Q. 7.6 - An Introduction to Thermal Physics

Question

Sketch the heat capacity of copper as a function of temperature from $0$ to $5 \mathrm{K}$, showing the contributions of lattice vibrations and conduction electrons separately. At what temperature are these two contributions equal?

Step-by-Step Solution

Answer

The temperature at which both the contributions are equal is $3.7 \mathrm{K}$

Step 1. Given information

The total heat capacity at low temperature is equal to the sum of the electronic heat capacity lattice vibrational heat capacity.

$C=\gamma T+\alpha T^3$

Step 2. Putting the value of γ   a n d   α in above equation we get

$\text{ Here, }\gamma =\frac{{\pi }^{2}N{k}_B^2}{2{\epsilon }_F},\alpha =\frac{12N{\pi }^4{k}_B}{5T_D^3}\text{ and }T\text{ is the temperature. }$

At low temperature, the electronic contribution to the heat capacity is directly proportional to the temperature.

$C_{\text{electronic }}=\gamma T$

The contribution of lattice vibrations to the heat capacity has a cubic dependence on temperature at the lower temperature.

$C_{\text{vibration }}=\alpha T^3$

Step 3. Now, solving for the value of γ   a n d   α .

$\text{Firstly, rearranging the equation }C=\gamma T+\alpha T^3\text{ for }\frac{C}{T}\text{. }$

$\frac{C}{T}=\gamma +\alpha T^2$

$\text{ Here, }\alpha \text{ is the slope on }\frac{C}{T}\text{ versus }{T}^2\text{ plot and }\gamma \text{ is the intercept. }$

$\text{ The slope of the graph between }\frac{C}{T}\text{ versus }{T}^2\text{ of the Copper is as follows: }$

$\alpha =\frac{0.9 \mathrm{mJ}/{\mathrm{K}}^2}{18{ \mathrm{K}}^2}$

$=5\times {10}^{-5} \mathrm{J}/{\mathrm{K}}^4$

$\text{ The intercept }\gamma \text{ for the graph between }\frac{C}{T}\text{ versus }{T}^2\text{ of the Copper is as follows: }$

$\gamma =0.7 \mathrm{mJ}/{\mathrm{K}}^2$

Step 4. Calculating the heat capabilities.

The temperature at which the electronic and the lattice vibration contributions of the heat capacities can be calculated by equating the electronic contribution of the heat capacity to the lattice vibration heat capacity.

$C_{\text{electronic }}=C_{\text{vibration }}$

$\text{ Substitute }\gamma T\text{ for }{C}_{\text{electronic }}\text{ and }\alpha T^3\text{ for }{C}_{\text{vibration }}\text{. }$

$\gamma T=\alpha T^3$

$T^2=\frac{\gamma }{\alpha }$

$T=\sqrt{\frac{\gamma }{\alpha }}$

$\text{ Substitute }0.7 \mathrm{mJ}/{\mathrm{K}}^2\text{ for }\gamma \text{ and }5\times {10}^{-5} \mathrm{J}/{\mathrm{K}}^4\text{ for }\alpha \text{. }$

$T=\sqrt{\frac{0.7 \mathrm{mJ}/{\mathrm{K}}^2\left(\frac{{10}^{-3} \mathrm{J}}{1 \mathrm{mJ}}\right)}{5\times {10}^{-5} \mathrm{J}/{\mathrm{K}}^4}}$

$=3.7 \mathrm{K}$

$\text{ At this temperature, both heat capacities are as follows: }$

$C_{\text{electronic }}=\gamma T$

$\text{ Substitute }0.7 \mathrm{mJ}/{\mathrm{K}}^2\text{ for }\gamma \text{ and }3.7 \mathrm{K}\text{ for }T$

$C_{\text{electronic }}=\left(\left(0.7 \mathrm{mJ}/{\mathrm{K}}^2\right)\left(\frac{{10}^{-3} \mathrm{J}}{1 \mathrm{mJ}}\right)\right)(3.7 \mathrm{K})$

$=0.0026 \mathrm{J}/\mathrm{K}$.

Step 5. Using the table that shows the data for the temperature and the electronic heat capacity lattice vibrational heat capacity for the copper.

The table we have,

$\begin{array}{|lll|}\hline T(\text{ in K) }& C_{\text{clectronic }}=\gamma T& C_{\text{vibation }}=\alpha T^3\\ 0& 0& 0\\ 1& 0.0007& 0.00005\\ 2& 0.0014& 0.0004\\ 3& 0.0021& 0.00135\\ 4& 0.0028& 0.0032\\ 5& 0.0035& 0.00625\\ \hline\end{array}$

Step 6. Plotted the graph between C V and temperature for the lattice vibration and electron contributions.

So, the required plot we have is shown below