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Statistical Relationship Between Helmoltz Free Energy and Entropy

  1. Jul 16, 2012 #1
    1. The problem statement, all variables and given/known data
    From Hill's "Introduction to Statistical Thermodynamics", question 3-4 reads:

    (note "STR" denotes the case of most probable distribution and should be read as C*)

    2. Relevant equations
    The most probable distribution for a system of independent indistinguishable molecules:
    [itex]\eta_i = \bar{C_j} /N =C^{str}/N = e^{-\beta \epsilon_j} / \sum_i e^{-\beta \epsilon_i}[/itex]

    Formula for accessible quantum states for an ensemble of independent indistinguishable molecules:
    [itex]\Omega(C) = N! /( \prod_j C_j) ![/itex]

    Relationship between S and A
    S = [itex]- (\partial A / \partial E) = kT ( \partial ln(Q) / \partial T )+ k ln(Q) [/itex]


    3. The attempt at a solution

    First, I computed C in terms of the given equations. This is pretty straight forward:

    [itex] C^{str}_j = Ne^{-\beta \epsilon_j} /{ \sum_i e^{-\beta \epsilon} }[/itex]

    Then, I computed the number of accessible quantum states with the above expression for C
    [itex] \Omega (C^{str}) = { N! }/{\prod C^{str} } [/itex]

    Skipping a lot of algebra I arrived as this expression for ln(Ω(C)):
    [itex] ln(C^{str}) = ln(N!)- \sum(ln(e^{-\beta \epsilon}) )- ln(\sum(e^{-\beta \epsilon}) [/itex]

    Which can be reduced to:
    [itex]ln(C^{str}) = ln(N!) + \sum(\beta \epsilon) - ln(Q) = ln(N!) + E \beta -ln(Q)[/itex]

    I am not sure exactly where I went wrong, as the formula I derived is very similar to the relationship between A and S. I think an approximation may have to be applied somewhere or perhaps the natural logs should not have been broken down into their operations. I am not sure, hence why I am posting here!

    Thanks for the help.

    PS, if needed I can post a small nomenclature of the variables.

    EDIT:
    Just realized I did not take into account the N factor in the equation for C, I believe this makes N! become (N-1)! in the following equations. Also, I did not show the additional formula if one were to expand [itex] \partial ln(Q) / \partial T[/itex]
     
    Last edited: Jul 16, 2012
  2. jcsd
  3. Jul 17, 2012 #2
    Bad question?
     
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