# Finding the partition function

1. Nov 13, 2013

### S_Flaherty

1. The problem statement, all variables and given/known data
Consider a solid of N localized, non-interacting molecules, each of which has three quantum states with energies 0, ε, ε, where ε > 0 is a function of volume.

Question: Find the internal energy, Helmholtz free energy, and entropy.

2. Relevant equations
Z = Ʃe-E(s)/kT
U = -N(dlnZ/dβ)
S = U/T + NklnZ
F = U - TS = -NkTlnZ

3. The attempt at a solution
For the internal energy would I just multiply N by the average energy?
So U = N((0 + ε + ε)/3) = 2Nε/3?

I also know there are equations for the three of these values that require the partition function, Z.

I know Z = Ʃe-E(s)/kT so would this just be
Z = 1 + e-ε/kT + e-ε/kT = 1 + 2e-ε/kT
U = -N(dlnZ/dβ)
lnZ = ln(1) + ln(2e-ε/kT) = ln2 - εβ, so U = Nε...

Which one of those solutions for U is the correct one? Or am I wrong in both cases?

2. Nov 13, 2013

### qbert

Second is right technique, although you've mathed wrongly.

for three states: 0, 1, 2
U = avg energy = E(0) P(0) + E(1) P(1) + E(2) P(2)

For E(0)=0, E(1)=E(2) =ε,
and P(energy) = exp(-energy/kT)/Z
P(0) = 1/Z, P(1)=P(2)=exp(-ε/kT)/Z

3. Nov 13, 2013

### S_Flaherty

So, I get U = 2εe-ε/kT/Z

I just figured out something I did wrong with Z. dlnZ/dβ = (-2εe-εβ)/(1 + 2εe-εβ) so U = (2Nεe-εβ)/(1 + 2εe-εβ) ? Or did I mess something up?

Last edited: Nov 13, 2013