1.22.7: Volumes: Neutral Solutes: Limiting Partial Molar Volumes
A given aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains solute \(j\), having molality \(\mathrm{m}_{j}\). The chemical potential of solute \(j\), \(\mu_{j}(\mathrm{aq})\) is related to \(\mathrm{m}_{j}\) using equation (a).
\[\begin{aligned}
&\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \\
&=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp}
\end{aligned} \nonumber \]
But
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left(\frac{\partial \mu_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Also \(\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)\) is, by definition, independent of pressure. From equation (a),
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
In equation (c), there is no term explicitly in terms of molaity \(\mathrm{m}_{j}\). From the definition of \(\gamma_{j}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \nonumber \]
\(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the limiting partial molar volume of solute \(j\) in aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In other words \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the partial molar volume of solute \(j\) in the (ideal) solution where there are no solute-solute interactions and characterizes solute-water interactions. Because \(\gamma_{j}\) tends to unity as \(\mathrm{m}_{j}\) tends to zero, \(\gamma_{j}\) is sometimes called an asymmetric activity coefficient [1]. [Contrast rational activity coefficients where \(\mathrm{f}_{1} \rightarrow 1 \text { as } \mathrm{x}_{1} \rightarrow 1\).]
At the risk of being repetitive we distinguish between the two possible reference states for substance \(j\) such as urea. One reference state is the pure solid chemical substance \(j\) at ambient pressure and \(298.2 \mathrm{~K}\). Another reference state is the ideal solution where \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\) at ambient pressure and \(298.2 \mathrm{~K}\). The properties of urea in the two states, pure solid and solution standard state are clearly quite different. Indeed, we can compare \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; 298.2 \mathrm{~K} ; \text { ambient p) }\) and \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~s} ; 298.2 \mathrm{~K} ; \text { ambient } \mathrm{p})\). We can also compare, for example, \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{j}=\text { urea; sln} 298.2 \mathrm{~K} ; \text { ambient p) }\) in a range of solvents [2]. These points are also nicely illustrated by the volumetric properties of water [3,]. At \(298.2 \mathrm{~K}\) and ambient pressure \(\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{H}_{2} \mathrm{O}\right)\) is \(18.07 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\) but for water as a solute in three solvents, \(\mathrm{V}^{\infty}\left(\mathrm{H}_{2} \mathrm{O} ; \operatorname{sln}\right)=18.47(\mathrm{MeOH}), 14.42(\mathrm{EtOH}) \text { and } 17.00(\mathrm{THF}) \mathrm{cm}^{3} \mathrm{~mol}^{-1}\) [4]. There is, of course, no reason why we should expect anything different. A water molecule in liquid water is surrounded by many millions of other water molecules. But a water molecule at infinite dilution in solvent ethanol is surrounded by many millions of ethanol molecules [5,6].
In the analysis of experimental results , we may express the composition of the solution in terms of mole fraction of solute \(\mathrm{x}_{j}\). Then
\[\begin{aligned}
&\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \\
&=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0} ; \mathrm{x}-\mathrm{scale}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}} \, \mathrm{f}_{\mathrm{j}}^{*}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp}
\end{aligned} \nonumber \]
But mole fraction \(\mathrm{x}_{j}\) is independent of pressure.
\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\mathrm{f}_{\mathrm{j}}^{*}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
From the definition of \(\mathrm{f}_{\mathrm{j}}^{*}\),
\[\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \nonumber \]
The limiting value of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is identical on the molality and mole fraction scales. If we use the concentration scale a problem arises in that the concentration of solute \(j\), \(\mathrm{c}_{j}\) is dependent on pressure because the volume of the solution is pressure dependent.
Footnote
[1] W. L. Masterton and H. K. Seiler, J. Phys. Chem., 1968, 72 , 4257.
[2] For \(j =\) urea at \(298.2 \mathrm{~K}\) and ambient pressure, \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{s} \ln ) / \mathrm{cm}^{3} \mathrm{~mol}^{-1}=\) 44.24 (water), 36.97 (methanol), 40.75 (ethanol) and 41.86 (DMSO).
[3] \(\mathrm{V}_{\mathrm{j}}^{\infty}(298.15 \mathrm{~K} ; \mathrm{j}=\text { water })=18.57 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}(\text { solvent }=\text { octan-1-ol })\) and \(31.3 \mathrm{cm}^{3} \mathrm{~mol}^{-1} \text { (solvent }= \mathrm{CCl}_{4})\); P. Berti, S. Cabani and V. Mollica, Fluid PhaseEquilib., 1987, 32 , 1.
[4] M. Sakurai and T. Nakagawa, Bull. Chem. Soc. Jpn., 1984, 55 , 195; J. Chem. Thermodyn., 1982, 14 , 269; 1984, 16 , 171.
[5] A similar contrast exists (H. Itsuki, S. Terasawa, K. Shinohara and H. Ikezwa, J. Chem. Thermodyn., 1987, 19 , 555) between the molar volume of a hydrocarbon and its limiting partial molar volume in another hydrocarbon; \(\mathrm{V}^{*}\left(\ell ; \mathrm{C}_{6} \mathrm{H}_{14}\right)=131.61 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\), but \(\mathrm{V}^{\infty}\left(\mathrm{C}_{6} \mathrm{H}_{14} ; \text { sln; solvent }=\mathrm{C}_{16} \mathrm{H}_{34}\right)=130.2 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\) at \(298 \mathrm{~K}\) and ambient pressure.
In this context the limiting enthalpies of solution water in monohydric alcohols depend on the alcohol at \(298.2 \mathrm{~K}\); (S.-O. Nilsson, J. Chem. Thermodyn., 1986, 18 , 1115).
[6] The partial molar volumes of fullerene in solution is \(401 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\) in cis- decalin and \(389 \mathrm{~cm}^{3} \mathrm}{mol}^{-1}\) in 1,2-dichlorobenzene both values being significantly less than the predicted volume of the pure liquid \(\mathrm{C}_{60}\); (P. Ruelle, A. Farina-Cuendet and U. W. Kesselring, J. Chem. Soc. Chem. Commun., 1995, 1161).