Stat. mech. (GRE)

1. Oct 19, 2006

Euclid

This is a GRE question.

A classical model of a diatomic molecule is a springy dumbbell, where the dumbbell is free to rotate about axes perpendicular to the spring. In the limit of high temperature. what is the specific heat per mole at constant volume?

The answer is 7R/2. This site
recommends to use a brute force approach to solve the problem. I believe the problem is much easier than suggested here, but since I have no experience with statistical mechanics, I would like help solving it this way.

1) How does one write down the partition function for this model? Taking into account only vibrational states, is it given by $$Z=\sum \limits_{n=0}^{\infty} \exp(-\beta (n+1/2)h\omega)$$?

If it is, the energy <E> should be given by
$$<E> = -\frac{1}{Z} \frac{\partial Z}{\partial \beta} = \frac{h \omega}{2} + \frac{1}{Z}\sum \limits_n^\infty n \exp(-\beta (n+1/2)h\omega)$$

The specific heat should then be given by
$$c=\frac{\partial <E>}{\partial T} =\frac{h\omega}{kT^2}(<n E> - <E><n>)$$,

where I have used shorthand that is hopefully clear. Evaluating the limit T--> infty seems unpleasant.

2) Now, what if I want to solve the problem completely, by taking into account the quantization of rotational states? How do I write down the partition function in this case?

How do I take into account translational energy?

Last edited: Oct 19, 2006
2. Oct 20, 2006

Euclid

Actually, I think I have (1) now. It is better to evaluate the sum first.

3. Oct 22, 2006

Galileo

Use the equipartition theorem. Each degree of freedom which appears as a quadratic in the hamiltonian contributes 1/2kT to the total energy.

I only count 6 degrees of freedom (3 translational, 2 rotational and 1 vibrational) which comes to 6/2RT per mole. So maybe I`m missing one degree somewhere.

4. Oct 22, 2006

Euclid

I understand that method is easier. There are actually two vibrational terms (one kinetic and one potential). That's how I get 7/2. But anyway, since I haven't studied statistical mechanics, I thought it would be helpful to go through this problem the hard way. I have a feeling my question is ill-phrased, so feel free to answer a related question that seems more appropriate.