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Weird form of entropy using grand partition function for a system

  1. Feb 19, 2014 #1
    1. The problem statement, all variables and given/known data
    Hey guys,

    Here's the question. For a distinguishable set of particles, given that the single particle partition function is [itex]Z_{1}=f(T)[/itex] and the N-particle partition function is related to the single particle partition function by [itex]Z_{N}=(Z_{1})^{N}[/itex] find the following:

    (a) The grand canonical partition function
    (b) The entropy
    (c) Prove that the entropy is given by
    [itex]\frac{S}{k}=N[\frac{Tf'(T)}{f(T)}-\log z]-\log(1-zf(T))[/itex] where [itex]z=e^{\beta\mu}[/itex] is the fugacity.


    2. Relevant equations
    Grand particle partition function
    [itex]Z=\sum_{N=0}^{\infty}z^{N}Z_{N}[/itex]

    Entropy
    [itex]S=(\frac{\partial(kT \log Z)}{\partial T})_{\beta,V}[/itex]
    (i found this myself so it might not be 100% right)

    3. The attempt at a solution
    So ive done everything but im struggling with part C:

    (a) [itex]Z=\frac{1}{1-zf(T)}[/itex]
    (b) Using that formula I found, i get [itex]\frac{S}{k}=\frac{Tzf'(T)}{1-zf(T)}-\log (1-zf(T))[/itex]

    for part (c), i dont know how im meant to get from what I have to what's required. Basically, i dont see how

    [itex]\frac{Tzf'(T)}{1-zf(T)}=N[\frac{Tf'(T)}{f(T)}-\log z][/itex]

    Thats pretty much all i need help with...but if you guys need more info just let me know! thanks a lot!
     
  2. jcsd
  3. Feb 20, 2014 #2
    Hi.
    All of the above is correct down to (a). For the rest:
    In (b) don't forget that z is a function of T when taking the derivative, you're missing a term.
    In (c), to obtain the expected expression you'll need to use the summation form of Z (your first "relevant equation") when taking the derivative of logZ and see what you get... (hint: <N>= N)
     
  4. Feb 20, 2014 #3
    OMGEEEE THANKS SO MUCH! Yea i got it now :D :D :D !!!
     
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