Temperature as a function of vibration energy quanta

[physicstudent_B]

Homework Statement

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

Homework Equations

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 ?

 

[TJGilb]

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.