1.9.6: Entropies- Liquid Mixtures
The chemical potential of liquid component 1 in a binary liquid mixture (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), close to the standard pressure \(\mathrm{p}^{o}\)) is related to the mole fraction \(\mathrm{x}_{1}\) using equation (a).
\[\mu_{1}(\operatorname{mix} ; \mathrm{id})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
But
\[\mathrm{S}_{1}(\operatorname{mix})=-\left[\partial \mu_{1}(\operatorname{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
Then,
\[\mathrm{S}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right) \nonumber\]
Hence the molar entropy of mixing of an ideal binary liquid mixture (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) is given by Equation \ref{d}.
\[\Delta_{\text {mix }} S_{m}(\text { id })=-R \,\left[x_{1} \, \ln \left(x_{1}\right)+x_{2} \, \ln \left(x_{2}\right)\right] \label{d}\]
The chemical potential of component 1 in a real binary liquid mixture (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), near the standard pressure) is given by Equation \ref{e}.
\[\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \label{e}\]
Then
\[\mathrm{S}_{1}(\mathrm{mix})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)-\mathrm{R} \, \ln \left(\mathrm{f}_{1}\right)-\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
\[S_{1}(\operatorname{mix})=S_{1}(\operatorname{mix} ; \text { id })-R \, \ln \left(f_{1}\right)-R \, T \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p} \nonumber \]
Similarly,
\[\mathrm{S}_{2}(\mathrm{mix})=\mathrm{S}_{2}(\mathrm{mix} ; \mathrm{id})-\mathrm{R} \, \ln \left(\mathrm{f}_{2}\right)-\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \nonumber \]
The extent to which the partial molar entropies for each liquid component in a given liquid mixture differs from that in the corresponding ideal mixture depends on the rational activity coefficient and its dependence on temperature. Hence we define excess partial molar entropies for both liquid components.
\[S_{1}^{E}=-R \, \ln \left(f_{1}\right)-R \, T \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p} \nonumber \]
and
\[S_{2}^{E}=-R \, \ln \left(f_{2}\right)-R \, T \,\left[\partial \ln \left(f_{2}\right) / \partial T\right]_{p} \nonumber \]
For the binary mixture,
\[\begin{aligned}
S_{m}^{E}=-R\left\{x_{1} \, \ln \left(f_{1}\right)\right.&+x_{1} \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p} \\
&\left.+x_{2} \, \ln \left(f_{2}\right)+x_{2} \,\left[\partial \ln \left(f_{2}\right) / \partial T\right]_{p}\right\}
\end{aligned} \nonumber \]