# Entropy of a mole of a crystalline solid as a function of temperature

1. Apr 9, 2014

### NobodyMinus

1. The problem statement, all variables and given/known data
5. The nuclei of atoms in a certain crystalline solid have spin one. Each nucleus can be in any one of three quantum states labeled by the quantum number m, where m = −1,0,1. This quant number measures the projection of the nuclear spin along a crystal axis of the solid. Due to the ellipsoidal symmetry, a nucleus has the same energyε for in the state m = −1 and the state m = 1, compared with an energy E = 0 in the state of m = 0.
(a) Find an expression as a function of T of the nuclear contribution to the average internal energy of the solid
per mol.
(b) Find an expression as a function of T of the nuclear contribution to the entropy of the solid per mol

2. Relevant equations
U=∑EiPi
Pi=$e^{-Ei/kT}/Z$
Z=∑$e^{-Ei/kT}$
Where the sums are over all available states

3. The attempt at a solution
I solved part a by using the first equation and solving for Z. I got

Z=$1+2e^{-ε/kT}$
U=$\frac{2ε}{2+e^{ε/kT}}$

To get the energy per mole as a function of temperature, I simply multiplied by Avagadro's number

$\frac{U}{mol}$=$\frac{2εN_{A}}{2+e^{ε/kT}}$

From here, I get stuck trying to find entropy as a function of T. I'm not quite certain what to do. I've tried S=$\int TdU$ but it gives me a gruesome mess that can't be solved analytically by Mathematica. Any suggestions?

Last edited: Apr 9, 2014
2. Apr 9, 2014