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?