Starting with the result of Problem, find a formula for the temperature of an Einstein solid in the limit q ≪ N. Solve for the energy as a function of temperature to obtain U = Nϵe−ϵ/kT (where ϵ is the size of an energy unit).

Problem:

Use the methods of this section to derive a formula, similar to below equation, for the multiplicity of an Einstein solid in the “low-temperature” limit, q≪ N.

Equation:

ANSWER:

Step 1:-

The number of microstates is given as,

-----------------------(1)

The total energy of the system is,

----------------------------(2)

--------------------------(3)

Step 2:-

So, now the number of microstates of the system we can write in terms of the internal energy.

The entropy of the system is,

--------------------(4)