Derive Eqs. (16.34) and (16.35). [begin{equation*} S=N kleft(ln frac{Q}{N}+1 ight)+N k Tleft(frac{partial ln Q}{partial T} ight)_{V} tag{16.34}

Derive Eqs. (16.34) and (16.35).

\[\begin{equation*}
S=N k\left(\ln \frac{Q}{N}+1\right)+N k T\left(\frac{\partial \ln Q}{\partial T}\right)_{V} \tag{16.34}
\end{equation*}\]

\[\begin{align*}
Q= & \sum_{i} \sum_{J} \sum_{n} \sum_{l} g_{i} g_{J} g_{n} g_{l} \exp \left[-\frac{1}{k T}\left(\varepsilon_{i_{\text {trans }}}+\varepsilon_{J_{\text {rot }}}+\varepsilon_{n_{\text {vib }}}+\varepsilon_{l_{\mathrm{el}}}\right)\right] \\
Q= & {\left[\sum_{i} g_{i} \exp \left(-\frac{\varepsilon_{i_{\text {trans }}}}{k T}\right)\right]\left[\sum_{J} g_{J} \exp \left(-\frac{\varepsilon_{J_{\text {rot }}}}{k T}\right)\right] } \\
& \times\left[\sum_{n} g_{n} \exp \left(-\frac{\varepsilon_{n_{\text {vib }}}}{k T}\right)\right]\left[\sum_{l} g_{l} \exp \left(-\frac{\varepsilon_{l_{\mathrm{el}}}}{\mathrm{kT}}\right)\right] \tag{16.36}
\end{align*}\]