24.14: Problems

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1. The partition function, \(Z\), for a system of \(N\), distinguishable, non-interacting molecules is \(Z=z^N\), where \(z\) is the molecular partition function, \(z=\sum{g_i}{\mathrm{exp} \left({-{\epsilon }_i}/{kT}\right)\ }\), and the \({\epsilon }_i\) and \(g_i\) are the energy levels available to the molecule and their degeneracies. Show that the thermodynamic functions for the \(N\)-molecule system depend on the molecular partition function as follows:

(a) \(E=NkT^2{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V\)

(b) \(S=NkT{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V+Nk{ \ln z\ }\)

(d) \(P_{\mathrm{system}}=NkT{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T\)

(e) \(H=NkT^2{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V+NkTV{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T\)

(f) \(G=-NkT{ \ln z\ }+NkTV{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T\)

2. When the number of available quantum states is much larger than the number of molecules, the partition function, \(Z\), for a system of \(N\), indistinguishable, non-interacting molecules is \(Z={z^N}/{N!}\), where \(z\) is the molecular partition function, \(z=\sum{g_i}{\mathrm{exp} \left({-{\epsilon }_i}/{kT}\right)\ }\), and the \({\epsilon }_i\) and \(g_i\) are the energy levels available to the molecule and their degeneracies. Show that the thermodynamic functions for the N-molecule system depend on the molecular partition function as follows:

(a) \(E=NkT^2{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V\)

(b) \(S=Nk\left[T{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V+{ \ln \frac{z}{N}\ }+1\right]\)

(d) \(P_{\mathrm{system}}=NkT{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T\)

(e) \(H=NkT^2{\left(\frac{\partial { \ln z\ }}{\partial T}\right)}_V+NkTV{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T\)

(f) \(G=-NkT\left[{\mathrm{1+ln} \frac{z}{N}\ }+V{\left(\frac{\partial { \ln z\ }}{\partial V}\right)}_T\right]\)

3. The molecular partition function for the translational motion of an ideal gas is

\[z_t= \left(\frac{2\pi mkT}{h^2}\right)^{3/2}V \nonumber \]

The partition function for a gas of \(N\), monatomic, ideal-gas molecules is \(Z={z^N_t}/{N!}\). Show that the thermodynamic functions are as follows:

(a) \(E=\frac{3}{2}NkT\)

(b) \(S=Nk\left[\frac{5}{2}+{ \ln \frac{z}{N}\ }\right]\)

(d) \(P_{\mathrm{system}}=\frac{NkT}{V}\)

(e) \(H=\frac{5}{2}NkT\)

(f) \(G=-NkT{ \ln \frac{z}{N}\ }\)

4. Find \(E\), \(S\), \(A\), \(H\), and \(G\) for one mole of Xenon at \(300\) K and \(1\) bar.

Notes

\({}^{1}\) Data from the Handbook of Chemistry and Physics, 79\({}^{th}\) Ed., David R. Linde, Ed., CRC Press, New York, 1998.