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Temperature as a function of vibration energy quanta

  1. Nov 30, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider a small cluster of four copper atoms. Assume, that each atom can oscillate around its equilibrium position independently of the other atoms. Let us model each direction of vibration as a harmonic oscillator, whose energy is quantized
    Evib = ¯hω(n +1/2), where ω = sqrt(k/m), k is the ’spring constant’ of the bond between the atoms (k ≈ 120 N/m) and m is the mass of the copper atom. Compute as the function of vibration energy quanta n = 0,1,2,3,... the (a) number of microstates Ω
    (b) entropy S
    (c) temperature T
    (d) heat capacity C / atom

    2. Relevant equations

    3. The attempt at a solution
    I have already functions for the number of microstates and the entropy S:
    Ω(n) = ((n+11)!) : (n!11!)
    S(n) = k*ln (((n+11)!) : (n!11!)
    Now I try to find the function T(n) and started with:

    1/T = dS/dEint (dS = change in entropy, dEint = change in internal energy)

    Can I put this equations as T = dEint/dS ?
  2. jcsd
  3. Dec 4, 2016 #2
    Yes you can do that, but you'll have to put E(int) in terms of S. It may help if you're allowed to make the assumption that n >> # of oscillators.
    Last edited: Dec 4, 2016
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