91Ó°ÊÓ

We are given that in the low-temperature limit $(kT<<∈)$, each term in the rotational partition function is much smaller than the one before.

Expanding the partition function by neglecting all higher order terms in the low temperature limit $(kT<<∈)$.

The average energy of the system is given as follows:

$$\begin{gathered} {\overline{E}}_{rot}=-\frac{1}{Z_{rot}}\frac{∂Z_{rot}}{∂\mathrm{β}} \\ =-\frac{1}{1+3e^{-2\mathrm{β}∈}}\frac{∂}{∂\mathrm{β}}(1+3e^{-2\mathrm{β}∈}) \\ =-\frac{1}{1+3e^{-2\mathrm{β}∈}}\left(3e^{-2\mathrm{β}∈}\right(-2∈\left)\right) \\ =\frac{6∈e^{-2\mathrm{β}∈}}{1+3e^{-2\mathrm{β}∈}} \end{gathered}$$

Neglecting the term $3e^{-2\mathrm{β}∈}$in the denominator of $\frac{6∈e^{-2\mathrm{β}∈}}{1+3e^{-2\mathrm{β}∈}}$,

Since the function $3e^{-2\mathrm{β}∈}$approaches to zero as $\mathrm{β}→∞$because $e^{-∞}=0$.

Therefore, the average energy of the system is $6∈e^{-2\mathrm{β}∈}$.

The specific heat capacity of the system is given by,

Therefore, the heat capacity of the system in the low temperature limit is $C=3k{\left(\frac{2∈}{kT}\right)}^2{e}^{-2∈/kT}$.

In the low temperature limit, the heat capacity of the system decreases exponentially to zero.

The low and high temperature limit of $\frac{C}{k}$are plotted against the dimensionless parameter $\frac{kT}{∈}$.

The graph is as follows,