1.12.21: Expansions- Solutions- Partial Molar Isobaric and Isentropic
The starting point is equation (a).
\[\mathrm{E}_{\mathrm{p}}=\mathrm{E}_{\mathrm{S}}+\frac{\mathrm{K}_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}} \nonumber \]
We differentiate this equation with respect to the amount of solute \(\mathrm{n}_{j}\) at fixed \(\mathrm{T}\), \(\mathrm{p}\) and amount of solvent \(\mathrm{n}_{1}\).
\[\mathrm{E}_{\mathrm{pj}}=\mathrm{E}_{\mathrm{Sj}}+\frac{1}{\mathrm{~T}} \,\left[\frac{\mathrm{K}_{\mathrm{T}}}{\mathrm{E}_{\mathrm{p}}} \, \mathrm{C}_{\mathrm{pj}}+\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{E}_{\mathrm{p}}} \, \mathrm{K}_{\mathrm{Tj}}-\frac{\mathrm{K}_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\left(\mathrm{E}_{\mathrm{p}}\right)^{2}} \, \mathrm{E}_{\mathrm{pj}}\right] \nonumber \]
Or,
\[\mathrm{E}_{\mathrm{pj}}=\mathrm{E}_{\mathrm{Sj}}+\frac{1}{\mathrm{~T}} \, \frac{\mathrm{K}_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{E}_{\mathrm{p}}} \,\left[\frac{\mathrm{C}_{\mathrm{p} j}}{\mathrm{C}_{\mathrm{p}}}+\frac{\mathrm{K}_{\mathrm{T} \mathrm{J}}}{\mathrm{K}_{\mathrm{T}}}-\frac{\mathrm{E}_{\mathrm{p} j}}{\mathrm{E}_{\mathrm{p}}}\right] \nonumber \]
We convert to volume intensive variables.
\[\mathrm{E}_{\mathrm{pj}}=\mathrm{E}_{\mathrm{S} \mathrm{j}}+\frac{1}{\mathrm{~T}} \, \frac{\kappa_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\alpha_{\mathrm{p}} \, \mathrm{V}} \,\left[\frac{\mathrm{V} \, \mathrm{C}_{\mathrm{pj}}}{\mathrm{C}_{\mathrm{p}}}+\frac{\mathrm{V} \, \mathrm{K}_{\mathrm{T} j}}{\mathrm{~K}_{\mathrm{T}}}-\frac{\mathrm{V} \, \mathrm{E}_{\mathrm{p} j}}{\mathrm{E}_{\mathrm{p}}}\right] \nonumber \]
Or,
\[E_{p j}=E_{S j}+\frac{1}{T} \, \frac{K_{T} \, \sigma}{\alpha_{p}} \,\left[\frac{C_{p j}}{\sigma}+\frac{K_{T_{j}}}{K_{T}}-\frac{E_{p j}}{\alpha_{p}}\right] \nonumber \]
But
\[\varepsilon=K_{\mathrm{T}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}} \nonumber \]
Then,
\[\frac{\mathrm{E}_{\mathrm{pj}}-\mathrm{E}_{\mathrm{S} j}}{\varepsilon}=-\frac{\mathrm{E}_{\mathrm{pj}}}{\alpha_{\mathrm{p}}}+\frac{\mathrm{K}_{\mathrm{T} j}}{\kappa_{\mathrm{T}}}+\frac{\mathrm{C}_{\mathrm{pj}}}{\sigma} \nonumber \]
Hence for an aqueous solution in the limit of infinite dilution,
\[\frac{\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})-\mathrm{E}_{\mathrm{sj}}^{\infty}(\mathrm{aq})}{\varepsilon_{1}^{*}(\ell)}=-\frac{\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\mathrm{K}_{\mathrm{T} j}^{\infty}(\mathrm{aq})}{\kappa_{\mathrm{T} 1}^{*}(\ell)}+\frac{\mathrm{C}_{\mathrm{pj}}^{\infty}}{\sigma_{1}^{*}(\ell)} \nonumber \]
We start with the equation,
\[\mathrm{E}_{\mathrm{s}}=-\frac{\mathrm{K}_{\mathrm{s}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}} \nonumber \]
The latter equation is differentiated with respect to the amount of solute \(\mathrm{n}_{j}\) in a solution at fixed \(\mathrm{T}\), fixed \(\mathrm{p}\) and fixed amount of solvent, \(\mathrm{n}_{1}\).
\[\begin{aligned}
\left(\frac{\partial E_{S}}{\partial n_{j}}\right)_{T, p, n(1)}=-\frac{C_{p}}{T \, E_{p}} \,\left(\frac{\partial K_{s}}{\partial n_{j}}\right)_{T, p, n(1)}-\frac{K_{s}}{T \, E_{p}} \,\left(\frac{\partial C_{p}}{\partial n_{j}}\right)_{T, p, n(1)} \\
&+\frac{K_{s} \, C_{p}}{T \,\left(E_{p}\right)^{2}} \,\left(\frac{\partial E_{p}}{\partial n_{j}}\right)_{T, p, n(1)}
\end{aligned} \nonumber \]
Or,
\[E_{S_{j}}=-\frac{C_{p}}{T \, E_{p}} \, \frac{K_{s}}{K_{s}} \, K_{s_{j}}-\frac{K_{s}}{T \, E_{p}} \, \frac{C_{p}}{C_{p}} \, C_{p j}+\frac{K_{s} \, C_{p}}{T \,\left(E_{p}\right)^{2}} \, E_{p j} \nonumber \]
We rewrite the latter equation in terms of volume intensive variables.
\[\mathrm{E}_{\mathrm{Sj}}=-\frac{1}{\mathrm{~T}} \, \frac{\sigma}{\alpha_{\mathrm{p}}} \, \frac{\kappa_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \, \mathrm{K}_{\mathrm{Sj}}-\frac{1}{\mathrm{~T}} \, \frac{\kappa_{\mathrm{S}}}{\alpha_{\mathrm{p}}} \, \frac{\sigma}{\sigma} \, \mathrm{C}_{\mathrm{pj}}+\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2}} \, \mathrm{E}_{\mathrm{pj}} \nonumber \]
But
\[\alpha_{\mathrm{s}}=-\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}} \nonumber \]
Then
\[E_{\mathrm{S}_{\mathrm{j}}}=\alpha_{\mathrm{s}} \, \frac{\mathrm{K}_{\mathrm{s} \mathrm{j}}}{\kappa_{\mathrm{s}}}+\alpha_{\mathrm{s}} \, \frac{\mathrm{C}_{\mathrm{p} j}}{\sigma}-\alpha_{\mathrm{s}} \, \frac{\mathrm{E}_{\mathrm{pj}}}{\alpha_{\mathrm{p}}} \nonumber \]
Therefore (with a change of order)
\[\frac{E_{\mathrm{s} j}}{\alpha_{p}}=-\frac{E_{p j}}{\alpha_{p}}+\frac{K_{S j}}{K_{s}}+\frac{C_{p j}}{\sigma} \nonumber \]