Gases Unit 6: Gases & The Kinetic Molecular Theory Dr. Jorge L. Alonso Miami-Dade College – Kendall Campus Miami, FL CHM 1045 : General Chemistry and Qualitative Analysis Textbook Reference : •Module # 8
Gases
Unit 6:Gases & The Kinetic
Molecular Theory
Dr. Jorge L. AlonsoMiami-Dade College –
Kendall CampusMiami, FL
CHM 1045: General Chemistry and Qualitative Analysis
Textbook Reference:
•Module # 8
Gases
Characteristics of Gases
Condensed phases
• Unlike liquids and solids, gases . . . . Are highly compressible. Expand to fill their containers. Have extremely low densities.
Gases
Characteristics of Gases
• Variables affecting the behavior of gases Amount = number of moles () Pressure (P) Volume (V) Temperature (T in Kelvin)
{PropGases*}
Gases
• Pressure is the amount of force applied to an area.
Pressure
• Atmospheric pressure is the weight of air per unit of area.
P =FA
Force = mass x acceleration
Newton = 1kg . m/sec2
Approx. 12 miles105 Newtons
meter2= = 101.325 kPa
= (104kg)(10 m/sec2)105 Newtons
Gases
Units of PressureTorricelli’s
(Normal atmospheric pressure at sea level).
• Atmosphere1.00 atm = 760 mm Hg (torr) = 101.325 kPa
760 mm Hg = weight of equal surface area of the atmosphere
Gases
Barometer
33 ft H2O = weight of equal surface area of the atmosphere
Gases
Manometerinstrument used to measure the difference in pressure between atmospheric pressure and that of a gas in a vessel.
{Manometer}
Gases
Manometer
Used to measure the difference in pressure between atmospheric pressure and that of a gas in a vessel.
Pgas = 760 torr Pgas = 760 + 6 torrs Pgas = 760 - 6 torrs
Gases
Gas Laws
1. Boyle’s Law Compared: P versus V ( & T are held constant).
2. Charles’s Law Compared: V versus T ( & P are held constant).
3. Avogadro’s Law Compared: V versus η (P & T are held constant).
4. Combined Gas Law Compared: P vs V vs. T ( is held constant).
5. Ideal Gas Law Compared: P vs V vs. η vs T (no variable held constant).
6. Dalton’s Law of Partial Pressure Compared: individual pressures of gases in a mixture
Variables affecting gases: moles (η), pressure (P), volume (V) and Temperature (T)
Gases
Boyle’s Law: Pressure-Volume Relationship
The volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure.
{Boyle’s Law}
( & T are held constant).
V 1P
2 x
V x ½
V ?
Gases
P & V: inversely proportional
Also, P ↑ V ↓ = k
{PV.Graphs}
This means a plot of V versus 1/P will be a straight line.
V 1P
V =kP
OR
Gases
Boyle’s Law
Gases
• The volume of a fixed amount of gas at constant pressure is directly proportional to its absolute temperature.
( & P are held constant).
{*Charles’s Law Liq N2}
Charles’s Law: Temp. – Volume Relationship
V T
T 2 x
V 2 x
V ?
Gases
Charles’s Law• The volume of a gas is directly proportional to its
absolute temperature.
A plot of V versus T will be a straight line.
orVT
= k
V = k T
V T
Gases
Charles’s Law
Gases
Avogadro’s Law: Moles-Volume Relationship
• The volume is directly proportional to the number of moles of the gas.
{AvogLaw}
(P & T are held constant).
{Avogadro’s Law}
V n
2 x
V 2 x
V ?
Gases
Avogadro’s Law
or, V = k n
• Mathematically, this means V n
{*Avogadro’s Law in Reactions}
Gases
Standard Temperature & Pressure (STP) and Molar Volume
• Standard Temperature: 00C or 273 K• Standard Pressure: 760 torr (1 atm)
At STP the Molar Volume of any gas is 22.4 L
1 mole = 6.022 x 1023 part. = gMM = 22.4 L
(11.1 in)3 or (28.2cm)3
Gases
Standard Temperature & Pressure (STP) and Molar Volume
CO g 75 mL ? 2
g 44.0
CO 1 2
At STP the Molar Volume of any gas is 22.4 L
1 mole = 6.023 x 1023 part. = gMM = 22.4 L
Problem: At STP, what volume in mL would 75g of CO2 occupy?
H2 = 2.0g
O2 = 32.0g
CO2 = 44.0 g
2CO 1
L4.22
L1
mL 1000 mL 10x 3.8 4
Gases
Ideal-Gas Equation
V 1/P (Boyle’s law)V T (Charles’s law)V n (Avogadro’s law)
The Gas Laws:
V nTP
k =nTP Vor
Combining these, we get
V =knTP
or
Gases
Ideal-Gas Equation
The relationship
then becomes
PV = nRT
R =P VnT
k =nTP V
=1R
Gases
PV = nRT
Ideal-Gas Equation:
(torr) (L) = (mol) R (K)Units:
{PV= nRT RapVideo}
Kmol
torrL
RapVideoLinkYouTube
Useful for pure gas under one set of conditions.
Gases
Ideal Gas Law Problems
What volume (in mL) would a 2.20 g sample of hydrogen gas (H2) at 50.00C occupying at 443 torr?
PV = nRT V = nRTP
V =
g 2.0
1 g 2.20
K36.62
torr L K0.50 273
torr 443
R= 50.0 L
L1
mL1000
Gases
Ideal-Gas Equation: Densities of Gases
RT MW
g# PV
MW-g
mole 1 g)(# (n) moles
MW V
TR g# P
PV = nRTSince
Then
For Ideal Gas Equation:
Where d = Density of Gas
Dividing both sides of the equation on the left by V we get
( ) MW
TR d P
PV
RT g# MW and
If we solve the equation for density, we get……..
Gases
T
(MW) P d
R
P
T d MW
R
Problem: What is the density of the oxygen in a tank in an AC room (25°C) and whose pressure gauge reads 25.0 atm
Problem: A gas whose density is 0.0131 g/mL and is in a container at room temperature and whose pressure gauge reads 1.9 x 104 mmHg. What is its MW?
Ideal-Gas Equation: Densities and Molecular Weigh of Gases
K) (298
)(32.0g/ atm) (25.0
KatmL 0.0821 L
g7.32
torrs10 x 1.9
( L
mL 1000 mL
g0.0131
4
K) 298
K
torrL36.62
g8.12
Gases
Ideal-Gas Equation: Densities & Molecular Weigh of Gases Problems
What is the density (in g/mL) of SO2 at STP?
RT MW
g# PV
MW V
TR g# P ( )
PV = nRT
MW
TR d P
T
(MW) P d
R =
K0273
K0
torr L 36.62
torr 760 )g 0.64( = 2.62 g/L
mL1000
L1
mLg00262.0
Gases
Ideal-Gas Equation: Densities & Molecular Weigh of Gases Problems
What is the molecular weight of a gas whose density @ STP is 7.78 g/L?
RT MW
g# PV
MW V
TR g# P ( )
PV = nRT
MW
TR d P
=
K0273
K0
torr L 36.62
torr 760P
T d MW
R
)Lg .787(g 741
Gases
2006 A
Gases
Gases
Combined Gas Law Equation
V 1/P (Boyle’s law)
V T (Charles’s law)
The Gas Laws:
Combining, we can get
P1V1
T1
P2V2
T2
The Combined Gas Law
Useful for a constant amount of a pure gas under two different conditions.
k=
Gases
Combined Gas Law EquationConstant
P1V1
T1
=P2V2
T2
Gases
Combined Gas Law Problem
P1V1
T1
=P2V2
T2
A scuba diver takes a gas filled 1.0 L balloon from the surface where the temperature is 34 0C down to a depth of 66 ft (33 ft H2O = 1 atm). What volume will the gas balloon have at that depth if the temperature is 15 0C?
V2
K273 15
) ( atm 3
K273 34
L 1 atm 100
L .313
atm 3 K307
K288 L 1 atm 1
0
0
V2
Gases
Dalton’s Law ofPartial Pressures
• The total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone.
• In other words,
Ptotal = P1 + P2 + P3 + …
Pair = P N2 + PO2 + PH2O + …
Gases
Partial Pressures• When one collects a gas over water, there is water vapor mixed
in with the gas.
• To find only the pressure of the desired gas, one must subtract the vapor pressure of water from the total pressure.
P of atmP of gas
Gases
Vapor Pressure of Water
Vapor Press (torr)
Vapor
Press (torr)
-10 2.15 40 55.3
0 4.58 60 149.4
5 6.54 80 355.1
10 9.21 95 634
11 9.84 96 658
12 10.52 97 682
13 11.23 98 707
14 11.99 99 733
15 12.79 100 760
20 17.54 101 788
25 23.76 110 1074.6
30 31.8 120 1489
37 47.07 200 11659
Ptotal = Pgas + PH2O
Pgas = Ptotal - PH2O
• To find only the pressure of the desired gas, one must subtract the vapor pressure of water from the total pressure.
• Daltons Law:
{Press on can}
Gases
Evaporation vs Boiling in terms of Vapor Pressure
Vapor Pressure (v.p. or Pvap)
++
=
Patm
Pvap
Patm
Pvap
Patm
Pvap
• Caused by the tendency of solids & liquids to evaporate to gaseous form. It is temperature (K.E.) dependent.
Gases
Stoichiometry with Gases
L 2.0
torr) 24 - 775(
298
36.62H η082.0 0
2
K
Kmol
torrL
Vo
Mg g 2.0 H η ? 2
Mg (s) + 2HCl (aq) MgCl2 (aq) + H2 (g)
Problem: If 2.0 g of Mg are reacted with excess HCl, what volume of H2 will be produced at 250C and 775 torr? At STP?
PV = nRT
Mg 3.24
Mg 1
g
Mg 1
H 1 2
2H 082.0
Gases
Kinetic-Molecular TheoryA model that aids in our understanding of what happens to gas particles as environmental variables change.
1. Gases consist of large numbers of molecules that are in continuous, random motion.
Main Tenets:
2. Collisions between gas molecules and between gas molecules and the walls of the container must be completely elastic (energy may be transferred between molecules, but none is lost).
Gases
Kinetic-Molecular Theory
Main Tenets:
3. Attractive and repulsive forces between gas molecules are negligible.
4. The combined volume of all the molecules of the gas is negligible (excluded volume) relative to the total volume in which the gas is contained.
Gases
Kinetic-Molecular TheoryMain Tenets:
{KE T(K)}
6. The average kinetic energy (KE=½mv2) of the molecules is proportional to the absolute temperature.
@ 100 0C5. Energy can be transferred
between molecules during collisions, but the average kinetic energy of the molecules does not change with time, as long as the temperature of the gas remains constant.
Gases
EffusionThe escape (diffusion) of gas molecules through a tiny hole into an evacuated space.
DiffusionMovement of
molecules from an area of high concentration to an area of low concentration until equilibrium is reached (homogeneity).
Gases
Effect of Molecular Mass on Rate of Effusion and Diffusion
AMW
1 ARate
Thomas Graham (1846): rate of diffusion is inversely proportional to the square root of its molar mass
2
2
1mvKE
Kinetic Energy per individual molecule:
Gases
Rate of Diffusion & Effusion
{BrDiffusion}
Comparing the rates of two gases:
Graham’s Law of Diffusion and Effusion of Gases
A
B
MW
MW
B
A
Rate
Rate
HCl Rate
NH Rate 3 {GasDiff}
Dropper with Br (l)
AMW
1 ARate
Thomas Graham (1846): rate of diffusion is inversely proportional to the square root of its molar mass
g/mol 17
g/mol 361.4 2.1
Gases
Effusion and Diffusion• This is the most widespread uranium
enrichment method. Uranium is reacted with fluorine to make uranium hexafluoride gas: 235UF6 & 238UF6
• The physical principle is that the diffusion speed of a gas molecule depends on the mass of the molecule: the lighter ones diffuse faster and get through a porous material easier.
• In gas diffusion units, uranium-
hexafluoride gas diffuses through an etched foil made of either an aluminum alloy or teflon, due to artificially maintained difference in pressure. The lighter molecules (i.e. those containing 235U) get through easier to the other side, therefore the gas accumulating there will be richer in 235U.
Gases
Gas Centrifugation
The gas centrifuge is essentially a bowl, in which there is a rotor spinning round at a very high speed. The gas (UF6) directed to the centrifuge is forced to spin by the rotor. Due to the centrifugal force the heavier molecules (those which contain 238U) will accumulate near the wall of the bowl, while the lighter molecules containing 235U will
stay closer to the center of the centrifuge.
Gases
Boltzmann DistributionsThe Maxwell–Boltzmann distribution is the statistical distribution of molecular speeds in a gas. It corresponds to the most probable speed distribution in a collisionally-dominated system consisting of a large number of non-interacting particles. {Boltzman Plot}
Gases
Kinetic Energy of Gas Molecules
2 2
1vmKE
Kinetic Energy per individual molecule:
Kinetic Energy per mole:
m
rms M
RTu
3
Combining above equations and solving for velocity we get:
• The root-mean square velocity of gases is a very close approximation to the average gas velocity.
• To calculate this correctly: The value of R = 8.314 kg m2/s2 K mol Mm = molar mass, and it must be in kg/mol.
RTKE2
3
☺
☺
☺
Gases
The Kinetic-Molecular Theory
• Example: What is the root mean square velocity of N2 molecules at room T, 25.0oC?
kg/mol 028.0
K 298molK sec
m kg8.3143
u2
2
rms
mi/hr 1159=m/s 515
m
rms M
RTu
3
• To calculate this correctly: The value of R = 8.314 kg m2/s2 K mol And M must be in kg/mol.
Gases
The Kinetic-Molecular TheoryProblem: What is the root mean square velocity of He atoms
at room T, 25.0oC?
You do it!You do it!
m
rms M
RTu
3
• To calculate this correctly: The value of R = 8.314 kg m2/s2 K mol And M must be in kg/mol.
kg/mol
K mol K sec
m kg8.3143
u2
2
rms 004.0
298
mi/hr 3067= m/s 1363
• Can you think of a physical situation that proves He molecules have a velocity that is so much greater than N2 molecules?
• What happens to your voice when you breathe He(g) or SF6 (g)?
mi/hr 1159=m/s was 515N 2 Gas MWHe 4N2 28SF6 146
Gases
Ideal vs.
Real Gases
In the real world, the behavior of gases only conforms to the ideal-gas equation at relatively high temperature and low pressure.
Gases
Deviations from Ideal Behavior
Two particular assumptions made in the kinetic-molecular model break down at high pressure and/or low temperature:
(1) attractive forces and (2) excluded volume.
Gases
2003 A
Gases
Corrections for Non-ideal Gas Behavior
• The ideal-gas equation can be adjusted to take these deviations from ideal behavior into account.
• The corrected ideal-gas equation is known as the van der Waals equation.
Gases
The van der Waals Equation
) (V − nb) = nRTn2aV2(P +
Gases
Real Gases:Deviations from Ideality
• van der Waals’ equation accounts for the behavior of real gases at low temperatures and high pressures.
nRT nVV
n + P
2
2
• The van der Waals constants a and b take into account two things:1. a accounts for intermolecular attraction2. b accounts for volume of gas molecules
• At large volumes a and b are relatively small and van der Waal’s equation reduces to ideal gas law at high temperatures and low pressures.
a b
Gases
Real Gases:Deviations from Ideality
• Example: Calculate the pressure exerted by 84.0 g of ammonia, NH3, in a 5.00 L container at 200. oC using the ideal gas law.
You do it!You do it!
mol g 17.0
mol NH g 84.0 = n 3 94.4
1
V
nRT = P
L 5.00
K K mol
atm Lmol 4730821.094.4
atm 4.38
Gases
Real Gases:Deviations from Ideality
molL0.0371=b
mol
atm L 4.17 =a mol 4.94 = n
2
2
• Example: Calculate the pressure exerted by 84.0 g of ammonia, NH3, in a 5.00 L container at 200. oC using the van der Waal’s equation You do it!You do it!
nRTnb-VV
an + P
2
2
2
2
V
an
nb-V
nRT=P
2
2
00.5
17.494.4
)0371.0)(94.4(00.5
4730821.094.4
L
mol
mol L
K mol P
2
2
molatm L
molL
K molatm L
atm atm atm atm L
atm L P 7.35)1.48.39(07.4
817.4
8.191
behavior ideal from difference 7.6% a is thisatm, 38.4 toCompared