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Homework Help: Helmholtz Free Energy

  1. Nov 8, 2006 #1
    [itex] \tau = k_{B} T [/itex]
    a) Find the expression for the free energy as a function of the temperature of the system with two states - one iwth eneryg zero and one with energy [itex] \epsilon_{0} [/itex]

    b) From the free energy find the expressions for the energy and entropy of the system

    c) Plot the average energy and the entropy as a function of tau. [itex] \tau = k_{B} T [/itex]


    Ok for a) wek now that
    [tex] F = U - T \sigma [/tex]

    Partition fun ction [itex] Z = \sum_{s} \exp(-\epsilon_{s}/\tau) = 1 + \exp(-\epsilon_{0}/\tau) [/itex]

    so then
    [tex] U = \frac{\epsilon_{0} \exp(-\epsilon_{s}/\tau)}{1 + \epsilon_{0} \exp(-\epsilon_{s}/\tau)} [/tex]

    but im not quite sure how to proceed with the calculation of the entropy, sigma ...

    for b)
    for entropy use this
    [tex] \sigma = \left(\frac{\partial F}{\partial \tau}\right)_{V} [/tex]

    but not sure about how to find the nergy for hte system... is it simply the expression wh9ich doesnt involve tau??

    I was thiking a bit more

    isnt helmholtz free enryg given by simple
    [tex] F- F(0) = -\tau \log Z [/tex]??
     
    Last edited: Nov 8, 2006
  2. jcsd
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