From equation (9-51), show that the specific heat (per mole) of a crystalline solid varies as ${\mathit{T}}^{\mathbf{3}}$ for $\mathit{T}\mathbf{\ll }{\mathit{T}}_{\mathbf{D}}$.

If the internal structure of a solid has the regular arrangement of atoms such that a fundamental structure of the atom, known as a cell, is repeated again and again. Then the solids are known as crystalline solids. They have a sharp melting point and the intermolecular forces between atoms are also uniform.

On the other hand, the amorphous solids don’t have any fixed internal structure. They generally form fragments, when broken. An example is a glass.

The internal vibrational energy of a solid per mole is given as-

$U_{permol}=9R\frac{T^4}{T_D^3}{\int }_0^{\raisebox{1ex}{$T_D$}\!\left/ \!\raisebox{-1ex}{$T$}\right.}\frac{x^3}{{\displaystyle e^x-1}}x$

Here, T is the temperature of the solid, is the Debye temperature and R is the gas constant$T\ll T_D$

The integral approaches zero as x becomes very large and for the condition the upper limit becomes very large and approach infinity.

So the integral in the above equation becomes-

$$\begin{gathered} {\int }_0^{\raisebox{1ex}{$T_D$}\!\left/ \!\raisebox{-1ex}{$T$}\right.}\frac{{\displaystyle x^3}}{{\displaystyle e^x-1}}dx={\int }_0^{\infty }\frac{{\displaystyle x^3}}{{\displaystyle e^x-1}}dx,\left(T\ll T_D\right) \\ =\frac{{\pi }^4}{15} \end{gathered}$$

Thus, we can write equation (1) as-

role="math" localid="1660142189309" $$\begin{gathered} U_{permol}=9R\frac{T^4}{T_D^3}\frac{{\pi }^4}{15} \\ {U}_{permol}\propto \frac{T^4}{T_D^3} \end{gathered}$$

Further we can write,

$U_{permol}\propto \frac{T^4}{T_D^3}$

The specific heat per mole is just the derivative of internal vibrational energy, with respect to T. the specific heat will be proportional to the third power of temperature.

Hence, the specific heat per mole of a solid varies as $T^3$.