Question

Starting with the result of Problem 3.5, calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature.

Short answer

The required expression is $C_v=\frac{N{\epsilon }^2}{k{T}^2}{e}^{-\left(\frac{\epsilon }{kT}\right)}$.

The graph of the predicted heat capacity as a function of temperature can be sketched as follows:

Step-by-step solution

Given

The equation for Einstein solid at low temperature is calculated as:

$U=N\epsilon e^{-\left(\frac{\epsilon }{kT}\right)}\text{............(1)}$

Here, $N$is number of oscillator, $\epsilon$is the amount of energy quanta, $k$is Boltzmann constant and $T$is temperature.

Calculation of heat capacity

Heat capacity at constant volume is given as:

$C_v={\left(\frac{\partial U}{\partial T}\right)}_{N,V}$

Where, $U$is internal energy.

By substututing the value of $U$in the above equation, we get,

$$\begin{gathered} C_v=\frac{\partial }{\partial T}\left(N\epsilon e^{-\left(\frac{\epsilon }{kT}\right)}\right) \\ {C}_v=\frac{N{\epsilon }^2}{k{T}^2}{e}^{-\left(\frac{\epsilon }{kT}\right)} \end{gathered}$$

Graph of the heat capacity as a function of temperature

Consider the equation which gives the relation of the heat capacity as a function of temperature:

$C_v=\frac{N{\epsilon }^2}{k{T}^2}{e}^{-\left(\frac{\epsilon }{kT}\right)}$

Now, by considering the rest other factors as a constant, heat capacity as a function of temperature can be given as:

$C_v\propto \frac{1}{T^2}{e}^{-\left(\frac{1}{T}\right)}$

Based on the above relation, the graph can be plotted as below:

Final answer

The required expression is derived as $C_v=\frac{N{\epsilon }^2}{k{T}^2}{e}^{-\left(\frac{\epsilon }{kT}\right)}$and the graph showing the heat capacity as a function of temperature can be made as follows: