# Entropy, energy levels and harmonic oscillation.

## Homework Statement

A model to describe the vibrations of atoms in a solid is to assume that the atoms are isotropic harmonic oscillators and that the vibrations are independent of the vibrations of the other atoms. We use this model to describe the entropy and heat capacity of Bohrium (B). The entropy is given as 9.797 J/(mol K) at T = 400K

What is the heat capacity of B at T = 600K?

## Homework Equations

Energy level of a harmonic oscillator: En = hw(1/2 + n) (1 dimension)
Z = the partition function = $$\sum$$e^(-En/k*T) = 1/(2*sinh(hw/k*T))
F = -k*T*ln(Z)

The partition function above is valid for a single atom which is only moving in one dimention but to get the correct parition function for motion in 3D one simply does Zcorrect = Z³ which gives the free energy as

Fcorrect = -3*k*T*ln(Z)

S = -($$\partial$$F/$$\partial$$T)
Cv = T($$\partial$$S/$$\partial$$T)

## The Attempt at a Solution

From the partition function i can derive the function for S in a straight forward way. I am given the entropy at T = 400 and what i need to know to derive the heat capacity Cv from S is the value for hw. But i cant figure out how to get hw from S given the entropy. I have tried solving the function by the Newton Rhapson method using MatLab but it doesnt want to work. So im stuck and im wondering if im even on the right track. Is there a simple way to find the energy levels of a vibration as in this case when given the entropy?

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Ok i think i got a clue (while looking in a book of all places!). What i need is the Einstein temperature of Bohrium and all will be solved nicely. At least i think it will.

I think you can make an approximation sinh(x)=x since x=hw/kT is generally very small for T=400K.

This just reduces to the classical expression 3k.

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I also noticed that it is Boron im working with and not Bohrium as i wrote. I blame the fact that i had to translate the problem to swedish.