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Homework Help: Entropy of N particle system

  1. Jan 18, 2010 #1
    1. The problem statement, all variables and given/known data
    A system of particles is in equilibrium at temperature T. Each particle may have energy 0, epsilon, or 2 epsilon. Find the entropy of the system.


    2. Relevant equations

    [tex]F=-\tau log(Z)[/tex]

    [tex]\sigma=-(\frac{\partial\sigma}{\partial\tau})|_{V}[/tex]

    3. The attempt at a solution

    [tex]

    Z = 1+exp(\frac{-\epsilon}{\tau})+exp(\frac{-2\epsilon}{\tau})
    [/tex]

    [tex]F=-\tau log(1+exp(\frac{-\epsilon}{\tau})+exp(\frac{-2\epsilon}{\tau}))[/tex]

    [tex]\sigma=-log(1+exp(\frac{-\epsilon}{\tau})+exp(\frac{-2\epsilon}{\tau}))-\tau\frac{[\epsilon\tau^{-2}exp(-\epsilon/\tau)+2\epsilon\tau^{-2}exp(-2\epsilon/\tau)]}{1+exp(\frac{-\epsilon}{\tau})+exp(\frac{-2\epsilon}{\tau})}[/tex]

    Hows that look? I'm really rusty with my thermal physics so even though this is not very complicated, I just have no confidence. One thing I was worried about was my partition function. When is the function I used applicable, and when do I need to use

    [tex]Z_{N}=\frac{Z^{N}_{1}}{N!}?[/tex]

    As I type this I am becoming increasing doubtful that I used the right partition function. Thanks for reading!
     
    Last edited: Jan 18, 2010
  2. jcsd
  3. Jan 19, 2010 #2
    Is my question ill posed? Please if you read this say anything, say whatever you think even if you don't know.
     
  4. Jan 19, 2010 #3
    Now I am feeling rather sure that all I need to do is replace my Z with Z^N and I have my answer. Any thoughts?
     
  5. Jan 20, 2010 #4
    Anyone? Anything?

    I just noticed a mistake in my first post, the second equation should be sigma=(partial F)/(partial tau). But if you can help me you probably knew that already...
     
  6. Jan 20, 2010 #5
    Using Z = Z_1^N/N! , you should get the correct expression.
     
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