Kinetic theory of gases

Maxwell law
Vander waals' equation
Mean free path
Root mean square velocity

1. Kinetic Theory of Gases
Dr.S.Radha
Assistant Professor of
Chemistry
SaivaBhanuKshatriya College
Aruppukottai

2. Postulates of Kinetic Theory Of
Gases
Gases are mostly empty space
Gases contain molecules which have random
motion
The molecules have kinetic energy
The molecules act independently of each other –
there are no forces between them
Molecules strike the walls of the container – the
collisions are perfectly elastic
Exchange energy with the container
The energy of the molecules depends upon the
temperature

3. Large collections are very
predictable
Fluctuations in behaviour of a small group of
particles are quite noticeable
Fluctuations in behaviour of a large group (a mole)
of particles are negligible
Large populations are statistically very reliable

4. Pressure and momentum
Pressure = force/unit area
Force = mass x acceleration
Acceleration = rate of change of velocity
Force = rate of change of momentum
Collisions cause momentum change
Momentum is conserved

5. Elastic collision of a particle with the wall
Momentum lost by particle = -2mv
Momentum gained by wall = 2mv
Overall momentum change = -2mv + 2 mv = 0
Momentum change per unit time = 2mv/Δt
momentum change no collisions 1
collision time area
P x x=

6. Factors affecting collision
rate
1.Particle velocity – the faster the particles the more
hits per second
2.Number – the more particles – the more collisions
3.Volume – the smaller the container, the more
collisions per unit area
2
2
2
o o
vN mv N
P mv
V V
= · =

7. Making refinements
We only considered one wall – but there are six
walls in a container
Multiply by 1/6
Replace v
2
by the mean square speed of the
ensemble (to account for fluctuations in velocity)
2
1
3
o
m v N
P
V
=
2 2
21
6 3
o o
mv N mv N
P x
V V
= =

8. Boyle’s Law
Rearranging the previous equation:
Substituting the average kinetic energy
Compare ideal gas law PV = nRT:
The average kinetic energy of one mole of
molecules can be shown to be 3RT/2
2
2 2
3 2 3
o o k
m v
PV N N E
æ ö
ç ¸= =
ç ¸
è ø
3
2
k
E nRT=

9. Root mean square speed
Total kinetic energy of one mole
But molar mass M = N
om
Since the energy depends only on T, v
RMS
decreases
as M increases
21 3
2 2
k o
E N m v RT= =
M
RT
mN
RT
v
o
RMS
33
==

10. Speed and temperature
Not all molecules move at the same speed or in the
same direction
Root mean square speed is useful but far from
complete description of motion
Description of distribution of speeds must meet two
criteria:
Particles travel with an average value speed
All directions are equally probable

11. Maxwell meet Boltzmann
The Maxwell-Boltzmann
distribution describes the
velocities of particles at a
given temperature
Area under curve = 1
Curve reaches 0 at v = 0
and ∞
2
/ 22
3/ 2
( )
4
2
B
mv k T
B
F v Kv e
m
K
k T
p
p
-
=
æ ö
=ç ¸
è ø

12. M-B and temperature
As T increases v
RMS
increases
Curve moves to right
Peak lowers in height
to preserve area

13. Boltzmann factor: transcends
chemistry
Average energy of a particle
From the M-B distribution
The Boltzmann factor – significant for any and all
kinds of atomic or molecular energy
Describes the probability that a particle will adopt a
specific energy given the prevailing thermal energy
1
exp
exp
B
B
k T
k T
e
e
æ ö
- =ç ¸
æ öè ø
ç ¸
è ø
2
2
mv
e=
( ) exp
B
P
k T
e
e
æ ö
m -ç ¸
è ø

14. Applying the Boltzmann
factor
Population of a state at a level ε above the ground
state depends on the relative value of ε and k
B
T
When ε << k
B
T, P(ε) = 1
When ε >> k
BT, P(ε) = 0
Thermodynamics, kinetics, quantum mechanics
( ) exp
B
P
k T
e
e
æ ö
m -ç ¸
è ø

15. Collisions and mean free path
Collisions between
molecules impede
progress
Diffusion and effusion
are the result of
molecular collisions

16. Diffusion
The process by which gas molecules become
assimilated into the population is diffusion
Diffusion mixes gases completely
Gases disperse: the concentration decreases with
distance from the source

17. Effusion and Diffusion
The high velocity of molecules leads to rapid mixing
of gases and escape from punctured containers
Diffusion is the mixing of gases by motion
Effusion is the escape of a gas from a container

18. Graham’s Law
The rate of effusion of a gas is inversely proportional
to the square root of its mass
Comparing two gases
m
Rate
1
µ
1
2
1
2
2
1
m
m
m
m
Rate
Rate
==

19. Living in the real world
For many gases under most conditions, the ideal
gas equation works well
Two differences between the ideal and the real
Real gases occupy nonzero volume
Molecules do interact with each other – collisions are
non-elastic

20. Consequences for the ideal gas
equation
1.Nonzero volume means actual pressure is larger
than predicted by ideal gas equation
Positive deviation
1.Attractive forces between molecules mean
pressure exerted is lower than predicted – or
volume occupied is less than predicted
Negative deviation
Note that the two effects actually offset each other

21. Van der Waals equation:
tinkering with the ideal gas equation
Deviation from ideal is more apparent at high P, as
V decreases
Adjustments to the ideal gas equation are made to
make quantitative account for these effects
( )nRTnbV
V
an
P =-
÷
÷
ø
ö
ç
ç
è
æ
+
2
2
Correction for
intermolecular
interactions
Correction for
molecular
volume

22. Real v ideal
At a fixed temperature
(300 K):
 PV
obs
< PV
ideal
at low P
PV
obs
> PV
ideal
at highP

23. Effects of temperature on deviations
For a given gas the
deviations from ideal
vary with T
As T increases the
negative deviations
from ideal vanish
Explain in terms of van
der Waals equation
( )nRTnbV
V
an
P =-
÷
÷
ø
ö
ç
ç
è
æ
+
2
2

24. Interpreting real gas
behaviors
First term is correction for
volume of molecules
Tends to increase P
real
Second term is correction
for molecular interactions
Tends to decrease P
real
At higher temperatures,
molecular interactions are
less significant
First term increases relative to
second term
( )
2
2
2
2
V
an
nbV
nRT
P
nRTnbV
V
an
P
-
-
=
=-
÷
÷
ø
ö
ç
ç
è
æ
+