# 24.14: Problems

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

(c) $$A=-NkT{ \ln z\ }$$

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

(c) $$A=-NkT\left[{\mathrm{1+ln} \frac{z}{N}\ }\right]$$

(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

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

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

(c) $$A=-NkT\left[1+{ \ln \frac{z}{N}\ }\right]$$

(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.