Chapter 2: Problem 5
Classical Maxwell-Boltzmann statistics considers particles to be distinguishable with no limit on the number of particles in each energy state. A physical example is a solid composed of localized atoms at distinguishable lattice sites. The thermodynamic probability in this case is given by $$ W_{M B}=N ! \prod_{j} \frac{g_{j}^{N_{i}}}{N_{j} !} $$ where \(N_{j}\) is the number of particles and \(g_{j}\) is the degeneracy of the \(j\) th energy level. a. Using the methods of statistical thermodynamics, show that the equilibrium particle distribution is $$ N_{j}=g_{j} e^{-\alpha} e^{-\beta \varepsilon_{j}} \text {. } $$ b. Defining the molecular partition function \(Z=\sum_{j} g_{j} e^{-\beta \varepsilon_{j}}\), show that $$ S=k(\beta E+N \ln Z) . $$ c. Using classical thermodynamics, verify that \(\beta=1 / k T\). Hence, the entropy for classical Maxwell-Boltzmann statistics becomes $$ S=\frac{U}{T}+k N \ln Z \text {. } $$ Compare this expression with that for corrected Maxwell-Boltzmann statistics. Explain the difference. d. Show that the probability of a particle being in the \(i\) th energy state is given by $$ P_{i}=\frac{N_{i}}{N}=\frac{e^{-\varepsilon_{i} / k T}}{Z}, $$ where the partition function \(Z=\sum_{i} e^{-\varepsilon_{i} / k T}\). e. Demonstrate that the entropy can be directly related to the probabilities \(P_{i}\) of the various energy states accessible to the system, i.e., $$ S=-k N \sum_{i} P_{i} \ln P_{i} . $$ Discuss the significance of this result.
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