The Debye temperature of copper is 45K .

(a) Estimate its molar heat capacity at 100 K using the plot in Figure 9.33(b) .

(b) Determine its corresponding specific heat and compare it with the experimental value of $\mathbf{0}\mathbf{.}\mathbf{254}\mathbf{\text{J/g·K}}$.

The Debye temperature of copper is 45K.

The temperature considered is 100K.

In statistical mechanics, we consider the mechanical vibrations in a solid as free bosons having zero spin. these quantum of vibrations are called Phonons. They can travel from one point to another inside the solid.

The molar heat capacity is given by-

${\mathit{C}}_{\mathbf{V}}\mathbf{=}\frac{\mathbf{12}{\mathbf{\pi }}^{\mathbf{4}}}{\mathbf{5}}\mathit{R}{\left(\frac{T}{T_D}\right)}^{\mathbf{3}}\mathbf{}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\left(1\right)$

Where, R is ideal gas constant $\left(R=8.314\raisebox{1ex}{$J$}\!\left/ \!\raisebox{-1ex}{$mol$}\right.\right)$, Tis temperature and ${\mathbf{T}}_{\mathbf{D}}$is Debye temperature.

The relationship between the molar heat capacity ${\mathbf{\text{C}}}_{\mathbf{V}}$, and specific heat ${\mathbf{\text{c}}}_{\mathbf{\text{m}}}$ is given by,

${\mathbf{\text{c}}}_{\mathbf{\text{m}}}\mathbf{=}\frac{{\mathbf{\text{C}}}_{\mathbf{\text{V}}}}{\mathbf{\text{M}}}\mathbf{}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\left(2\right)$

Where, M is molar mass of the material, molar mass of copper is$\mathbf{63}\mathbf{.}\mathbf{546}\mathbf{}\mathbf{\text{g}}\mathbf{/}\mathbf{\text{mol}}$

a)

The Debye temperature ( $T_D$) of copper is 345 K.

Using equation (1), for the Debye temperature 345K , we get-

$$\begin{gathered} C_V=\frac{12{\pi }^4}{5}\left(8.314J/mol\right){\left(\frac{100\text{K}}{345\text{K}}\right)}^3 \\ =\left(1939.72\times {(0.29)}^3\right)J/mol·K \end{gathered}$$

The molar heat capacity of copper is $47.30J/mol·K$.

b)
From the solution of part (a), The molar heat capacity of copper at 100K is known to be $47.30J/mol·K$.

Substituting the numerical values in equation (2)

localid="1660140897696" $$\begin{gathered} {\text{c}}_{\text{m}}=\frac{{\displaystyle 47.23J/mol·K}}{{\displaystyle 63.546\times {10}^{-3}kg/mol}} \\ =743.23J/kg·K \\ =\text{0.743}J/g·K \end{gathered}$$

The specific heat capacity of copper at is found to be$=\text{0.743}J/g·K$.

The experimentally calculated value of specific heat is $=\text{0.254}J/g·K$, which is less than the calculated value