1.22.3: Volume: Partial and Apparent Molar
In descriptions of the volumetric properties of solutions, two terms are extensively used. We refer to the partial molar volume of solute \(j\) in, for example, an aqueous solution \(\mathrm{V}_{j}(\mathrm{aq})\) and the corresponding apparent molar volume \(\phi\left(\mathrm{V}_{j}\right)\). Here we explore how these terms are related. We consider an aqueous solution prepared using water, \(1 \mathrm{~kg}\), and \(\mathrm{m}_{j}\) moles of solute \(j\). The volume of this solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is given by equation (a).
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
The chemical potential of solvent, water, in the aqueous solution \(\mu_{1}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (b) where \(\mu_{1}^{*}(\ell)\) is the chemical potential of pure water(\(\ell\)), molar mass \(\mathrm{M}_{1}\), at the same \(\mathrm{T}\) and \(\mathrm{p}\).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Here practical osmotic coefficient \(\phi\) is defined by equation (c).
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1 \quad \text { at all T and } \mathrm{p} \nonumber \]
But
\[\mathrm{V}_{1}(\mathrm{aq})=\left[\partial \mu_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
Then[1]
\[\mathrm{V}_{1}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}} \nonumber \]
For the solute, the chemical potential \(\mu_{j}(\mathrm{aq})\) is related to the molality of solute \(\mathrm{m}_{j}\) using equation (f) where pressure \(\mathrm{p}\) is close to the standard pressure.
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
where
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{j}=1 \quad \text { at all } \mathrm{T} \text { and } \mathrm{p} \nonumber \]
Then[2]
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
\[\operatorname{Limit}\left(m_{j} \rightarrow 0\right) V_{j}(a q)=V_{j}^{0}(a q)=V_{j}^{\infty}(a q) \nonumber \]
Combination of equations (a), (e) and (h) yields equation (j).
\[\begin{gathered}
\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right] \\
\quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\right\}
\end{gathered} \nonumber \]
An important point emerges if we re-arrange equation (j).
\[\begin{aligned}
&\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell) \\
&\quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}-\mathrm{R} \, \mathrm{T} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right\}
\end{aligned} \nonumber \]
Equation (k) has an interesting form in that the brackets {….} contain \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) and terms describing the extent to which the volumetric properties of the solution are not ideal in a thermodynamic sense. It is therefore convenient to define an apparent molar volume of solute \(j\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) using equation (l).
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}-\mathrm{R} \, \mathrm{T} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}} \nonumber \]
Then
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \]
Therefore we obtain equation (n).
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Interest in equation (n) arises from the fact that the for a given solution \(\mathrm{V}(\mathrm{aq})\) can be measured {using the density \(\rho(\mathrm{aq})\)} and hence knowing \(\mathrm{V}_{1}^{*}(\ell)\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is obtained. If we measure \(\phi\left(\mathrm{V}_{j}\right)\) as a function of \(\mathrm{m}_{j}\), equation (m) indicates how one obtains \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\). Moreover the difference \(\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right)-\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\right]\) signals the role of solute - solute interactions.
Footnotes
[1]
\[\begin{array}{r}
\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \phi}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\
=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]
\end{array} \nonumber \]
[2]
\[\begin{aligned}
\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\
=& {[\mathrm{N} \mathrm{m} \mathrm{mol}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right.}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] }
\end{aligned} \nonumber \]