- #1
S_Flaherty
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Homework Statement
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.
Homework Equations
Z = Ʃe-E(s)/kT
U = -N(dlnZ/dβ)
S = U/T + NklnZ
F = U - TS = -NkTlnZ
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?