23.2: The Ensemble Entropy and the Value of ß
- Page ID
- 151971
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\[\begin{align*} \hat{N}S &=k\hat{N}{ \ln \hat{N}-k\hat{N}\ } + k\sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_i\left({ \ln Z+{ \ln {\hat{N}}^{\textrm{⦁}}_i\ }\ }+\beta E_i-{ \ln \hat{N}\ }\right)}-k\sum^{\infty }_{i=1}{\left({\hat{N}}^{\textrm{⦁}}_i{ \ln {\hat{N}}^{\textrm{⦁}}_i-{\hat{N}}^{\textrm{⦁}}_i\ }\right)} \\[4pt] &=\hat{N}k{ \ln Z\ }+k\beta \sum^{\infty }_{i=1}{\hat{N}^{\textrm{⦁}}_iE_i} \end{align*} \nonumber \]
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)