Entropy of N particle system

  • #1
585
2

Homework Statement


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.


Homework Equations



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

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

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:

Answers and Replies

  • #2
585
2
Is my question ill posed? Please if you read this say anything, say whatever you think even if you don't know.
 
  • #3
585
2
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?
 
  • #4
585
2
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...
 
  • #5
1,838
7
Using Z = Z_1^N/N! , you should get the correct expression.
 

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