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Help simplifying this using Stirlings' Formula?

  1. Mar 6, 2015 #1
    Hey guys!

    So I'm self studying Daniel Schroeder's Intro to Thermal Physics and we're to use Stirlings Approximation to simplify the following (assume that BOTH q and N are large, but we don't have a relationship between q and N):

    [tex]
    \Omega \left( N,q\right) \approx \dfrac {\left( \dfrac {q+N} {q}\right) ^{q}\left( \dfrac {q+N} {N}\right) ^{N}} {\sqrt {2\pi q\left( q+N\right) / N}}
    [/tex]

    The formula to find the multiplicity of an Einstein solid is:

    [tex]
    \Omega \left( N,q\right) =\dfrac {\left( N-1+q\right) !} {q!\left( N-1\right) !}
    [/tex]

    The hint in the text says to show that first:

    [tex]
    \Omega \left( N,q\right) =\dfrac {N} {q+N}\dfrac {\left( q+N\right) !} {q!N!}
    [/tex]

    With q and N being large, I thought I could immediately say that

    [tex]
    \Omega \left( N,q\right) \approx \dfrac {\left( q+N\right) !} {q!N!}
    [/tex]

    But I don't know where you get the

    [tex]
    \dfrac {N} {q+N}
    [/tex]

    I tried immediately applying Stirlings Approximation but got nothing like the hint or the answer we're looking for.

    I'm a little lost on where to begin here. Any thoughts?

    Thanks so much.
     
  2. jcsd
  3. Mar 6, 2015 #2

    Mentallic

    User Avatar
    Homework Helper

    If you cancel the common factor of N in the numerator and N in the N! expression in the denominator, what do you get?
     
  4. Mar 6, 2015 #3
    oh nice! got it.

    Man, i'm dumb! Thanks so much.
     
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