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

[NobodyMinus]

Homework Statement

5. The nuclei of atoms in a certain crystalline solid have spin one. Each nucleus can be in anyone 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

Homework Equations

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

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?

 

[TSny]

Your energy calculation looks correct.

For getting the entropy from Z, take a look at

http://employees.csbsju.edu/cgearhart/Courses/Thermal/Lecture/Entropy_Canonical.pdf

or

http://en.wikipedia.org/wiki/Partit...echanics)#Relation_to_thermodynamic_variables

Or search for "entropy from partition function".