic Gases
Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gases This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3int
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Monatomic GasesMonatomic Gases In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
Change in internal energy depends only on temperature Change in internal energy depends only on temperature for an ideal gas and therefore are the same for the for an ideal gas and therefore are the same for the constant volume process and for constant pressure constant volume process and for constant pressure process process
CCPP ndash C ndash CVV = R = R
VPTnCWQE P int
TnRTnCTnC PV
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Monatomic GasesMonatomic Gases
CCPP ndash ndash CCVV = = RR This also applies to any ideal gasThis also applies to any ideal gas
CCPP = 52 = 52 RR = 208 Jmol = 208 Jmol K K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
We can also define We can also define
Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
A A 100-mol 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) at at 300 K300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
A A 100-mol 100-mol sample of air (a diatomic ideal gas) at sample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume of volume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas after after 440 kJ440 kJ of energy is transferred to the air by heat of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is valid is valid
The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
= = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Equipartition of EnergyEquipartition of Energy
With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Equipartition of EnergyEquipartition of Energy
Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to the compared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Equipartition of EnergyEquipartition of Energy
The molecule can The molecule can also vibratealso vibrate
There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Equipartition of EnergyEquipartition of Energy
The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Agreement with ExperimentAgreement with Experiment Molar specific heat is a function of temperatureMolar specific heat is a function of temperature At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Agreement with ExperimentAgreement with Experiment
At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Complex MoleculesComplex Molecules
For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
The number of degrees of freedom is largerThe number of degrees of freedom is larger The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Quantization of EnergyQuantization of Energy
To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the frequencyrepresenting the frequency
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Quantization of EnergyQuantization of Energy
This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Quantization of EnergyQuantization of Energy
The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Quantization of EnergyQuantization of Energy
Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Quantization of EnergyQuantization of Energy
At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molar Specific Heat of SolidsMolar Specific Heat of Solids
Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
DuLong-Petit LawDuLong-Petit Law
At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R3R
This occurs above This occurs above 300 K300 K
The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT and and CCvv = 3 = 3 RR
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Molar Specific Heat of SolidsMolar Specific Heat of Solids
As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c JmolKc JmolK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
Alcohol (ethyl)Alcohol (ethyl) 2424 111111
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
The molar mass of copper is 635 gmol Use the Dulong-Petit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per mole
The Dulong-Petit lawgives c in terms of R
c= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar mass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Boltzmann Distribution LawBoltzmann Distribution Law
The motion of molecules is extremely chaoticThe motion of molecules is extremely chaotic Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Number Density and Boltzmann Number Density and Boltzmann Distribution LawDistribution Law
From statistical mechanics the number density From statistical mechanics the number density is is
nnV V ((E E ) = ) = nnooe e ndashndashE E kkBBTT
This equation is known as the Boltzmann This equation is known as the Boltzmann distribution lawdistribution law
It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy divided by exponentially as the energy divided by kkBBTT
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Distribution of Molecular SpeedsDistribution of Molecular Speeds
The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown equilibrium is shown at rightat right
NNVV is called the is called the
Maxwell-Boltzmann Maxwell-Boltzmann speed distribution speed distribution functionfunction
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Distribution FunctionDistribution Function
The fundamental expression that describes the The fundamental expression that describes the distribution of speeds in distribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Most Probable SpeedMost Probable Speed
The average speed is somewhat lower than the The average speed is somewhat lower than the rmsrms speed speed
The most probable speed The most probable speed vvmpmp is the speed at is the speed at
which the distribution curve reaches a peakwhich the distribution curve reaches a peak
B Bmp
2141
k T k Tv
m m
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Speed DistributionSpeed Distribution The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is infinity and the highest is infinity
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Speed DistributionSpeed Distribution
The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
The speed distribution for liquids is similar to that The speed distribution for liquids is similar to that of gasesof gases
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
EvaporationEvaporation Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan others Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Mean Free PathMean Free Path
A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Mean Free PathMean Free Path
The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Mean Free PathMean Free Path
The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
We assume that the molecules are spheres of We assume that the molecules are spheres of diameter diameter dd
No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Mean Free PathMean Free Path
The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvΔΔtt traveled in a time interval traveled in a time interval ΔΔtt divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
Collision FrequencyCollision Frequency
The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn