Use a computer to sum the exact rotational partition function numerically, and plot the result as a function of$\frac{kT}{\in }$. Keep enough terms in the sum to be confident that the series has converged. Show that the approximation in equation 6.31 is a bit low, and estimate by how much. Explain the discrepancy

We are given the exact rotational partition function.

Use the values of $\frac{kT}{\in }$from zero to five and also the values of j from zero to $20$to determine the exact values of rotational partition function.

The expression for the rotational partition function for this case is,

$Z=\sum _{j=0}^{20}(2j+1)e^{\frac{-j(j+1)\in }{kT}}$

Use the above equation to find the values of the partition function. These values are tabulated as below.

Using the data in the above table, the graph between the rotational partition function and $\frac{kT}{\in }$is as follows,

From the above figure, the area under the smooth curve is

$$\begin{gathered} Z_{rot.exact}=(0.35)\left(1\right) \\ =0.35 \end{gathered}$$

The area under the linear curve (below the smooth curve) of the approximate plot indicates the approximate value of the total rotational partition function. From the above graph, the area under the linear curve is,

$$\begin{gathered} Z_{rot.app}=\frac{1}{2}(0.4)(0.35) \\ =0.07 \end{gathered}$$

From the above result. it is clear that the approximate value of partition function is a bit low compared to the exact value of rotational partition function. Hence, the given statement is proved.

The difference between the exact value of rotational partition function and the approximate value of partition function is,

$$\begin{gathered} Z_{rot.exact}-Z_{rot.app}=0.35-0.07 \\ =0.28 \\ \approx 0.3 \\ \approx \frac{1}{3} \end{gathered}$$

Rewriting the equation for,.

$Z_{rot.app}\approx Z_{rot.exact}-\frac{1}{3}$

Therefore, the approximate value $Z_{rot.app}$appears to be less by $\frac{1}{3}$in the exact value.