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Free energy of a rotational system.

  1. Sep 1, 2008 #1
    If one has a diatomic molecule with energy levels

    [tex]
    \epsilon_l = \frac{h^2 l(l+1)}{2I}
    [/tex]

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

    if the degneracy is given by [tex]g_l = (2l+1)[/tex]

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

    [tex]
    F = -3kT e^{-h^2 / IkT} + ...
    [/tex]

    I got as far as getting the partition function to be

    [tex]
    Z = \sum_{l=0}^{\inf} (2l+1)e^{-h^2 l(l+1)/2IkT}
    [/tex]
     
  2. jcsd
  3. Sep 1, 2008 #2

    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
     
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