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Finding the partition function

  1. Nov 13, 2013 #1
    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. jcsd
  3. Nov 13, 2013 #2
    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
     
  4. Nov 13, 2013 #3
    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
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