24.14: Problems
- Page ID
- 152812
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\[z_t= \left(\frac{2\pi mkT}{h^2}\right)^{3/2}V \nonumber \]
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) \(S=Nk\left[\frac{5}{2}+{ \ln \frac{z}{N}\ }\right]\)
(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.