Chapter 2: Problem 1
We have shown that the entropy for Bose-Einstein and Fermi-Dirac statistics is given by $$ S(E, V, N)=k(\beta E+\alpha N) \mp k \sum_{j} g_{j} \ln \left(1 \mp e^{-\alpha} e^{-\beta \varepsilon_{j}}\right), $$ where \(N=\sum_{j} N_{j}\) and \(E=\sum_{j} N_{j} \varepsilon_{j}\). Similarly, from classical thermodynamics, $$ d S(E, V, N)=\frac{1}{T} d E+\frac{P}{T} d V-\frac{\mu}{T} d N $$ for a single-component system. a. Prove that \(\beta=1 / k T\). b. Prove that \(\alpha=-\mu / k T\). c. Show that the pressure is given by $$ P=-\sum_{j} N_{j}\left(\partial \varepsilon_{j} / \partial V\right)_{E, N} . $$ d. We will soon demonstrate that the internal energy is independent of volume whereas the translational energy \(\varepsilon_{j, t r}=C_{j} V^{-2 / 3}\), where \(C_{j}\) is a constant for each translational energy level. Utilizing this information, show that $$ P V=\frac{2}{3} E_{t r}, $$ where \(E_{t r}\) is the translational portion of the total energy (external plus internal) for the particle assembly. e. Is this result limited to the dilute limit? Does it assume independent particles? Why should the pressure be related only to the translational energy? Explain.
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