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Q. 3.8

Expert-verified
Found in: Page 93

### An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

# 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.

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:

See the step by step solution

## Step 1: 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.

## Step 2: 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,

${C}_{v}=\frac{\partial }{\partial T}\left(N\epsilon {e}^{-\left(\frac{\epsilon }{kT}\right)}\right)\phantom{\rule{0ex}{0ex}}{C}_{v}=\frac{N{\epsilon }^{2}}{k{T}^{2}}{e}^{-\left(\frac{\epsilon }{kT}\right)}$

## Step 3: 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:

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:

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