1.7.11: Compression- Isentropic- Apparent Molar Volume
A given liquid system is prepared using \(\mathrm{n}_{1}\) moles of water, molar mass \(\mathrm{M}_{1}\), and \(\mathrm{n}_{j}\) moles of substance \(j\). The closed system is at equilibrium, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The volume of the system is given by equation (a).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{\mathrm{1}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Here \(V_{1}^{*}(\ell)\) is the molar volume of pure water and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is the apparent molar volume of substance \(j\) in the system; \(\mathrm{V}(\mathrm{aq})\) and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) depend on the composition of the system, but \(\mathrm{V}_{1}^{*}(\ell)\) does not.
The solution is perturbed to a local equilibrium state by a change in pressure along a path for which the entropy remains constant at \(\mathrm{S}(\mathrm{aq})\). At a specified molality \(\mathrm{m}_{j the change in volume is characterised by the isentropic compressibility, \(\mathrm{K}_{\mathrm{s}}(\mathrm{aq})\) defined in equation (b).
\[\kappa_{\mathrm{s}}(\mathrm{aq})=-\frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})} \nonumber \]
Hence,
\[\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq})=-\mathrm{n}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}-\mathrm{n}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})} \nonumber \]
The isentropic condition on the first partial differential in equation (c) refers to the entropy of an aqueous solution at molality, \(\mathrm{m}_{j}\). There is interest in relating this partial differential to the isentropic compressibility of the pure liquid substance 1 at the same \(\mathrm{T}\) and \(\mathrm{p}\), which is defined in equation (d).
\[\kappa_{\mathrm{s} 1}^{*}(\ell)=-\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)} \nonumber \]
For substance 1 the different isentropic conditions are related by equation (e).
\[\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\mathrm{aq}) \mathrm{m}(\mathrm{j})}=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}+\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) / \mathrm{m}(\mathrm{j})} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\mathrm{l})}\right)_{\mathrm{p}^{*}} \nonumber \]
In the latter equation we identify \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}\) and \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(1)}\right)_{\mathrm{p}^{*}}\) with, respectively, \(-\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{S} 1}^{*}(\ell)\) and \(\mathrm{T} \, \alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\), which are thermodynamic properties of water (\(\ell\)). Here \(\sigma_{1}^{*}(\ell)\) is the heat capacitance (or heat capacity per unit volume) of water (\(\ell\)) Using the same calculus operation, the remaining partial differential is related to an isothermal property in equation (f).
\[\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}=\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})} \,\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}^{*}} \nonumber \]
Since \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{pl}}^{*}(\ell),\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq})) \mathrm{m}(\mathrm{j})}=\mathrm{T} \, \frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})}\), and \(\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}^{*}}=\frac{\mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\mathrm{T}}\), we combine these results with equation (f) to express equation (e) as equation (g).
\[\begin{aligned}
&\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \\
&-\kappa_{\mathrm{S} 1}^{*}(\ell)-\mathrm{T} \, \frac{\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\sigma_{1}^{*}(\ell)}+\mathrm{T} \, \frac{\alpha_{\mathrm{p} 1}^{*}(\ell) \, \alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})}
\end{aligned} \nonumber \]
We return to equation (c). Using equation (a) for \(\mathrm{V}(\mathrm{aq})\), equation (c) yields equation (h).
\[\begin{aligned}
&-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \\
&{\left[\left(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right) \, \mathrm{V}_{1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq})+\left(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right) \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}}
\end{aligned} \nonumber \]
We note that \(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}=\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \mathrm{M}_{1}}\) And that density \(\rho_{1}^{*}(\ell)=\frac{\mathrm{M}_{1}}{\mathrm{~V}_{1}^{*}(\ell)}\). Then combining equations (g) and (h) leads to equation (i) after slight simplification.
\[\begin{aligned}
&-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \\
&{\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)} \\
&+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})}-\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\sigma_{1}^{*}(\ell)}\right]
\end{aligned} \nonumber \]
An equivalent derivation of equation (i) has been given [1].
Footnotes
[1] M. J. Blandamer, J. Chem. Soc., Faraday Trans., 1998, 94, 1057.