Chemistry 232
Kinetic Theory of Gases
Kinetic Molecular Theory of
Gases


Macroscopic (i.e., large quantity)
behaviour of gases – pressure, volume,
and temperature.
The kinetic molecular theory of gases
attempts to explain the behaviour of
gases on a molecular level.
Assumptions of Kinetic Theory

Total energy of the system
E   V

Intermolecular attractive interactions
are negligible.
Postulates of Kinetic Theory of
Gases



Gases consist of molecules of mass m
and diameter d.
Gas molecules are in constant, rapid,
straight-line motion. Collisions are
elastic.
The gas molecules interact only when
they collide.
Kinetic Theory Postulates
(Cont’d)


Average kinetic energy (K.E.) of
molecules depends on absolute
temperature (T) only.
All collisions are elastic.
Kinetic Theory of Gases
Explanation of Pressure


Gas pressure - collisions of gas
molecules with the container walls.
The force of a collision depends on
• the number of collisions per unit time
• how hard gas molecules strike the container
wall!

The greater the momentum of gas
molecules, the greater the effect of the
impact on the walls.
Force/A = P
Fx i
d 2 x i 
d v x i 
 mi 
  mi  2 
 dt 
 dt 
The Momentum Change During a
Collision

Particle of mass mi collides with the wall
with only the x component of the
momentum changing.
+ m vix
- m vix
Not All Particles Reach the Wall!

How many particles actually reach the
wall during a specified time interval t?
+vJ,xt
These molecules don’t
reach the wall!
These molecules come
into contact with the wall!
The Total Momentum Change

The total momentum change is
calculated form the sum of the
momentum changes for the individual
particles.
n A M J v J ,x  t
2
total momentum change 
V
The Definition of Pressure

The pressure exerted by the gas is
calculated as follows
F n M J v J ,x  t
P 
A
V
2
Distribution of Molecular Speeds


This speed in the above equation should
be an average speed (some will always
be fast, some slow).
Replace with the ensemble average
F n M J v J ,x
P 
A
V
2
t
The Mean Square Speed


Kinetic Molecular Theory of Gases
allows us to relate macroscopic
measurements to molecular quantities
P, V are related to the molar mass and
mean square seed
1
2
PV  n M i v i  nRT
3
The Root Mean Square Speed
1/3 MJ<vJ>2 = RT
<vJ>2 = 3RT / MJ
(<vJ>2 )1/2 = vrms = (3RT/MJ)1/2
vrms = the root mean square speed
The Maxwell Probability Distribution


In kinetic theory, we are interested in the
fraction of molecules having a particular
range of speeds.
The probability distribution of speeds
3
2

 MJ 
k BT
g (v )  
e


 2 RT 
The Maxwell Distribution for Typical
Gases
Other Speed Equations

In addition to the root-mean-square
speed, we have the
1
• Most probable speed
 2RT  2
v mp  

 MJ 
• The mean speed
 8RT 
v 

 MJ 
1
2
The Root Mean Square Speed
Collisions With Walls and
Surfaces

Rate at which molecules collide with a
wall of area A
Zw 
pN Avo
 2 RTM J 
1
2
Effusion

Rate at which molecules pass through a
small hole of area Ao, r
r  Z w Ao 
pN Avo Ao
 2 RTM J 
1
2
Effusion (Cont’d)

Effusion.
• A gas under pressure goes (escapes) from
one compartment of a container to another by
passing through a small opening.
Effusion
The Effusion Equation


Graham’s Law - estimate the ratio of the
effusion rates for two different gases.
Effusion rate of gas 1 r1.
r1  Z w ,1 Ao 
pN A v oAo
1
2
2RTM1 
Effusion Equation (Cont’d)

Effusion rate of gas 2  r2.
r2  Z w ,2 Ao 
pN A v oAo
1
2
2RTM2 
Effusion Ratio

Ratio of effusion rates.
r2
r1

pN A v oAo
1
2
2RTM 2 
pN A v oAo
1
2
2RTM1 
M


1


 M2 
1
2
Intermolecular Collisions in Hardsphere Gases

Quantitative picture of the events that
take place in a collection of gaseous
molecules.
• Frequency of collisions?
• Distance between successive collisions?
• Rate of collisions per unit volume?
The Definition of a Collision

A pair of molecules will collide whenever
the centres of the two molecules come
within a distance d (the collision
diameter) of one another.
No collision.
Collision occurs.
d
The Collision Cylinder
d
2d
Stationary particles inside the
collision tube.
The Number Density

For N-1 stationary particles, the number
of molecules per unit volume
N 1  N
Nd  

 V  V
Collision Frequency

We count the total number of molecules
with centres inside the collision tube.
# Inside tube = Nd<v>t
Collision Frequency (cont’d)

For N-1 stationary particles
• The collision frequency - z1
2
z 1  v N d  v d N d

Examine the case where all the
molecules inside the collision tube are
moving.
Collision Frequency (Cont’d)

Relative speed of the colliding particles.
v
z1  v
rel
 2 v
1
2
rel
 16RT 
2
Nd  
 d Nd
 M J 
The Mean Collision Time


The mean collision time is average time
elapsed between successive collisions.
Define
• coll = 1/z1
 coll
1
2
 M J 
1


2
 16RT   d N d
The Mean Free Path


Gas molecules encounter collisions with
other gas molecules and with the walls
of the container
Define the mean free path as the
average distance between successive
molecular collisions
• Note -  - the collision cross section
 = d2
The Mean Free Path

The mean free path - the average
distance traveled between successive
collisions.
  v  coll 
1
1
2 d Nd
2
The Mean Free Path
The Collision Density

We define the collision density as the
total rate of collisions per unit volume.
Z 11  1
2
z 1N d
1
2
 4RT 
2
2

 d Nd
 MJ 
Collisions in Heteronuclear Systems

Modify the above discussion to include
collisions between unlike molecules.
• The mean collision diameter.
• The reduced mass of the colliding molecules.
• The collision zone.
The Mean Collision Diameter

Define in terms of the collision diameters
of the colliding species.
d1
d2
Mean collision
diameter
d12 = ½ (d1+d2)
The Collision Zone

For a collision occurring along the x and
y axis.
x
Impact Zone
X1=tc<v1>
tc = time yet to
elapse before the
collision occurs
y2=tc<v2>
y
Mean Relative Speed

The mean relative speed.
v
rel

 v1
 8RT
 
 
2
 v2
2

1
1
1  



M1 M2  
2
1
2
The Reduced Mass

The reduced mass of two particles 1 and
2 is defined as follows
1
1
1


12
M1 M2
Mean Free Paths in Heteronuclear
Collisions

For substance 1 colliding with substance
2
1 2  
v1
1
v 12 d12  Nd 2 
 M2
 
 M1  M2
2



1
2
1
d12  Nd 2 
2
Mean Free Paths (Cont’d)

For substance 2 colliding with substance
1
 2 1 
v2
1
v 12 d12  Nd 1
 M1
 
 M1  M2
2



1
2
1
d12  Nd 1
2
Heteronuclear Collision Frequencies

The collision frequency of molecule 1
with molecule 2 is given by
 8 RT
1
z 1 2  
 
 1 2   12
1
2

2
  d 12  N d 2 

Heteronuclear Collision Frequencies
(cont’d)

The collision frequency of molecule 2
with molecule 1 is given by
 8RT
1
z 2 1 
 
 2 1  12
1
2

2
  d 12  N d 1

Heteronuclear Collision Density

The total rate of heteronuclear collisions
per unit volume
Z 12  z 1 2 N d 1  z 2 1N d 2 
 8RT
 
 12
1
2

2
  d 12  N d 1N d 2 
