# Free energy of a rotational system.

1. Sep 1, 2008

### Narcol2000

If one has a diatomic molecule with energy levels

$$\epsilon_l = \frac{h^2 l(l+1)}{2I}$$

l = 0,1,2,3,4,5...

if the degneracy is given by $$g_l = (2l+1)$$

How does one show that the Helmholtz free energy at low temperature ($$h^2/Ikt$$ large)
is given by

$$F = -3kT e^{-h^2 / IkT} + ...$$

I got as far as getting the partition function to be

$$Z = \sum_{l=0}^{\inf} (2l+1)e^{-h^2 l(l+1)/2IkT}$$

2. Sep 1, 2008

### Count Iblis

Take the first two terms of the summation and then use that
F = - k T Log(Z). The term Log(Z) is of the form

Log(1 + small term) = small term - small term^2/2 = approximately small term