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Entropy of a Fermi dirac ideal gas

  1. Apr 23, 2014 #1
    Hello
    1. The problem statement, all variables and given/known data
    From the expression of the partition function of a fermi dirac ideal gas
    ln(Z)=αN + ∑ ln(1+exp(-α-βEr))
    show that
    S= k ∑ [ <nr>ln(<nr>)+(1-<nr>)ln(1-<nr>)


    2. Relevant equations
    S=k( lnZ+β<E>)
    <nr>=-1/β ∂ln(Z)/∂Er
    <E>=-∂ln(Z)/∂β

    3. The attempt at a solution
    I tried to start with
    S=k( lnZ+β<E>)
    But I don't know how we get to introduce the nr in that ?

    Edit:
    S=-dF/dT with F=-kTln(Z)
    I wrote ln(Z) in terms of nr
    ln(Z)=α ∑<nr>-∑ln(<nr>)
    But I don't get to the result
    I get at the end
    S= kα∑<nr>-k∑ln(<nr>)+kα∑(1-<nr>)

    Thanks
     
    Last edited: Apr 23, 2014
  2. jcsd
  3. Apr 23, 2014 #2

    rude man

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    Homework Helper
    Gold Member

    I suggest moving this to the advanced physics forum.
     
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