IB PHYSICSSUBSIDIARY LEVEL
THERMAL PHYSICS
NOTES
3.1
Thermal Concepts3.1.1
Molecular theory of solids, liquids and gases
Whether a substance is a solid, liquid or gas at room
temperature, depends on the forces between its particles, the
distance between the particles and how fast they are moving.
The strength of the forces between particles depends on:
*whether they are ions, molecules or atoms. For molecules, the
forces depend on which atoms are present and the structure of the
molecule.
*the distance between the particles the greater the distance,
the weaker the force.
*the speed of the particles the faster the speed, the greater
the distance between them and the weaker the force.
The table below summarises the properties of the states of
matter in terms of molecular motion.
StatePropertyParticle behavior
solidFixed shape and volume.
Specific melting point when pure.
Most expand when heated, contract
when cooled.
Can't be compressed.Particles held close together by strong
forces and are stacked neatly in a crystal lattice. Particles
vibrate on the spot with average kinetic energy much less than the
binding energy - energy that must be absorbed for a particle to
break free.
liquidShape varies to fit lowest part of the container.
Specific boiling point when pure.
Most expand when heated, contract
when cooled.
Specific melting point when pure.
Can't be compressed.Particles close but slightly further apart
than in a solid. Forces between particles are weak allowing
movement of particles around each other. Average kinetic energy of
the particles is similar in size to the binding energy.
gasNo shape, will expand to completely fill any container.
Exert a pressure on the walls of the container that depends on
the number of particles, the volume and the temp.
Can be compressed considerably.Particles are far apart compared
to their size and exert virtually no force on each other except
during collisions. Average kinetic energy much greater than the
energy binding them together when in liquid state.
3.1.2
Temperature and absolute temperature
Temperature is a measure of the average kinetic energy of the
vibrations called thermal motion of the particles in a substance.
The higher the temperature, the faster the thermal motion. When
moving particles collide, energy is transferred from the one with
more KE to the one with less KE. This is why a warm object cools
down and a cool object warms up when they are in contact until the
two are at the same temperature. This is called thermal
equilibrium. Thermal energy is transferred from the object at the
higher temperature to the object at the lower temperature until the
two are in thermal equilibrium.
A temperature scale is constructed by choosing two reproducible
events such as the melting of ice and the boiling of water. The
temperature values of these two events are then chosen. For the
Celsius scale these are 0oC and 100oC. For the Fahrenheit scale
these are 32oF and 212oF. A thermometer is constructed that
contains a liquid with a freezing point below that of ice and a
boiling point above that of water. Mercury is suitable. It is
placed in a reservoir with thin glass walls that enable rapid
thermal energy transfer from an object to the mercury. Up from the
reservoir is a thin tube that will cause a large change in the
level of the mercury over the temperature range. The thermometer is
put in melting ice and the level of mercury marked. The thermometer
is put in boiling water and the level marked again.
2.
These marks are named 0oC and 100oC or 32oF and 212oF depending
on the scale chosen. The distance between the marks is divided up
into 100 equal divisions for the Celsius thermometer and 180
divisions for the Fahrenheit thermometer.
The Celsius scale is based on the temperature of melting ice
being assigned the value zero. This is an arbitrary zero. At 0oC,
the H2O molecules still have thermal motion. Try dropping an ice
cube into liquid air. It causes the liquid air to boil! Recall that
temperature is related to the thermal motion of particles so it
follows that absolute zero of temperature is when thermal motion of
particles ceases. The Kelvin temperature scale starts at absolute
zero and each degree is the same as the Celsius scale. Zero Kelvin
is -273.14 oC but this is conveniently rounded off to -273 oC. The
word degree is omitted when using the Kelvin scale.
The following equivalences apply:
0K = -273 oC and 273K = 0oC
Thus Kelvin = Celsius + 2733.1.3
Internal energy
The particles of a substance move and rotate. They have kinetic
energy of motion and rotation. Forces between particles means
potential energy is stored in them. The total of the potential and
kinetic energies of the particles is called the internal energy of
the substance.
Previously it was stated that temperature of a substance is
related to the motion of its particles More specifically,
temperature is a measure of the average kinetic energy of the
particles of a substance and will be proven to be true later in
this study. The temperature of a substance determines if internal
energy will be transferred to or from its surroundings.
Internal energy is the total of the kinetic and potential
energies of the particles of the substance.
Conduction transfers internal energy within a substance and from
one substance to another in contact when a temperature difference
exists.
During collisions, the particles with the greater kinetic energy
slow down and the ones with smaller kinetic energy speed up. The
higher temperature substance cools and the lower temperature
substance warms up until the temperatures are the same. Solids are
the best conductors because the particles are stacked close
together and collisions occur readily between them. Gases are the
poorest conductors because the particles are far apart and
collisions between them occur less often than in solids.
Heat is infra-red electromagnetic waves that travel at the speed
of light. Thermal energy is the energy transferred to or from a
substance when heat is absorbed or radiated away. When heat is
absorbed, the internal energy of a substance increases. When heat
is radiated away, its internal energy decreases. 3.1.4
Specific heat capacityThermal Capacity is the internal energy
increase needed to cause a 1oC increase in the temperature of an
object. The value depends on the mass of the object and the
substance that makes up the object.
The bigger the mass, the more particles it contains and the more
internal energy needed to increase their average kinetic energy. It
takes longer to boil the water in the electric jug when it is
full!
The forces between the particles of a substance are different
for different substances. This means the ease of increasing the
average kinetic energy of the particles varies. Less internal
energy increase occurs when 100 g of oil has its temp raised 1 oC
than for 100 g of water.
The unit of Heat Capacity is Joule per oC, written as J oC-1.
The change in the internal energy of an object is equal to its Heat
Capacity times its change in temperature.
i.e. H = HC x T
3.
The Specific Heat Capacity is the Heat Capacity of a specific
mass of sample. This is either the gram or the kilogram. Strictly
speaking, SI units should involve the kilogram but the gram is in
common usage with Specific Heat Capacities. Thus the Specific Heat
Capacity (SHC) is the change in internal energy when 1 kg of a
substance has its temperature changed by 1 oC. The unit of SHC is
Joule per kilogram per oC or Joule per gram per oC, written as J
kg-1 oC-1 and J g-1 oC-1.The change in the internal energy of an
object is equal to its specific heat capacity times its mass times
its change in temperature.
i.e. H = SHC x m x T
Some examples of substances and their Specific Heat Capacities
are:
water4200 J kg-1 oC-1
ice 2100 J kg-1 oC-1 lead130 J kg-1 oC-1
The diagram opposite shows a direct way of determining the
Specific Heat Capacity of water. An electric current raises the
temperature of the heating coil that causes an increase in the
internal energy of the water. The electrical energy transferred to
heat energy can be determined from the readings on the meters and
the heating time. The mass of the water and its rise in temperature
are measured. Assuming complete transfer of electrical energy to
internal energy of the water, a value for the Specific Heat
Capacity of water can be calculated.
The steel can and the parts of the calorimeter dipping in the
water also have their internal energy raised. Corrections for these
must be introduced if an accurate value for the SHC of water is to
be obtained.
e.g. The arrangement above caused a 6.4oC rise in temp. when 100
g of water was heated for 2.0 min. Calculate the SHC of water.
Electrical energy transfer = 7.0 J C-1 x 3.5 C s-1 x 120 sec
= 2900 J
Internal energy increase = SHC x m x T
2900 = SHC x 0.10 x 6.4
SHC = 4600 J kg-1 oC-1The accepted value for the SHC of water is
4186 J kg-1 oC-1. A larger value was obtained because the internal
energy of the calorimeter was increased. Consequently, the
temperature rise of the water was less than if only the water's
internal energy was increased. This produced the larger value for
the SHC.
The error can be reduced, by including the increase in the
internal energy of the calorimeter. The mass of the steel in the
calorimeter was 80 g and its SHC is 450 J kg-1 oC-1.
Internal energy increase = SHCwater x m x T + SHCsteel x m x
T
2900 = SHCwater x 0.10 x 6.4 + 450 x 0.080 x 6.4
SHCwater = 4200 J kg-1 oC-1
4.The diagram opposite shows an indirect way of finding the SHC
of copper. The cube has been heated and its temperature measured.
The cube was placed in a calorimeter containing a known mass of
water at a known temperature. The temperature at equilibrium can be
used to calculate the SHC of copper.
e.g. The calorimeter used contained 80 mL of water at 15.0oC. A
150 g
cube of copper was placed in a beaker of boiling water for
several minutes.
The cube was removed and without delay, placed in the
calorimeter. The
temperature at equilibrium was 26.5oC.
Internal energy decrease of Cu = SHCCu x 0.15 x (100.0 -
26.5)
IE incr. of H2O + IE incr. of steel = SHCH2O x 0.080 x
(26.5-15.0) + 450 x 0.080 x (26.5-15.0)
= 4200 x 0.080 x 11.5 + 450 x 0.080 x 11.5
= 4.3 x 103 J
Assuming the internal energy decrease of Cu is equal to the
internal energy increase of the H2O and the steel:
SHCCu x 0.15 x (100.0 - 26.5) = 4.3 x 103
SHCCu = 390 J kg-1 oC-1
The thermal capacity of the 150 g cube of copper = 390 x
0.150
= 58.5 J oC-1
3.1.5
Phase changeAfter it has been raining, the water on the ground
evaporates. Evaporation of a liquid can occur even though it is not
boiling. A few of its particles have kinetic energy greater than
the binding energy and escape into the air. The temperature of the
liquid can increase as it evaporates.
When the liquid is boiling, a large number of its particles have
kinetic energy greater than the binding energy and escape.
Evaporation occurs at the fastest rate but the temperature remains
constant,
In a liquid, the average kinetic energy of the particles is
similar in size to the binding energy.
Some of the particles have more kinetic energy than the average
and some have less than the average. Particles with more KE than
the binding energy, move to the top of the liquid and escape from
its surface. This will occur even if the liquid is not boiling.
When the particles with more KE than average leave the liquid, the
average KE of the particles remaining becomes lower. Temperature is
a measure of the average KE so evaporation of a liquid causes its
temperature to drop. This is why evaporation of perspiration cools
your body.
ChangeSubstance behaviorParticle behavior
meltingAs temperature increases, the solid
expands and when a certain temperature is reached, the solid
loses its fixed shape and the material becomes runny.As temp
increases, particles vibrate faster. Distance between particles
increases slightly and forces between particles weaken. Binding
energy and av. KE become similar.
evaporatingAs temp. increases, the volume of liquid gradually
decreases. The amount of the gaseous state increases in the area
above the liquid.As temp. increases, particles vibrate faster. When
the KE of a particle exceeds the binding energy, it moves off by
itself.
sublimingIf the pressure of the air above a solid is
sufficiently low it changes straight to a gas.At low air pressure,
the KE of particles exceeds their binding energy at low temps.
When a substance is changing state, its temperature remains the
same. The internal energy increases during the process, but this is
hidden by the temperature remaining constant. This hidden internal
energy increase is called latent heat. The table on the next page
summarises the processes of internal energy increases for the
melting of ice and the evaporation of water.
5.
ChangeSubstance behaviorParticle behavior
meltingWhen ice is melting, the temperature of the ice/water
mixture remains at 0oC until all the ice has meltedHeat energy must
be absorbed if ice is to melt. The internal energy of the molecules
increases during the melting process. The internal energy absorbed
does not increase their average KE. It increases the potential
energy stored between the molecules.
evaporatingWhen water is boiling, its temperature remains at
100oC. If the evaporated steam is trapped in a sealed container and
heating continues, the temperature will rise above 100oC. A
pressure cooker does this.Heat energy must be absorbed if water is
to evaporate. The internal energy of the molecules increases during
the evaporating process. The internal energy absorbed does not
increase their average KE. It increases the potential energy stored
between the molecules.
For a typical substance surrounded by normal air pressure, the
graph of its temperature versus
heat energy added is shown below.
The state of a substance depends on both its temperature and the
pressure of the air around it.
The graph of pressure versus temperature shows the conditions
when it will be a solid a liquid
or a gas.
The triple point marks the only temperature and pressure that
the solid, liquid and gas states can
exist together. The critical point marks the temperature that if
exceeded, increasing the pressure
will not cause the gas to become a liquid. The molecules might
be very close but no horizontal
surface forms. Deposition is when a gas changes straight to a
solid. It occurs in clouds
when water vapour turns into snowflakes when the temperature and
pressure are low enough.
6.
For carbon dioxide, the triple point is at -56 OC and 5.1
atmosphere. Therefore when placed on
the table and warming up at 1 atmosphere of pressure, it will
sublime.
For water, the triple point is at 0 OC and 0.006 atmosphere.
Therefore when ice is placed on
the table and warming up at 1 atmosphere of pressure, it will
melt.
Freeze dried food is created by freezing the food so the water
content turns into solid ice. Then,
the air pressure around the food is dropped sufficiently and
warmed slightly until the ice
sublimes and leaves the food as water vapour. This makes the
food light for hikers to carry but
free from bacteria that would multiply if the water content were
removed by strongly heating it.3.1.6
Specific latent heat.
Specific latent heat is the amount of internal energy increase
needed to cause 1 kg of a substance to change its state without
changing its temperature. Specific latent heat of Fusion is for
melting and Specific latent heat of Vaporization is for
evaporating. The unit is J kg-1.
LH of Fusion of ice = 3.33 x 105 J kg-1 LH of Vaporization of
water = 2.26 x 106 J kg-1The LH of V is greater than the LH of F
due to a much greater increase in the distance between the
molecules during evaporation and hence a larger increase in the
potential energy between the molecules.
From the definition of Latent Heat it follows that:
Change in IE during a change of state J = Latent Heat of
substance J kg-1 x mass in kg
A saucepan contains 2.0 kg of water at 20oC. When heated on the
stove, the internal energy of the water increases by 9.0 x105 J.
How much of the water evaporated?
The LHV of water is 2.3 x 106 J kg-1.
Increase in IE = Increase for water 20 to 100oC + Increase for
water 100oC to steam 100oC
= SHCwater x mwater x T + LHVwater x mwater evaporated
9.0 x105 = 4186 x 2.0 x 80 + 2.3 x 106 x m
9.0 x105 = 6.7 x 105 + 2.3 x 106 x m
m = 0.1 kg
3.2
Modelling a gas3.2.1
Pressure.The pressure on a surface is equal to the force on the
surface divided by its area. P = F/A. The unit of pressure is N
m-2. A pressure of 1 N m-2 is called 1 Pascal Pa.
Imagine a very thin sheet of plywood with a mass of 100 g and an
area of 1 m2. The wood weighs about 1 N. When the wood lies flat on
the floor it applies a pressure of 1 Pa on the floor. The Pascal is
a small unit. Air applies a force of about 10 N on each cm2. This
amounts to about105 N m-2 or 105 Pa. The unit kiloPascal kPa is
used frequently. Normal air pressure is about 101 kPa.
The walls of a container are continually bombarded by fast
moving gas molecules. When a molecule bounces off the wall, it
pushes outward against the wall and the wall pushes inwards on the
molecule. Each molecule is very tiny and many collisions occur each
second. It seems that the container is receiving a constant outward
force. The force per square metre of area is the pressure of the
gas.
3.2.2
Kinetic model of an ideal gas.
An ideal gas consists of tiny particles that are small compared
to their separation. The particles
only apply forces on each other during collisions. Most gases
are close to being ideal.
7.
The kinetic model of an ideal gas assumes:
*that a gas consists of a large number of very small particles
that are far apart and are moving randomly in all directions with a
range of speeds. The average kinetic energy of the particles is a
measure of the gas's temperature.
*the particles obey the laws of mechanics.
*The particles exert no force on each other except during
collisions.
*When a particle collides with the container wall or another
particle it bounces off making an elastic collision. During impact
the particle applies a force to the container and the container
applies a force back on the particle. The net outward force of all
the particles on the container wall gives rise to the pressure of
the gas.
Real gas exert molecules small forces on each other between
collisions. When a gas is compressed or its temperature is low the
gas is is close to being a liquid, it does not show ideal behavior.
3.2.3
Mole, molar mass and the Avogadro constantOne mole of a
substance is the mass of the substance that contains the same
number of particles as there are in 12.00 grams of Carbon 12.
Carbon 12 is the isotope containing 6 protons and 6 neutrons in the
nuclei. The mass of a carbon atom is 1.99 x 10-23 g so in 12.0 g
there are 12.0/1.99 x 10-23 or 6.02 x 1023 atoms of carbon. Thus a
mole of a substance is the mass of 6.02 x 1023 particles of the
substance. This number is called Avogadro's constant NA. The word
particle can stand for a single He atom of helium or a single CO2
molecule of carbon dioxide.
Consider a molecule of CO2. It consists of a carbon atom and two
oxygen atoms. A carbon atom has 12 particles in its nucleus and an
oxygen atom has 16 particles in its nucleus. In a CO2 molecule
there are 12 + 16 x 2 or 44 particles in its nucleus. There are 12
particles in the nucleus of carbon atoms and 12 g of carbon is 1
mole and contains 6.02 x 1023 carbon atoms. For CO2 there are a
total of 44 particles in the 3 nuclei making up the molecule. To a
good approximation, 44 g of CO2 is 1 mole of CO2 and contains 6.02
x 1023 molecules of carbon dioxide.
The numbers 12 and 16 are the mass numbers of carbon and oxygen.
The number 44 is the molecular mass of carbon dioxide. More
accurate mass numbers of elements can be found in periodic tables
e.g. C 12.00, O 15.99 and Cl 35.45. These values can be used to
calculate accurate molecular masses. e.g. 1 mole of chlorine Cl2 is
35.45 x 2 or 70.9 g.
To sum up the mass in gram of one mole of a substance is
approximately equal to the total number of protons and neutrons in
a particle of the substance and accurately equal to the sum of the
mass numbers of the atoms in a particle of the substance.
The mole concept was discussed in this section because you will
see soon there is an advantage to measure the mass of a gas in
mole.
3.2.4
Equation of state for an ideal gas.If a gas is in a sealed
container with a fixed volume, the pressure of the gas increases
when its temperature increases. When the temperature increases, the
average KE of the particles increases. More collisions occur with
the walls each second and the collisions are more forceful. The
outward force on the container increases.
When the pressure on a gas is increased, the volume of the gas
decreases. When the volume decreases, the molecules travel less
distance before colliding with the walls. More collisions occur
with the walls each second. The outward pressure on the container
increases to match the pressure outside the container.
8.
The pressure of a gas is also affected by the mass of the gas.
The greater the mass of gas, the more particles present and the
more collisions that occur with the walls of the container each
second. If the volume remains the same as in pumping up a tyre, the
pressure increases. If the pressure remains the same, the volume
increases.
A bike pump is a cylinder of gas with a piston. Putting your
finger over the hole and pushing in the piston increases the
pressure on the air in the pump. When the pressure on the air is
increased, the volume of air decreases. When the volume decreases,
the molecules travel less distance before colliding with the walls.
More collisions occur with the walls each second. The outward force
on the container increases and the pressure of the air inside the
pump increases.
The movement of the piston down on the gas does work on the gas.
The gas molecules bounce off the moving piston with a faster speed
just like a ball that is hit by a moving tennis racquet. The
Temperature of the gas increases as the average kinetic energy of
the gas molecules has increased. Conduction of heat energy to the
surroundings results in a decrease in internal energy of the gas.
When the piston is pulled back (finger still on the hole) the
temperature of the gas is lower than its initial value. This is the
principle that fridges use to operate.
All the discussions above indicate there is a connection between
the pressure, temperature, volume and mass of a gas.
The walls of a container are continually bombarded by fast
moving gas molecules. When a
molecule bounces off the wall, it pushes outward against the
wall and the wall pushes inwards
on the molecule. Each molecule is very tiny and many collisions
occur each second. It seems
that the container is receiving a constant outward force. The
force per square metre of area is the
pressure of the gas.
The pressure on a surface is equal to the force on the surface
divided by its area. P = F/A. The
unit of pressure is N m-2. A pressure of 1 N m-2 is called 1
Pascal Pa. The Pascal is a small unit.
Air applies a force of about 10 N on each cm2. This amounts to
about105 N m-2 or 105 Pa. The
unit kiloPascal kPa is used frequently. Normal air pressure is
about 101 kPa.
If the temperature of a gas remains constant
the volume of the gas is inversely
proportional to its pressure. V ( 1/P or
PV = constant. This relationship is called
Boyle's Law, as it was discovered by Robert
Boyle in the mid 1600's.
The diagram shows a gas inside a container that has a
piston free to slide. The piston will stop in the
position where the gas pressure inside the container
is the same as atmospheric pressure. When the gas
is heated, its pressure will remain constant and its
volume increases as the piston slides.
Jacques Charles discovered around 1800 that the volume of the
gas increased in equal amounts
for equal rises in temperature. This is now described as volume
is directly proportional to its
absolute Temperature (measured in K) V (T when P constant. This
is called Charles' Law.
If a gas is in a sealed container with a fixed volume, the
pressure of the gas increases when its
temperature increases. When the temperature increases, the
average KE of the particles
increases. More collisions occur with the walls each second and
the collisions are more forceful.
The outward force on the container increases. The air pressure
in a cold car tyre is lower than it
pressure after the car has been driven and the tyre has
increased temperature. Recommended
tyre pressures apply when the tyre is cold.
9.
Joseph Gay-Loussac found that for a gas with constant volume,
its pressure is directly
proportional to its absolute temperature i.e. P ( T when V
constant. This is Gay-Loussac's Law.
The three Gas Laws just described are not true laws in that they
are only true for ideal gases.
They are good approximations when applied to a real gas that is
not close to becoming a liquid
when pressure is high or temperature is low.
Combining the relationships discussed so far gives:
PV T
PV = KT
Adding more molecules to a sample of gas means more collisions
with the walls and pressure
would increase. For the pressure and temperature to remain the
same, the volume increase to
reduce the frequency of molecules hitting the walls of the
container. In the last equation, if P
and T remain constant and the number of molecules increases, the
value of K must increase
also. Thus K is proportional to the number of molecules N.
K = kN
PV = kNT
where N/V is the number of molecules per m3.
The table shows the number of molecules per m3 when at STP -
Standard Temperature and
Pressure - At STP, pressure is 1 atmosphere, 1.01 x 105 N m-2
and temperature is 0oC, 273 K.
Gas
No. of molecules per m3
Hydrogen
2.68 x 1025
Helium
2.68 x 1025
Oxygen
2.68 x 1025
When at the same temperature and pressure, the number of
molecules per m3 is the same for all
gases. Therefore the constant k is the same for all gases i.e.
it is a universal constant.
k is called Boltzmann's constant
Imagine N molecules of an ideal gas are in a cubic container
with
volume V and area of sides A.
A molecule with mass m is moving directly at one wall with
velocity v.
Assuming the molecule bounces off the wall at the same
speed:
v = 2 v
The change in momentum of the molecule is given by:
p = 2m v
10.
In the time t it takes the molecule to reach the wall, if n
molecules hit the wall, the average
impulse applied to the wall is:
F t = 2m v x n
The volume of the gas between the dotted line and the right hand
wall is vt x A
This volume is the fraction vtA/V of the total volume. Therefore
the number of molecules in
this volume is N x vtA/V.
However half the molecules in this volume will be moving away
from the wall. Therefore the
number of molecules n that collide with the wall in time t is
0.5NvtA/V. Substituting the value
of n into the impulse equation gives:
F t = 2mv x 0.5NvtA/V
F = Amv2 x N/V
The pressure P of the gas on the wall is force F divided by area
A. Therefore:
P = mv2 x N/V
This last line was calculated assuming all the molecules were
travelling directly towards the
wall. Call this the x direction. The last equation becomes:
P = mvx2 x N/V
However, the molecules will be moving in random directions.
Molecules will have velocity
components in the x, y and z directions.
For a molecule with velocity v, its components have magnitudes
vx, vy and vz. Applying
Pythagoras in 3-D gives:
v2 = vx2 + vy2 + vz2
Given the randomness of the motion of a large number of
molecules, the average values of vx2,
vy2 and vz2 should be equal. Therefore:
v2av = 3 vx2av
hence vx2av = 1/3 v2av
The expression for pressure now becomes:
P = 1/3 mv2av x N/V
= 2/3 x mv2av x N/V
= 2/3 x KEav x N/V
hence PV = 2/3 x KEav x N
Earlier it was established that PV = kNT. Therefore:
2/3 x KEav x N = kNT
hence KEav = 3/2kT
The average kinetic energy of the gas molecules is directly
proportional to its absolute
temperature. The opposite viewpoint is the temperature of a gas
is a measure of the average
Kinetic Energy of its molecules.
11.
What is the average speed of an air molecule at 20oC? Most of
the air is nitrogen. The mass
of a nitrogen molecule N2 is 4.65 x 10-26 kg. The absolute
temperature is 293 K.
KEav = 3/2kT
mvav2 = 3/2kT
x 4.65 x 10-26 x vav2 = 1.5 x 1.38 x 10-23 x 293
vav = 511 m s-1
Ever wondered why the hair of people sitting behind the
windscreen of an opened topped car is
blown forward!!!? The air molecules are rushing into the space
behind the windscreen faster
than the car and its passengers are travelling. Wear a hat or
you will have a sore scalp.
The pressure of a gas is affected by the mass of the gas. When V
( 1/P, V ( T and V ( m are
combined, then PV ( mT. As an equation PV = mT where (kappa) is
a constant.
Consider the following information about gases at STP - 0oC and
1 atmosphere:
Gas
mass kg
Volume m3
no. of mole
oxygen
0.032
0.0224
1
hydrogen
0.002
0.0224
1
carbon dioxide0.044
0.0224
1
Each gas has the same volume 0.0224 m3, the same temperature 273
K, the same pressure
1.01 x 10-5 N m-2 but a different mass in kg. Hence if the
information for each gas is substituted
into the equation PV = mT, a different value for would be
obtained for each gas.
However, the number of mole is the same for each gas. If the
mass is measured in number of
mole n, a new constant R will apply. i.e. PV = RnT. All gases in
the table have the same values
for P, V, n and T so R is a universal constant.
PV = RnT
1.01 x 105 x 0.0224 = R x 1 x 273
R = 8.3 J mol-1 K-1 called the universal gas constant
By tradition the ideal gas equation is written as PV = nRT.
In a closed container, the mass of gas remains constant.
Therefore:
PV/T = nR
= constant for the same no. of mole
Previously it was established that PV = 2/3 KEav x N where KEav
is the average kinetic energy of
a single molecule and N is the total number of molecules.
Comparing this to the ideal gas equation PV = RnT means:
2/3 KEav x N = nRT
rearranging KEav = 3/2 x n/N x RT
The total kinetic energy of all the molecules is N times the
last equation i.e.
KEtotal = 3/2nRT
One mole is the mass of 6.023 x 1023 molecules of the gas. The
number 6.023 x 1023 is called
Avogadro's constant NA. If 3.012 x 1023 molecules are present,
the mass is 0.5 mol. Hence the
number of mole is equal to the number of molecules present
divided by Avogadro's constant
i.e. n = N/NA. Rearranging, n/N = 1/NA. Substituting into the
second last equation gives:
KEav = 3/2 RT/NA
12.
What is the average kinetic energy of a gas molecule at
20oC?
Using KEav = 3/2kT
Using KEav = 3/2 RT/NA
KEav = 3/2 x 1.38 x 10-23 x 293
KEav = 3/2 x 8.3 x 293/6.023 x 1023
= 6.1 x 10-21 J
= 6.1 x 10-21 J
In a sealed container, the mass of the gas remains constant.
This means PV/T = nR i.e. PV/T is a
constant. For two different sets of P, V and T values, this is
usually written as:
P1V1/T1 = P2V2/T2
You will be expected to solve problems involving the ideal gas
equation. Pressure gauges
measure the pressure difference between the gas pressure inside
a container and the air outside.
If a gauge reads 1atmosphere, the pressure inside the container
is one atmosphere higher than
the air outside. The pressure of the gas inside the container is
2 atmospheres. This is called the
absolute pressure. Thus absolute pressure equals gauge pressure
plus 1 atmosphere.
3.2.5
Differences between real and ideal gases
The table below describes differences between real and ideal
gases.
Real
Ideal
Smallish number of particles.
Very large number of particles.
Molecules have size and shape.
Molecules are point particles.
Molecules can be close or far apart.
Molecules are far apart compared with size.
The volume of the molecules issignificant
The volume of the molecules is insignificant
compared with the container.
compared with the container.
Molecules exert forces on each other
Molecules exert no forces on each other
between collisions.
between collisions.
Molecules have energy of rotation and
Molecules have no energy of rotation or
vibration within the molecule.
vibration.
During collisions some of the kinetic
Collisions are elastic. Total kinetic energy
energy is transferred to energy of rotation
before and after the collision is the same.
and vibration.
Do not obey the ideal gas laws when close
Always obey the ideal gas laws.
to turning into a liquid.
PV = nRT not always accurate.
PV = nRT always accurate.
Exchange energy with surroundings.
Do not exchange energy with surroundings.
Copyright B & G Scientific 2014. May be copied for student
use. Thermal Physics.