Phyisab****
Jan18-10, 06:58 PM
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
F=-\tau log(Z)
\sigma=-(\frac{\partial\sigma}{\partial\tau})|_{V}
3. 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!
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
F=-\tau log(Z)
\sigma=-(\frac{\partial\sigma}{\partial\tau})|_{V}
3. 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!