1.12.20: Expansions- Isentropic- Liquid Mixtures
A given binary liquid mixture has mole fraction \(x_{1}\left[=1-x_{2}\right]\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The system is at equilibrium at a minimum in Gibbs energy where the affinity for spontaneous change is zero. The molar volume and molar entropy of the mixtures are given by equations (a) and (b).
\[\mathrm{V}_{\mathrm{m}}=\mathrm{V}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}\right] \nonumber \]
\[\mathrm{S}_{\mathrm{m}}=\mathrm{S}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}\right] \nonumber \]
These two equations describe the properties of the system in the \(\mathrm{T}-\mathrm{p}\)-composition domain; i.e. a Gibbsian description. We consider two dependences of the volume on temperature under the constraint that the affinity for spontaneous change remains at zero; i.e equilibrium expansions.The isobaric expansion is defined by equation (c).
\[\mathrm{E}_{\mathrm{p}}(\operatorname{mix})=\left(\frac{\partial \mathrm{V}(\operatorname{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
The isentropic expansion is defined by equation (d)
\[\mathrm{E}_{\mathrm{S}}(\operatorname{mix})=\left(\frac{\partial \mathrm{V}(\operatorname{mix})}{\partial \mathrm{T}}\right)_{\mathrm{S}} \nonumber \]
In the latter case the system tracks a path with increase in temperature where the affinity for spontaneous change remains at zero and the entropy remains the same at that defined by equation (b). [\(\mathrm{NB} \mathrm{~E}_{p}(\operatorname{mix})\) and \(E_{S}(\operatorname{mix})\) as defined by equations (c) and (d) are extensive properties.] The two expansions are related through the (equilibrium) isobaric heat capacity \(\mathrm{C}_{\mathrm{p}} (\operatorname{mix})\)and the (equilibrium) isothermal compression \(\mathrm{K}_{\mathrm{T}}(\operatorname{mix})\) [1]. Thus
\[\mathrm{E}_{\mathrm{S}}(\operatorname{mix})=\mathrm{E}_{\mathrm{p}}(\operatorname{mix})-\frac{\mathrm{C}_{\mathrm{p}}(\operatorname{mix}) \, \mathrm{K}_{\mathrm{T}}(\operatorname{mix})}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}(\operatorname{mix})} \nonumber \]
In the context of the property \(\mathrm{E}_{\mathrm{p}(\operatorname{mix})\), the entropy of the system changes with an increase in temperature at constant pressure. But by definition the entropy does not change for an isentropic expansion, \(\mathrm{E}_{\mathrm{S}(\operatorname{mix})\).
For a binary liquid mixture having ideal thermodynamic properties,
\[E_{S}(\operatorname{mix} ; i d)=E_{p}(\operatorname{mix} ; i d)-\frac{C_{p}(\operatorname{mix} ; i d) \, K_{T}(\operatorname{mix} ; i d)}{T \, E_{p}(\operatorname{mix} ; i d)} \nonumber \]
In this comparison we note that \(\mathrm{E}_{\mathrm{p}(\operatorname{mix})\) and \(\mathrm{E}_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id})\) refer to the same pressure but the entropies referred to in \(\mathrm{E}_{\mathrm{S}}(\operatorname{mix})\) and \(\mathrm{E}_{\mathrm{S}}(\operatorname{mix} ; \mathrm{id})\) are not the same. The same contrast arises when we set out the two equations describing expansions of the pure liquids.
\[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)-\frac{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)}{\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)} \nonumber \]
\[\mathrm{E}_{\mathrm{S} 2}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)-\frac{\mathrm{C}_{\mathrm{p} 2}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}{\mathrm{T} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)} \nonumber \]
The subject is complicated by the galaxy of entropies implied by the phrase ‘at constant entropy’.
Footnote
[1] Using a calculus operation,
\[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{S}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{S}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
We note two Maxwell equations. From \(\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}], \quad \partial^{2} \mathrm{U} / \partial \mathrm{S} \, \partial \mathrm{V}=\partial^{2} \mathrm{U} / \partial \mathrm{V} \, \partial \mathrm{S}\) Then
\[\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{p}}{\partial \mathrm{S}}\right)_{\mathrm{V}} \nonumber \]
We invert the latter equation. Hence
\[\begin{aligned}
\mathrm{E}_{\mathrm{S}}=&\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \\
&=-\mathrm{K}_{\mathrm{S}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=-\mathrm{K}_{\mathrm{s}} \, \mathrm{C}_{\mathrm{p}} / \mathrm{T} \, \mathrm{E}_{\mathrm{p}}
\end{aligned} \nonumber \]
Similarly
\[\partial^{2} \mathrm{G} / \partial \mathrm{T} \, \partial \mathrm{p}=\partial^{2} \mathrm{G} / \partial \mathrm{p} \, \partial \mathrm{T} \nonumber \]
Then,
\[E_{p}=\left(\frac{\partial V}{\partial T}\right)_{p}=-\left(\frac{\partial S}{\partial p}\right)_{T} \nonumber \]
Also at equilibrium, \(\mathrm{S}=-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\)
But \(\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\). Then \(\mathrm{H}=\mathrm{G}-\mathrm{T} \,\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\)
\[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}=\frac{\partial \mathrm{G}}{\partial \mathrm{T}}-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)-\frac{\partial \mathrm{G}}{\partial \mathrm{T}} \nonumber \]
Further,
\[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
Based on equation (a), \(\mathrm{E}_{\mathrm{S}}=\mathrm{E}_{\mathrm{p}}-\mathrm{C}_{\mathrm{p}} \, \mathrm{K}_{\mathrm{T}} / \mathrm{T} \, \mathrm{E}_{\mathrm{p}}\)