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

**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=[itex]e^{-Ei/kT}/Z[/itex]

Z=∑[itex]e^{-Ei/kT}[/itex]

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=[itex]1+2e^{-ε/kT}[/itex]

U=[itex]\frac{2ε}{2+e^{ε/kT}}[/itex]

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

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

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=[itex]\int TdU[/itex] but it gives me a gruesome mess that can’t be solved analytically by Mathematica. Any suggestions?

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

http://ift.tt/1qiSsVh

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