1. Kinetic Theory of Gases
Chemistry 232
Kinetic Theory of Gases

2. 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.

3. Assumptions of Kinetic Theory
Total energy of the system
Intermolecular attractive interactions are negligible.

4. 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.

5. Kinetic Theory Postulates (Cont’d)
Average kinetic energy (K.E.) of molecules depends on absolute temperature (T) only.
All collisions are elastic.

6. Kinetic Theory of Gases

7. 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!

8. The greater the momentum of gas molecules, the greater the effect of the impact on the walls.
Force/A = P

9. 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

10. Not All Particles Reach the Wall!
How many particles actually reach the wall during a specified time interval t?
+vi,xt
These molecules don’t reach the wall!
These molecules come into contact with the wall!

11. The Total Momentum Change
The total momentum change is calculated form the sum of the momentum changes for the individual particles.

12. The Definition of Pressure
The pressure exerted by the gas is calculated as follows

13. 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

14. 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

15. The Root Mean Square Speed
1/3 Mi<vi>2 = RT
<vi>2 = 3RT / Mi
(<vi>2 )1/2 = vrms = (3RT/Mi)1/2
vrms = the root mean square speed

16. 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

17. The Maxwell Distribution for Typical Gases

18. Other Speed Equations
In addition to the root-mean-square speed, we have the
Most probable speed
The mean speed

19. The Root Mean Square Speed

20. Intermolecular Collisions in Hard-sphere 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?

21. 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

22. The Collision Cylinder
Stationary particles inside the collision tube.

23. The Number Density
For N-1 stationary particles, the number of molecules per unit volume

24. # Inside tube = Nd<v>t
Collision Frequency
We count the total number of molecules with centres inside the collision tube.
# Inside tube = Nd<v>t

25. Collision Frequency (cont’d)
For N-1 stationary particles
The collision frequency - z1
Examine the case where all the molecules inside the collision tube are moving.

26. Collision Frequency (Cont’d)
Relative speed of the colliding particles.

27. The Mean Collision Time
The mean collision time is average time elapsed between successive collisions.
Define
coll = 1/z1

28. 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

29. The Mean Free Path
The mean free path - the average distance traveled between successive collisions.

30. The Mean Free Path

31. The Collision Density
We define the collision density as the total rate of collisions per unit volume.

32. 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.

33. The Mean Collision Diameter
Define in terms of the collision diameters of the colliding species. d2 d1
Mean collision diameter
d12 = ½ (d1+d2)

34. The Collision Zone For a collision occurring along the x and y axis. x
Impact Zone x
X1=tc<v1>
tc = time yet to elapse before the collision occurs
y2=tc<v2> y

35. Mean Relative Speed
The mean relative speed.

36. The Reduced Mass
The reduced mass of two particles 1 and 2 is defined as follows

37. Mean Free Paths in Heteronuclear Collisions
For substance 1 colliding with substance 2

38. Mean Free Paths (Cont’d)
For substance 2 colliding with substance 1

39. Heteronuclear Collision Frequencies
The collision frequency of molecule 1 with molecule 2 is given by

40. Heteronuclear Collision Frequencies (cont’d)
The collision frequency of molecule 2 with molecule 1 is given by

41. Heteronuclear Collision Density
The total rate of heteronuclear collisions per unit volume