# 5.2: The Ensemble Entropy and the Value of ß

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

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

so that

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

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

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

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

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

Substituting, we find

$S=k\beta E+k{ \ln Z\ }$

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

${\left(\frac{\partial E}{\partial S}\right)}_V=T$

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

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

We have, for the $$N$$-molecule system

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

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

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