In Exercise 35, a simple two-state system is studied. Assume that the particles are distinguishable. Determine the molar specific heat ${\mathit{C}}_{\mathbf{v}}$of this material and plot it versus T. Explain qualitatively why it should behave as it does.

The molar-specific heat of a crystalline solid can be described as the rate of change of internal vibrational energy of a solid with respect to change in temperature. The formula for molar-specific heat,

${\mathit{C}}_{\mathbf{v}}\mathbf{=}\frac{\mathbf{\partial }{\mathbf{U}}_{\mathbf{\text{mole}}}}{\mathbf{\partial }\mathbf{T}}\mathbf{.}$

The average energy, one mole of the system needs, is-

$\overline{E}=\frac{E_n}{1+e^{E_n/k_{B}T}}$

To be able to find the molar-specific heat, the energy per mole of the system needs to be found first. That is just found by multiplying the average energy of the system by Avogadro's number $N_A$:

role="math" localid="1660143306024" $$\begin{gathered} U_{mol}=N_A\left(\frac{E_n}{1+e^{E_n/k_{B}T}}\right) \\ {U}_{mol}=\frac{N_A{E}_n}{1+e^{E_n/k_{B}T}}···································\left(1\right) \end{gathered}$$

The specific heat is described as -

$C_v=\frac{\partial U_{\text{mole}}}{\partial T}$

Using equation (1) in the formula for specific heat, we get-

role="math" localid="1660143586415" $$\begin{gathered} C_v=\frac{{\displaystyle \partial }\left(\frac{N_A{E}_n}{1+e^{E_n/k_{B}T}}\right)}{{\displaystyle \partial T}} \\ =N_A{E}_n{\left(-1\right)}^2\frac{e^{E_n/k_{B}T}}{{\displaystyle {\left(1+e^{E_n/k_{B}T}\right)}^2}}{\left(\frac{E_n}{k_{B}T}\right)}^2 \\ =N_A{k}_B\frac{e^{E_n/k_{B}T}}{{\left(1+e^{E_n/k_{B}T}\right)}^2}{\left(\frac{E_n}{k_{B}T}\right)}^2 \\ =R\frac{e^{E_n/k_{B}T}}{{\left(1+e^{E_n/k_{B}T}\right)}^2}{\left(\frac{E_n}{k_{B}T}\right)}^2··········································\left(2\right) \end{gathered}$$

Thus, the specific heat of the solid is given as$C_v=R{\left(\frac{E_n}{k_{B}T}\right)}^2\frac{e^{E_n/k_{B}T}}{{\left(1+e^{E_n/k_{B}T}\right)}^2}$

Using equation (2), we can plot a curve between the molar specific heat and absolute temperature as-

From the plot it is clear that there is a bump in the middle, this is formed because for low values of temperature, all the particles will occupy their ground state and $\overline{E}=0$. Therefore, no significant change will occur in internal energy.

For higher values of temperature, the occupancy of the two energy levels will be equal. So, $\overline{E}=\frac{1}{2}{E}_n$and hence no change occurs in this case as well. So, only when $k_{B}T$approaches $E_n$, there will be significant change in energy with increasing absolute temperature.