1.22.3: Volume: Partial and Apparent Molar
- Page ID
- 397781
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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).
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]
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 \(\mathrm{p}\).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]
Here practical osmotic coefficient \(\phi\) is defined by equation (c).
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1 \quad \text { at all T and } \mathrm{p}\]
But
\[\mathrm{V}_{1}(\mathrm{aq})=\left[\partial \mu_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}}\]
Then[1]
\[\mathrm{V}_{1}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\]
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.
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]
where
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{j}=1 \quad \text { at all } \mathrm{T} \text { and } \mathrm{p}\]
Then[2]
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]
\[\operatorname{Limit}\left(m_{j} \rightarrow 0\right) V_{j}(a q)=V_{j}^{0}(a q)=V_{j}^{\infty}(a q)\]
Combination of equations (a), (e) and (h) yields equation (j).
\[\begin{gathered}
\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right] \\
\quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\right\}
\end{gathered}\]
An important point emerges if we re-arrange equation (j).
\[\begin{aligned}
&\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell) \\
&\quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}-\mathrm{R} \, \mathrm{T} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right\}
\end{aligned}\]
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).
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}-\mathrm{R} \, \mathrm{T} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\]
Then
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]
Therefore we obtain equation (n).
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]
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]
\[\begin{array}{r}
\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \phi}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\
=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]
\end{array}\]
[2]
\[\begin{aligned}
\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\
=& {[\mathrm{N} \mathrm{m} \mathrm{mol}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right.}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] }
\end{aligned}\]