1. 14.5 Distribution of molecular speeds
For a continuum of energy levels,
where
and

2. Combining the above equations, one has
ε is a kinetics energy calculated through (1/2)mv2, thus dε = mvdv
The above equation can be transformed into (in class demonstration)

3. 14.6 Equipartition of energy
From Kinetics theory of gases
showing that the average energy of a molecule is the number of degrees of freedom (f) of its motion.
For a monatomic gas, there are three degrees of freedom, one for each direction of the molecule’s translational motion.
The average energy for a single monatomic gas molecule is (3/2)kT (in class derivation).
The principle of the equipartition of energy states that for every degree of freedom for which the energy is a quadratic function, the mean energy per particle of a system in equilibrium at temperature T is (1/2)kT.

4. 14.7 Entropy change of mixing revisited
From classical thermodynamics
Δs = - nR (x1lnx1 + x2lnx2)
where x1 = N1/N and x2 = N2/N
Now consider mixing two different gases with the same T and P, the increase in the total number of configurations available to the system can be calculated with

5. From Boltzmann relationship
Using Stirling’s approximation (see white board for details)
we get Δs = - nR (x1lnx1 + x2lnx2)
From statistical point of view, when mixing two of the same type of gases under the same T and V (i.e. non-distinguishable particles with the same Ej), there is no change in the total number of available microstates, thus Δs equals 0

6. 14.8 Maxwell’s Demon

7. Demon (II)
Figure 14.4 Maxwell’s demon in action. In this version the demon operates a valve, allowing one species of a two-component gas (hot or cold) through a partition separating the gas from an initially evacuated chamber. Only fast molecules are allowed through, resulting in a cold gas in one chamber and a hot gas in the other.

8. Problem 14.2: Show that for an assembly of N particles that obeys Maxwell-Boltzmann statistics, the occupation numbers for the most probable distribution are given by:
Solution

9. 14.3a) Show that for an ideal gas of N molecules,
Solution:

10. 14-3(b) For
calculate
Solution: