23.2: The Ensemble Entropy and the Value of ß

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At equilibrium, the entropy of the \(\hat{N}\)-system ensemble, \(S_{\text{ensemble}}\), must be a maximum. By arguments that parallel those in Chapter 20, \(\hat{W}\) is a maximum for the ensemble population set that characterizes this equilibrium state. Applying the Boltzmann definition to the ensemble, the ensemble entropy is \(S_{\text{ensemble}}=k{ \ln {\hat{W}}_{\text{max}}\ }\). Since all \(\hat{N}\) systems in the ensemble have effectively the same entropy, \(S\), we have \(S_{\text{ensemble}}=\hat{N}S\). When we assume that \({\hat{W}}_{\text{max}}\) occurs for the equilibrium population set, \(\left\{\hat{N}^{\textrm{⦁}}_1,\ {\hat{N}}^{\textrm{⦁}}_2,\dots ,\ {\hat{N}}^{\textrm{⦁}}_i,\dots \right\}\), we have

\[{\hat{W}}_{\text{max}}=\hat{N}!\prod^{\infty }_{i=1}{\frac{\Omega^{\hat{N}^{\textrm{⦁}}_i}_i}{\hat{N}^{\textrm{⦁}}_i!}} \nonumber \]

so that

\[S_{\text{ensemble}}=\hat{N}S=k \ln \hat{N}! +k \sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_i} {\ln \Omega_i} - k \sum^{\infty }_{i=1} { \ln \left(\hat{N}^{\textrm{⦁}}_i!\right) } \nonumber \]

From the Boltzmann distribution function, \({\hat{N}^{\textrm{⦁}}_i}/{\hat{N}}=Z^{-1}\Omega_i{\mathrm{exp} \left(-\beta E_i\right)\ }\), we have

\[{ \ln \Omega_i\ }={ \ln Z\ }+{ \ln {\hat{N}}^{\textrm{⦁}}_i\ }+\beta E_i-{ \ln \hat{N}\ } \nonumber \]

Substituting, and introducing Stirling’s approximation, we find

\[\begin{align*} \hat{N}S &=k\hat{N}{ \ln \hat{N}-k\hat{N}\ } + k\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_i\left({ \ln Z+{ \ln {\hat{N}}^{\textrm{⦁}}_i\ }\ }+\beta E_i-{ \ln \hat{N}\ }\right)}-k\sum^{\infty }_{i=1}{\left({\hat{N}}^{\textrm{⦁}}_i{ \ln {\hat{N}}^{\textrm{⦁}}_i-{\hat{N}}^{\textrm{⦁}}_i\ }\right)} \\[4pt] &=\hat{N}k{ \ln Z\ }+k\beta \sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i} \end{align*} \]

Since \(\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i}\) is the energy of the \(\hat{N}\)-system ensemble and the energy of each system is the same, we have

\[\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i}=E_{\text{ensemble}}=\hat{N}E \nonumber \]

Substituting, we find

\[S=k\beta E+k{ \ln Z\ } \nonumber \]

where \(S\), \(E\), and \(Z\) are the entropy, energy, and partition function for the \(N\)-molecule system. From the fundamental equation, we have

\[{\left(\frac{\partial E}{\partial S}\right)}_V=T \nonumber \]

Differentiating \(S=k\beta E+k{ \ln Z\ }\) with respect to entropy at constant volume, we find

\[1=k\beta {\left(\frac{\partial E}{\partial S}\right)}_V \nonumber \] and it follows that \[\beta =\frac{1}{kT} \nonumber \]

We have, for the \(N\)-molecule system

\[Z=\sum^{\infty }_{i=1}{\Omega_i}{\mathrm{exp} \left(\frac{-E_i}{kT}\right)\ } \nonumber \] (System partition function)

\[{\hat{P}}_i=Z^{-1}\Omega_i{\mathrm{exp} \left(\frac{-E_i}{kT}\right)\ } \nonumber \] (Boltzmann’s equation)

\[S=\frac{E}{T}+k{ \ln Z\ } \nonumber \] (Entropy of the N-molecule system)