We have to show that (a) the molar internal energy of a solid can be written, according to Einstein’s theory of heat capacities, as

in which , where is the Einstein temperature . (b) We have to verify that approaches its classical value of

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(a)

According to the Einstein’s theory of atomic oscillations, an oscillator with characteristic frequency at temperature T will have an average energy

As a molecule can oscillate in three independent directions, the total vibrational energy of a mole, , of molecules at temperature T will therefore be

We have used that the gas constant R and Avogadro number are related through the Boltzmann constant k as .

Einstein temperature is defined as

Therefore,

where we have substituted x for .

(b)

As