Free energy of a rotational system.

[Narcol2000]

If one has a diatomic molecule with energy levels

<br /> \epsilon_l = \frac{h^2 l(l+1)}{2I}<br />

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

<br /> F = -3kT e^{-h^2 / IkT} + ...<br />

I got as far as getting the partition function to be

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

 

[Count Iblis]

If one has a diatomic molecule with energy levels

<br /> \epsilon_l = \frac{h^2 l(l+1)}{2I}<br />

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

<br /> F = -3kT e^{-h^2 / IkT} + ...<br />

I got as far as getting the partition function to be

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

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