Starting with the result of Problem 1, calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature. (Note: Measurements of heat capacities of actual solids at low temperatures do not confirm the prediction that you will make in this problem. A more accurate model of solids at low temperatures is presented in Section 7.5.)
Starting with the result of Problem 2, 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).
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.
Step 1 </p>
The value of U is given is
U = N
= The size of the energy unit
Step 2 </p>
To calculate the heat capacity of the einstein solid in the law temperature limit consider the thermodynamic equation.
N and are constants so Cv become
CV = N
CV = N e-
CV = N e-
CV = e-