Entropy of N particle system

[Phyisab****]

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

$$F=-\tau log(Z)$$

$$\sigma=-(\frac{\partial\sigma}{\partial\tau})|_{V}$$

The Attempt at a Solution

$$Z = 1+exp(\frac{-\epsilon}{\tau})+exp(\frac{-2\epsilon}{\tau})$$

$$F=-\tau log(1+exp(\frac{-\epsilon}{\tau})+exp(\frac{-2\epsilon}{\tau}))$$

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

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

$$Z_{N}=\frac{Z^{N}_{1}}{N!}?$$

As I type this I am becoming increasing doubtful that I used the right partition function. Thanks for reading!

 

[Phyisab****]

Is my question ill posed? Please if you read this say anything, say whatever you think even if you don't know.

 

[Phyisab****]

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?

 

[Phyisab****]

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...

 

[Count Iblis]

Using Z = Z_1^N/N! , you should get the correct expression.