1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions
A given aqueous solution at temperature \(\mathrm{T}\) and near ambient pressure \(\mathrm{p}\) contains a solute \(j\) at molality \(\mathrm{m}_{j}\). The chemical potential \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (a).
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
Then
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
By definition, the partial molar isothermal compression of solute \(j\),[1]
\[K_{T_{j}}(a q)=-\left(\frac{\partial V_{j}(a q)}{\partial p}\right)_{T} \nonumber \]
Then
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\mathrm{K}_{\mathrm{TJ}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}^{2}\right]_{\mathrm{T}} \nonumber \]
Thus by definition,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq}) \nonumber \]
Hence the difference between \(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}(\mathrm{aq})\) and \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) depends on the second differential of \(\ln \left(\gamma_{j}\right)\) with respect to pressure.
Footnotes
[1] The formal definition of \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is given by equation (a).
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{K}_{\mathrm{T}}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
However,
\[\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{n}(\mathrm{j})} \nonumber \]
Then,
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}(\mathrm{j})} \nonumber \]
Or,
\[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
In other words, equation (c) shows that \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is a Lewisian partial molar property