91Ó°ÊÓ

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

$C=\mathrm{γ}T+\mathrm{Î\pm }{T}^3$

$\text{Here,}\mathrm{γ}=\frac{{\mathrm{Ï€}}^{2}N{k}_B^2}{2{\mathrm{Î\mu }}_F},\mathrm{Î\pm }=\frac{12N{\mathrm{Ï€}}^4{k}_B}{5T_D^3}\text{and}T\text{is the temperature.}$

$\text{Firstly, rearranging the equation}C=\mathrm{γ}T+\mathrm{Î\pm }{T}^3\text{for}\frac{C}{T}\text{.}$

$\frac{C}{T}=\mathrm{γ}+\mathrm{Î\pm }{T}^2$

in the above equation $\mathrm{Î\pm }\text{is the slope on}\frac{C}{T}\text{versus}{T}^2\text{plot and}\mathrm{γ}\text{is the intercept.}$

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

role="math" localid="1647622591195" $\mathrm{γ}=\frac{{\mathrm{Ï€}}^{2}N{k}_B^2}{2{\mathrm{Î\mu }}_F}$

Here, $N$is the number of the conduction electrons per mole of the metal, $k_B$is the Boltzmann's constant, and ${\mathrm{Î\mu }}_F$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 $7.28$.

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

$\mathrm{Î\pm }=\frac{12N{\mathrm{Ï€}}^4{k}_B}{5T_D^3}$

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

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

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

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.