# CHAPTER 12

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```					     Gases and Kinetic
12   Molecular Theory

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CHAPTER GOALS
1.   Comparison of Solids, Liquids, and Gases
2.   Composition of the Atmosphere and Some
Common Properties of Gases
3.   Pressure
4.   Boyle’s Law: The Volume-Pressure Relationship
5.   Charles’ Law: The Volume-Temperature
Relationship; The Absolute Temperature Scale
6.   Standard Temperature and Pressure
7.   The Combined Gas Law Equation
8.   Avogadro’s Law and the Standard Molar Volume

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CHAPTER GOALS
9. Summary of Gas Laws: The Ideal Gas Equation
10. Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
11. Dalton’s Law of Partial Pressures
12. Mass-Volume Relationships in Reactions Involving
Gases
13. The Kinetic-Molecular Theory
14. Diffusion and Effusion of Gases
15. Real Gases: Deviations from Ideality

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Comparison of Solids, Liquids,
and Gases
• The density of gases is much less than that
of solids or liquids.
Densities      Solid    Liquid       Gas
(g/mL)
H2O         0.917      0.998    0.000588

CCl4     1.70        1.59     0.00503

 Gas molecules must be very far apart
compared to liquids and solids.
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Composition of the Atmosphere and
Some Common Properties of Gases
Composition of Dry Air
Gas         % by Volume
N2            78.09
O2             20.94
Ar             0.93
CO2             0.03
He, Ne, Kr, Xe      0.002
CH4           0.00015
H2           0.00005     6
Pressure
• Pressure is force per unit area.
– lb/in2
– N/m2
• Gas pressure as most people think of it.

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Pressure
• Atmospheric pressure is measured using a
barometer.
• Definitions of standard pressure
–   76 cm Hg
–   760 mm Hg
–   760 torr
–   1 atmosphere
–   101.3 kPa

Hg density = 13.6 g/mL
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Boyle’s Law:
The Volume-Pressure Relationship
• V  1/P or
• V= k (1/P) or PV = k
• P1V1 = k1 for one sample of a gas.
• P2V2 = k2 for a second sample of a gas.
• k1 = k2 for the same sample of a gas at
the same T.
• Thus we can write Boyle’s Law
mathematically as P1V1 = P2V2
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Boyle’s Law:
The Volume-Pressure Relationship
Example 12-1: At 25oC a sample of He has a volume
of 4.00 x 102 mL under a pressure of 7.60 x 102 torr.
What volume would it occupy under a pressure of
2.00 atm at the same T?

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Boyle’s Law:
The Volume-Pressure Relationship
• Notice that in Boyle’s law we can use any
pressure or volume units as long as we
consistently use the same units for both P1 and
P2 or V1 and V2.
volume will go up or down as the pressure is
changed and vice versa.

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Charles’ Law:
The Volume-Temperature Relationship
35

30

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20                                                         Volume (L)
vs.
15                                                       Temperature (K)

10                          Gases liquefy
before reaching 0K
5

0
0    50   100   150   200   250   300   350   400

absolute zero = -273.15 0C
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Charles’ Law:
The Volume-Temperature Relationship
• Charles’s law states that the volume of a
gas is directly proportional to the absolute
temperature at constant pressure.
– Gas laws must use the Kelvin scale to be
correct.
• Relationship between Kelvin and

K = o C + 273
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Charles’ Law:
The Volume-Temperature Relationship

• Mathematical form of Charles’ law.

V V k
V  T or = = or
V  T or V V kT kT or k
T T
V1          V2
 k1 and     k 2 however the k's are equal so
T1          T2
V1 V2
   in the most useful form
T1 T2
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Charles’ Law:
The Volume-Temperature Relationship
Example 12-2: A sample of hydrogen, H2, occupies
1.00 x 102 mL at 25.0oC and 1.00 atm. What volume
would it occupy at 50.0oC under the same pressure?

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Standard Temperature and
Pressure
• Standard temperature and pressure is
given the symbol STP.
– It is a reference point for some gas
calculations.
• Standard P  1.00000 atm or 101.3 kPa
• Standard T  273.15 K or 0.00oC

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The Combined Gas Law
Equation
• Boyle’s and Charles’ Laws combined into one
statement is called the combined gas law
equation.
– Useful when the V, T, and P of a gas are changing.

Boyle' Law
Boyle' s s Law       Charles' Law
Charles' Law
V V
V11  V2
V P
P1VP11P2 V22V2             T2
1                    T1
T1    T2
2

For a given sample of gas : The combined gas law is :
PV                           P1 V1 P2 V2
k                           
T                            T1     T2
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The Combined Gas Law
Equation
Example 12-3: A sample of nitrogen gas, N2,
occupies 7.50 x 102 mL at 75.00C under a pressure
of 8.10 x 102 torr. What volume would it occupy at
STP?
V1 = 750 mL       V2 = ?
T1 = 348 K      T2 = 273 K
P1 = 810 torr    P2 = 760 torr
P1 V1 T2
Solve for V2 =
P2 T1


810 torr 750 mL 273 K 
760 torr 348 K 
 627 mL                    18
The Combined Gas Law
Equation
Example 12-4 : A sample of methane, CH4, occupies
2.60 x 102 mL at 32oC under a pressure of 0.500
atm. At what temperature would it occupy 5.00 x 102
mL under a pressure of 1.20 x 103 torr?
You do it!

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Standard Molar Volume

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Standard Molar Volume
• Avogadro’s Law states that at the same temperature and
pressure, equal volumes of two gases contain the same
number of molecules (or moles) of gas.
• If we set the temperature and pressure for any gas to be
STP, then one mole of that gas has a volume called the
standard molar volume.
• The standard molar volume is 22.4 L at STP.
– This is another way to measure moles.
– For gases, the volume is proportional to the number of moles.
• 11.2 L of a gas at STP = 0.500 mole
– 44.8 L = ? moles

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Standard Molar Volume
Example 12-5: One mole of a gas occupies 36.5 L
and its density is 1.36 g/L at a given temperature
and pressure. (a) What is its molar mass? (b)
What is its density at STP?

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Summary of Gas Laws:
The Ideal Gas Law
• Boyle’s Law - V  1/P (at constant T & n)
• Charles’ Law – V  T (at constant P & n)
• Avogadro’s Law – V  n (at constant T & P)
• Combine these three laws into one statement
V  nT/P
• Convert the proportionality into an equality.
V = nRT/P
• This provides the Ideal Gas Law.
PV = nRT
• R is a proportionality constant called the universal gas   23

constant.
Summary of Gas Laws:
The Ideal Gas Law
• We must determine the value of R.
– Recognize that for one mole of a gas at 1.00 atm, and
273 K (STP), the volume is 22.4 L.
– Use these values in the ideal gas law.

PV 1.00 atm  22.4 L
R =   
nT 1.00 mol  273 K 
L atm
 0.0821
mol K
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Summary of Gas Laws:
The Ideal Gas Law
• R has other values if the units are changed.
• R = 8.314 J/mol K
– Use this value in thermodynamics.
• R = 8.314 kg m2/s2 K mol
– Use this later in this chapter for gas velocities.
• R = 8.314 dm3 kPa/K mol
– This is R in all metric units.
• R = 1.987 cal/K mol
– This the value of R in calories rather than J.

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Summary of Gas Laws:
The Ideal Gas Law
Example 12-6: What volume would 50.0 g of ethane,
C2H6, occupy at 1.40 x 102 oC under a pressure of
1.82 x 103 torr?

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Summary of Gas Laws:
The Ideal Gas Law
Example 12-7: Calculate the number of moles in,
and the mass of, an 8.96 L sample of methane, CH4,
measured at standard conditions.
You do it!

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Summary of Gas Laws:
The Ideal Gas Law
Example 12-8: Calculate the pressure exerted by
50.0 g of ethane, C2H6, in a 25.0 L container at
25.0oC.
You do it!

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Determination of Molecular Weights and
Molecular Formulas of Gaseous Substances
Example 12-9: A compound that contains only
carbon and hydrogen is 80.0% carbon and 20.0%
hydrogen by mass. At STP, 546 mL of the gas has a
mass of 0.732 g . What is the molecular (true)
formula for the compound?

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Determination of Molecular Weights and
Molecular Formulas of Gaseous Substances
Example 12-10: A 1.74 g sample of a compound that
contains only carbon and hydrogen contains 1.44 g
of carbon and 0.300 g of hydrogen. At STP 101 mL
of the gas has a mass of 0.262 gram. What is its
molecular formula?
You do it!

30
Dalton’s Law of Partial
Pressures
• Dalton’s law states that the pressure
exerted by a mixture of gases is the sum
of the partial pressures of the individual
gases.
Ptotal = PA + PB + PC + .....

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Dalton’s Law of Partial
Pressures
Example 12-11: If 1.00 x 102 mL of hydrogen,
measured at 25.0 oC and 3.00 atm pressure, and
1.00 x 102 mL of oxygen, measured at 25.0 oC and
2.00 atm pressure, were forced into one of the
containers at 25.0 oC, what would be the pressure of
the mixture of gases?

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Dalton’s Law of Partial
Pressures
• Vapor Pressure is the pressure exerted by
a substance’s vapor over the substance’s
liquid at equilibrium.

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Dalton’s Law of Partial
Pressures
Example 12-12: A sample of hydrogen was collected
by displacement of water at 25.0 oC. The
atmospheric pressure was 748 torr. What pressure
would the dry hydrogen exert in the same container?

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Dalton’s Law of Partial
Pressures
Example 12-13: A sample of oxygen was collected
by displacement of water. The oxygen occupied 742
mL at 27.0 oC. The barometric pressure was 753
torr. What volume would the dry oxygen occupy at
STP?
You do it!

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Mass-Volume Relationships in
Reactions Involving Gases

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Mass-Volume Relationships
in Reactions Involving Gases
•In this section we are looking at
reaction stoichiometry, like in Chapter 3,
just including gases in the calculations.
MnO 2 &

2 KClO 3(s)   2 KCl(s) + 3 O2 (g)
2 mol KClO3 yields     2 mol KCl and 3 mol O2
2(122.6g)   yields    2 (74.6g) and 3 (32.0g)
Those 3 moles of O2 can also be thought of as:
3(22.4L) or 67.2 L at STP
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Mass-Volume Relationships in
Reactions Involving Gases
Example 12-14: What volume of oxygen
measured at STP, can be produced by the
thermal decomposition of 120.0 g of KClO3?
You do it!

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The Kinetic-Molecular Theory
• The basic assumptions of kinetic-
molecular theory are:
• Postulate 1
– Gases consist of discrete molecules that are
relatively far apart.
– Gases have few intermolecular attractions.
– The volume of individual molecules is very
small compared to the gas’s volume.
• Proof - Gases are easily compressible.

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The Kinetic-Molecular Theory
• Postulate 2
– Gas molecules are in constant, random,
straight line motion with varying velocities.
• Proof - Brownian motion displays
molecular motion.

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The Kinetic-Molecular Theory
• Postulate 3
– Gas molecules have elastic collisions with
themselves and the container.
– Total energy is conserved during a collision.
• Proof - A confined gas in an insulated and
sealed container exhibits no pressure drop
over time.

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The Kinetic-Molecular Theory
• Postulate 4
– The kinetic energy of the molecules is
proportional to the absolute temperature.
– The average kinetic energies of molecules
of different gases are equal at a given
temperature.
• Proof - Brownian motion increases as
temperature increases.

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The Kinetic-Molecular Theory
• The kinetic energy of the molecules is
proportional to the absolute temperature.
• Displayed in a Maxwellian distribution.

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The Kinetic-Molecular Theory
• The gas laws that we have looked at earlier in this chapter are
confirmations that kinetic-molecular theory is the basis of
gaseous behavior.
• Boyle’s Law
– P  1/V
– As the V increases the molecular collisions with container
walls decrease and the P decreases.
• Dalton’s Law
– Ptotal = PA + PB + PC + .....
– Because gases have few intermolecular attractions, their
pressures are independent of other gases in the container.
• Charles’ Law
– VT
– An increase in temperature raises the molecular velocities,
thus the V increases to keep the P constant.             44
The Kinetic-Molecular Theory

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The Kinetic-Molecular Theory
• The root-mean square velocity of gases is
a very close approximation to the average
gas velocity.
• Calculating the root-mean square velocity
is simple:
3RT
u rms 
Mm
• To calculate this correctly:
– The value of R = 8.314 kg m2/s2 K mol
– And molar mass (Mm) must be in kg/mol.
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The Kinetic-Molecular Theory
Example 12-17: What is the root mean square
velocity of N2 molecules at room T, 25.0oC?

47
The Kinetic-Molecular Theory
Example 12-18: What is the root mean square
velocity of He atoms at room T, 25.0oC?
You do it!

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Diffusion and Effusion of Gases
• Diffusion is the intermingling of
gases.
• Effusion is the escape of gases
through tiny holes.

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Diffusion and Effusion of Gases
• This is a demonstration of diffusion.

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Diffusion and Effusion of Gases
• The rate of effusion is inversely
proportional to the square roots of the
molecular weights (M) or densities (D).
R1          M2

R2          M1
or
R1         D2

R2         D1
51
Diffusion and Effusion of Gases
Example 12-15: Calculate the ratio of the rate of
effusion of He to that of sulfur dioxide, SO2, at the
same temperature and pressure.

52
Diffusion and Effusion of Gases
Example 12-16: A sample of hydrogen, H2, was
found to effuse through a pinhole 5.2 times as rapidly
as the same volume of unknown gas (at the same
temperature and pressure). What is the molecular
weight of the unknown gas?
You do it!

53
Real Gases:
Deviations from Ideality
•    Real gases behave ideally at ambient
temperatures and pressures.
•    At low temperatures and high pressures real
gases do not behave ideally.
•    The reasons for the deviations from ideality
are:
1. The molecules are very close to one another,
thus their volume is important.
2. The molecular interactions also become
important.
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Real Gases:
Deviations from Ideality
• van der Waals’ equation accounts for the
behavior of real gases at low temperatures
and high pressures.
      n 2a 
2 
P +           V  nb  nRT
       V 
• The van der Waals constants a and b take into
account two things:
1. a accounts for intermolecular attraction
2. 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.                                      55
Real Gases:
Deviations from Ideality
•  What are the intermolecular forces in
gases that cause them to deviate from
ideality?
1. For nonpolar gases the attractive
forces are London Forces
2. For polar gases the attractive forces
are dipole-dipole attractions or
hydrogen bonds.
56
Real Gases:
Deviations from Ideality
Example 12-19: 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!

57
Real Gases:
Deviations from Ideality
Example 12-20: Solve Example 12-19 using the
van der Waal’s equation.

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Synthesis Question
The lethal dose for hydrogen sulfide is 6.0 ppm. In
other words, if in 1 million molecules of air there
are six hydrogen sulfide molecules then that air
would be deadly to breathe. How many hydrogen
sulfide molecules would be required to reach the
lethal dose in a room that is 77 feet long, 62 feet
wide and 50. feet tall at 1.0 atm and 25.0 oC?

59
Group Question
Tires on a car are typically filled to a pressure of
35 psi at 3.00 x 102 K. A tire is 16 inches in radius
and 8.0 inches in thickness. The wheel that the
tire is mounted on is 6.0 inches in radius. What is
the mass of the air in the tire?

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Gases and Kinetic
12   Molecular Theory

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