1.12.11: Expansions- Isobaric- Apparent Molar- Neutral Solutes
For many aqueous solutions at ambient pressure and temperature, the dependence of \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) on molality of a neutral solute \(j\), \(\mathrm{m}_{j}\) is accounted for by an equation having the following general form. [The reason for choosing the molality scale is again the fact that \(\mathrm{m}_{j}\) is independent of \(\mathrm{T}\) and \(\mathrm{p}\) but concentration \(\mathrm{c}_{j}\) is not.]
\[\phi\left(E_{p j}\right)=a_{1}+a_{2} \,\left(m_{j} / m^{0}\right)+a_{3} \,\left(m_{j} / m^{0}\right)^{2}+a_{4} \,\left(m_{j} / m^{0}\right)^{3}+\ldots \nonumber \]
At low molalities, the linear term is dominant. Granted therefore that equation(a) accounts for the observed pattern, we need a quantitative description which accounts for this pattern. There are advantages in linking directly the apparent property \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) and the partial molar property \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\).
For an aqueous solution at fixed temperature and pressure,
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\phi\left(\mathrm{E}_{\mathrm{pj}}\right) \nonumber \]
Hence the partial molar isobaric expansion of solute \(j\) can be calculated from the apparent molar isobaric expansion and its dependence on molality, \(\mathrm{m}_{j}\). Hence if equation (a) satisfactorily describes the observed dependence of \(\phi\left(E_{p j}\right)\) on \(\mathrm{m}_{j}\),
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{a}_{1}+2 \, \mathrm{a}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+3 \, \mathrm{a}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\ldots \nonumber \]
Therefore,
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi \mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})=\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \nonumber \]
Consequently the parameter \(\mathrm{a}_{1}\) in equations (a) and (b) is the limiting partial molar isobaric expansion of solute \(j\). For dilute solutions, equation (c) takes the following simple form.
\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{S}_{\mathrm{Ep}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
In these terms we can identify the basis of the parameter \(\mathrm{a}_{2}\) in equations (a) and (c). Desrosiers et al [1] used a quadratic (cf. equation (a)) to express the dependence of \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) at \(298 \mathrm{~K}\) on molality of urea in aqueous solutions; \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=0.07 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\), coefficient \(\mathrm{a}_{2}\) being positive and coefficient \(\mathrm{a}_{3}\) being negative.
The majority of published information concerns the dependence on temperature of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\). A survey [2] based on a dilatometric study of 15 non-electrolytes in aqueous solution indicates that \(\left[\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}\right]\) is less than \(\left[\mathrm{dV}_{\mathrm{j}}^{*}(\ell) / \mathrm{dT}\right]\) for the pure liquid substance \(j\); the second derivative \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\) is positive. However for hydrophilic solutes \(\left[\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}\right]\) is larger than \(\left[\mathrm{dV}_{\mathrm{j}}^{*}(\ell) / \mathrm{dT}\right]\) and \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\) is negative [3]. A similar pattern is observed for sucrose and urea for which \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\) is negative. Indeed Hepler [4] classified solutes in aqueous solutions as either structure-breaking (negative) or structure forming (positive) on the basis of the sign for \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\). The dependence of \(\phi\left(V_{j}\right)^{\infty}\) on temperature for both glycine and alanine in \(\mathrm{NaCl}(\mathrm{aq})\) is small [5], For monosacchrides(aq) \(\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\) is positive.
Footnotes
[1] N. Desrosiers, G. Perron, J. G. Mathieson, B. E. Conway and J. E. Desnoyers, J Solution Chem.,1974, 3 ,789.
[2] J. I. Neal and D. A. I. Goring, J. Phys. Chem.,1970, 74 ,658.
[3] J. Sengster, T.-T. Ling and F. Lenzi, J Solution Chem,1976, 5 ,575.
[4] L.G. Hepler, Can J.Chem.,1969, 47 ,4613.
[5] B. S. Lark, K.Balat and S. Singh, Indian J Chem., Sect A, 25 ,534.
[6] S. Paljk, K. Balat and S. Singh, J. Chem. Eng Data, 1990, 35 .41.