1. Lecture 4 – Kinetic Theory of Ideal Gases
Ch 24
Notice: we are following a different route from the book
Getting to the same end-point though

2. Summary of lecture 3
The ideal Gas Law: PV=nRT is an equation of state deduced from experiments (i.e. Boyle’s and Charles’ Laws). It introduces pressure as a strictly macroscopic concept
Classical mechanical principles express pressure as a mechanical (microscopic) property in the Kinetic Theory of Gases:

3. Summary of lecture 3
Statistical thermodynamics provided yet another description of an ideal gas from the statistical mechanical point of view
All together, we have provided a microscopic interpretation of temperature in terms of the average kinetic energy per molecule:

4. Maxwell-Boltzmann speed distribution
Maxwell-Boltzmann speed distribution function provides the probability for each molecule in the gas having a certain speed
We will see how to use this distribution in the next exercise

5. Maxwell-Boltzmann speed distribution
What is the fraction of O2 molecules in a container held at T=1000K that have a speed of 1500m/s?
Use the expression:
Calculate the mass m for a single O2 molecule (in kilograms):

6. Average Kinetic Energy, Speed, Root-Mean-Square Speed
According to the basic principles of statistical mechanics, we can calculate the average of any quantity that depends on the speed c using the Maxwell-Boltzmann distribution function, e.g.

7. Average Kinetic Energy, Speed, Root-Mean-Square Speed
For example, let us calculate average kinetic energy:
Let us use the integral:

8. Average Kinetic Energy, Speed, Root-Mean-Square Speed
For us, a=2 and
Hence:

9. Average Kinetic Energy, Speed, Root-Mean-Square Speed
We have found again that the average kinetic energy:
Since:
This is yet another (statistical mechanical) derivation of the ideal gas law

10. Average Kinetic Energy, Speed, Root-Mean-Square Speed
Let us derive the energy of an ideal gas using the Boltzmann distribution function. The energy of an ideal gas is the average molecular kinetic energy:
Hence:

11. Average Kinetic Energy, Speed, Root-Mean-Square Speed
We can provide other quantities that will become useful in the next few lectures
For example, the root-mean squared speed (square root of the average squared speed)
The rms speed is not equal to the average speed:

12. Average Kinetic Energy, Speed, Root-Mean-Square Speed
Let us calculate the average speed; this integral again has standard form
With a=1 and

13. Average Kinetic Energy, Speed, Root-Mean-Square Speed
Note once again that:
This difference is very important

14. Collision Rates, Mean Free Path, Diffusion
How molecules in a gas move
The zig-zag motion of the molecules in a gas is called diffusion. It explains the characteristic slowness of gases (straight line speed for O2 at room T is about 400 m/s!)

15. Collision Rates, Mean Free Path, Diffusion
Statistical techniques and the velocity distribution function can be used to calculate properties of gases:
wall collision rate Zw
molecular collision rate Z1
mean-free path l
diffusion constant D
We will give only approximate derivations for these quantities; more thorough derivations yield only small correction factors.

16. Wall Collision Rates (rate at which molecules collide with wall)
We expect this to be proportional to:
density N/V
average speed
surface area
A better derivation accounting for the directionality of molecular motion would provide:

17. Wall Collision Rates
We can also derive this expression from M-B distribution by realizing that if a particle has speed c and is within a distance c(dt) from a surface A will hit the surface within time dt
All particle within a volume Acdt will hit the wall in time dt; how many particles are there?
The number of collisions per unit time will be:

18. Molecular Collision Rates
(the rate at which molecules collide with each other)
If a molecule has diameter d and speed c, it moved cdt within a time dt=1s tracing a cylindrical volume:
The number of collisions per unit time will be simply the number of molecules found in that volume

19. Molecular Collision Rates
(the rate at which molecules collide with each other)
An exact calculation (assuming the other molecules move as well, as is correct) yields

20. Mean Free Path
(how far a molecule travel before colliding with another)
This is simply the average speed divided by the collision rate

21. Diffusion coefficient
The zig-zag motion of the molecules in a gas is called diffusion. It explains the characteristic slowness of gases
The mean square displacement associated with this motion is given by:
D is called diffusion coefficient and has units of m2/s

22. Diffusion coefficient
D is proportional to the mean free path l and the average speed:
The proportionality constant is difficult to calculate, and depends on the properties of the gas; for a single component gas, for example: