Show that the coefficient \(\bar{b}_{2}\) for a quantum-mechanical Boltzmannian gas composed of "spinless" particles satisfies the following relations:

\[\begin{aligned}\bar{b}_{2} & =\lim _{J \rightarrow \infty}\left\{\frac{1}{(2 J+1)^{2}} \hbar_{2}^{S}(J)\right\}=\lim _{J \rightarrow \infty}\left\{\frac{1}{(2 J+1)^{2}} \hbar_{2}^{A}(J)\right\} \\& =\frac{1}{2}\left\{\hbar_{2}^{S}(0)+\hbar_{2}^{A}(0)\right\}\end{aligned}\]

Obtain the value of \(\hbar_{2}\), to the fifth order in \((D / \lambda)\), by using the Beth-Uhlenbeck expressions in equations (10.5.36) and (10.5.37), and compare your result with the classical value of \(\bar{b}_{2}\), namely, \(-(2 \pi / 3)(D / \lambda)^{3}\).