Calculation of vibrational heat capacity

The distribution of the oscillators is governed by an exponential law (for proof see Section 29.1) 

The average energy is obtained by multiplying the energy of each level by the number in that level, summing over all the levels, and dividing by the total number of molecules. 

Using this expression in Eq. (4.84) for e , we obtain, after setting a = IjkT, 

The average energy is made up of the zero point energy jhv, which is the lowest energy possible for the quantum oscillator, plus a term that depends on temperature. 

At very low temperature, hv/kT 1, hence exp (hv/kT) 1 so that the second term is very small, and 

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