Q. 7.59 - An Introduction to Thermal Physics

Question

Explain in some detail why the three graphs in Figure $7.28$ all intercept the vertical axis in about the same place, whereas their slopes differ considerably.

Step-by-Step Solution

Answer

The conduction electrons in copper, silver, and gold is one. The fermi energy for these elements is approximately equal. Hence, the intercept for copper, silver, and gold is equal on vertical axis in the figure 7.28.

Basically , the slopes of the graphs depend on the speed of sound $C_S$ in each material of the three metals. The metal copper has the largest value $C_s$, and hence largest Debye temperature, and smallest slope. On the other hand, Gold has the smallest value of $C_S$ and hence the largest slope. Silver is somewhere in between copper and the gold metal slope.

So,the slope of the graph between $\frac{C}{T}$ and $T^2$ in the figure $7.28$for copper, silver, and gold is not same.

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

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

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

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

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

Step 3. finding the intercept value for copper, silver, gold.

$\text{ The intercept of the graph between }\frac{C}{T}\text{ and }{T}^2\text{ in the figure }7.28\text{ for copper, silver, and gold is }$

$\gamma =\frac{{\pi }^{2}N{k}_B^2}{2{\epsilon }_F}$

Here, $N$ is the number of the conduction electrons per mole of the metal, $k_B$ is the Boltzmann's constant, and ${\epsilon }_F$is the fermi energy.

As the intercept in the graph between$\frac{C}{T}$ and $T^2$ is directly proportional to $N$ and indirectly proportional to the fermi energy.

So, the conduction electrons in copper, silver, and gold is one. The fermi energy for these elements is approximately equal . Hence, the intercept for copper, silver, and gold is equal on vertical axis in the figure $7.28$.

Step 4. finding the slope relation for copper, silver, gold.

$\text{ The slope of the graph between }\frac{C}{T}\text{ and }{T}^2\text{ in the figure }7.28\text{ for copper, silver, and gold is as }$

$\alpha =\frac{12N{\pi }^4{k}_B}{5T_D^3}$

$\text{ Here, }{T}_D\text{ is the Debye temperature. }$

So, the slope is indirectly proportional to the cube root of the Debye temperature. 

$T_{\mathrm{D}}=\frac{h{c}_s}{2k_B}{\left(\frac{6N}{\pi V}\right)}^{1/3}$

Here,$h$ is the Planck's constant, $c_s$ is the speed of the sound in the liquid, $N$ is the Avogadro number, $V$ is the volume, and $k$ is the Boltzmann's constant.

Hence, we can say that the slopes of the graphs depend on the speed of sound$C_s$ in each material of the three metals. The metal copper has the largest value $C_S$, and so largest Debye temperature, and smallest slope.

Similarly, Gold has the smallest value of $C_S$ and hence the largest slope. Silver is somewhere in between copper and the gold metal slope.

So, basically the slope of the graph between $\frac{C}{T}$ and $T^2$ in the figure $7.28$ for copper, silver, and gold is not same.