Gases and Kinetic-Molecular Theory:
Chapter 12
Chem 101
Fall 2004
Chapter Outline
• Comparison of Solids, Liquids, and Gases
• Composition of the Atmosphere and Some Common
Properties of Gases
• Pressure
• Boyle’s Law: The Volume-Pressure Relationship
• Charles’ Law: The Volume-Temperature Relationship;
The Absolute Temperature Scale
• Standard Temperature and Pressure
• The Combined Gas Law Equation
Chem 101
Fall 2004
Chapter Outline
• Avogadro’s Law and the Standard Molar Volume
• Summary of Gas Laws: The Ideal Gas Equation
• Determination of Molecular Weights and Molecular
Formulas of Gaseous Substances
• Dalton’s Law of Partial Pressures
• Mass-Volume Relationships in Reactions Involving Gases
• The Kinetic-Molecular Theory
• Diffusion and Effusion of Gases
• Real Gases: Deviations from Ideality
Chem 101
Fall 2004
1
Summary of the Ideal Gas Laws
Boyle's Law
P1V1 = P2 V2
Charles' s Law
V1 V2
=
T1 T2
Avogadro's Law
V1 V2
=
n1 n 2
Combined Gas Law
P1V1 P2 V2
=
(for fixed n)
T1
T2
Chem 101
Fall 2004
Dalton’s Law of Partial Pressures
• The pressure exerted by a single component in a
gas mixture
• “What the pressure would be if only that gas were
present in the container”
Chem 101
Fall 2004
Dalton’s Law of Partial Pressures
Ptot = P1 + P2 + P3 + ....
P1 = X1Ptot
mole fraction
where
X1 =
number of
moles of 1
n1
n tot
total number
of moles
Chem 101
Fall 2004
2
Dalton’s Law of Partial Pressures
• If the gases above are allowed to mix, what are the
partial pressures of the gases?
He
1.0 L
0.50 atm
Ne
1.0 L
0.40 atm
Ar
2.0 L
0.20 atm
Chem 101
Fall 2004
Dalton’s Law of Partial Pressures
•
•
•
•
Each gas has fixed n, R, T
So PV = constant, or PiVi = PfVf
For helium:Pi=0.5 atm,Vi = 1 L, Vf=4L
So: Pf = PiVi /Vf = (0.5)(1)/(4)
=0.125 atm
• For others: PNe = 0.1 atm = PAr
• Total P = sum = 0.325
Chem 101
Fall 2004
Gas Stoichiometry
• Chemical equations relate number of moles, as
always!
• Gas law relates number of moles to P, V, T
• Allows us to find amount of gas produced or
consumed in a reaction in terms of P, V, T.
Chem 101
Fall 2004
3
Gas Stoichiometry
• If we burn 10 L of ammonia with 15 L of oxygen
at 600 oC what volume of nitric oxide can we
form?
4NH3(g) + 5O2(g) → 4NO(g) + 6H2O(g)
• (Assume the gases are at the same temperature,
600 oC, and 1 atm pressure)
Chem 101
Fall 2004
Gas Stoichiometry: Air Bags
• Automobile air bag systems involve chemical
generation of gas to fill bag.
• Some requirements:
• Rapid generation of gas
(bag fills in 30 ms)
• “Safe” gas: cool, non-toxic, etc.
• Easily triggered in response to impact
Chem 101
Fall 2004
Air Bags: Chemistry
• Decomposition of sodium azide (NaN3) to
give sodium and nitrogen
• Reaction triggered on impact by a spark
• Problems?
Sodium product.
Also, NaN3 is very toxic
2 NaN3(s) → 2 Na(s) + 3 N2(g)
Chem 101
Fall 2004
4
Gas Stoichiometry: Air Bags
• Estimate the mass of solid sodium azide needed to
produce enough N2 to fill a typical air bag
• Typical air bag is about 67 liters, and inflates to a
maximum P of 1.3 atm (then rapid deflation)
• Assume ambient T= 25 °C
Chem 101
Fall 2004
Gas Stoichiometry: Air Bags
• P = 1.3 atm, V = 40 L, T = 25 oC
• Solve for n:
PV
(1.3)(40)
=
RT (0.0821)(298)
= 2.13 mol of gas
2 mol NaN 3 65 g NaN 3
2.13 mol N 2 x
x
3 mol N 2
1 mol NaN 3
n=
= 92.3 g NaN 3
Chem 101
Fall 2004
Gas Stoichiometry: Air Bags
• Final P = 1.3 atm, V = 67 L, T = 25 oC. Suppose
we wanted to store as gas. If V were 1 L, P would
need to be (67)x(1.3) = 87 atm, or 1270 psi
• 1L would be big tank in steering column. Also
high pressure & valve problems.
Chem 101
Fall 2004
5
Space Shuttle: Booster Engine
• 2 NH4ClO4 (s) → N2 (g) + Cl2(g) + 2 O2 (g) + 4 H2O (g)
• Volume of gas produced from
75,000 kg of NH4ClO4?
(assume final gas at 100°C
and 1 atm)
Chem 101
Fall 2004
Space Shuttle: Booster Engine
•
•
•
•
•
2 NH4ClO4 (s) → N2 (g) + Cl2(g) + 2 O2 (g) + 4 H2O (g)
HN4ClO4 = 117.5 g/mol
75,000 kg = 75x106 g = 6.38x105 mol
2 mol NH4ClO4 → 8 mol gas
Get 2.55 x 106 mol gas
• Now use the gas law
Chem 101
Fall 2004
Space Shuttle: Booster Engine
• 2.55 x 106 mol gas
• P = 1 atm, T = 100 C =398 K
nRT ( 2.55x10 6 )(0.0821)(398)
=
P
1
8.34x10 7 L
V=
Chem 101
Fall 2004
6
Kinetic Theory of Gases
• 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.
• Postulate 2
• Gas molecules are in constant, random, straight line motion
with varying velocities.
Chem 101
Fall 2004
Kinetic Theory of Gases
• Postulate 3
• Gas molecules have elastic collisions with themselves
and the container.
• Total energy is conserved during a collision.
• 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.
Chem 101
Fall 2004
Kinetic Energy
KE =
1
mv 2
2
mass
velocity
• SI Units: kg m2s-2 or Joules, J
Chem 101
Fall 2004
7
Kinetic Energy
Number of Molecules
1.0
0.8
0.6
0.4
0.2
0.0
0
20
40
60
80
Energy (kJ/mol)
Chem 101
Fall 2004
Average Kinetic Energy
Gas
Constant
KE avg =
3 RT
2 NA
Avogadro’s
Constant
• R=8.314 J mol-1 K-1
• NA=6.02x1023 mol-1
Chem 101
Fall 2004
Kinetic Energy and Temperature
KE avg =
3 RT
2 NA
• At a given temperature, T, different gases have
IDENTICAL average kinetic energy
• What is the average kinetic energy of an N2
molecule in the room?
Chem 101
Fall 2004
8
Average Molecular Speed
3 RT
2 NA
KE avg =
and KE avg =
1
2
mvavg
2
∴
1/ 2
 3 RT 
v avg = 

 MM 
molar mass
Chem 101
Fall 2004
Molecular Speeds
vavg (N2)
vavg (He)
Maxwell-Boltzmann
Distribution
Number of Molecules
1.0
•Average speed α (T/m)1/2
0.8
0.6
•Light faster than Heavy
0.4
•Hot faster than Cold
0.2
0.0
0
500
1000
1500
2000
2500
3000
Speed (m/s)
Chem 101
Fall 2004
Molecular Speeds
vavg (300K)
Number of Molecules
N2 at two different
temperatures
vavg (800K)
1.0
•Average speed α (T/m)1/2
0.8
•Light faster than Heavy
0.6
0.4
•Hot faster than Cold
0.2
0.0
Chem 101
Fall 2004
0
500
1000
1500
2000
2500
3000
Speed (m/s)
9
Molecular Speeds
Number of Molecules
1.0
0.8
The ‘tail’ of the
distribution is
extremely important
in chemical reactions
0.6
0.4
0.2
0.0
Chem 101
Fall 2004
0
500
1000
1500
2000
2500
3000
Speed (m/s)
Next Class:Gases and Kinetic-Molecular
Theory: Chapter 12
Chem 101
Fall 2004
10