1.7.18: Compressions- Isothermal- Apparent Molar Compression
The volume of a given solution prepared at fixed temperature and fixed pressure using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\) is given by equation (a).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
If the solution is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)),
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
\(\mathrm{M}_{1}\) is the molar mass of the solvent, water(\(\ell\)); \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of water and solute \(j\) respectively in the solution. As we change \(\mathrm{m}_{j}\) (for a fixed mass of solvent) so both \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) change. An important procedure rewrites equation (b) in the following form where \(\mathrm{V}_{1}^{*}(\ell)\) is the molar volume of pure solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). Thus,
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
\(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is the apparent molar volume of the solute \(j\). The system, an aqueous solution, is displaced by a change in pressure (at fixed \(\mathrm{T}\)) along a path where the affinity for spontaneous change is zero. In other words the system is subjected to an equilibrium displacement. The isothermal differential dependence of volume \(\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) is given by equation (d).
\[\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right.}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{M}_{1}^{-1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
By definition, the apparent molar (isothermal) compression of the solute, [1-3]
\[\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Similarly for equilibrium molar compression of the pure solvent,
\[\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
By definition,
\[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=-\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Hence,
\[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right) \nonumber \]
Moreover recalling that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq}) \nonumber \]
These equations combined with those yielding \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) from measured \(\kappa_{\mathrm{T} 1}^{*}(\ell)\) and \(\kappa_{\mathrm{T}}(\mathrm{aq})\) signal an attractive approach to the study of solvent-solute interactions via \(\mathrm{K}_{\mathrm{T} j}^{\infty}(\mathrm{aq})\). In this context Gurney [4] identified a cosphere of solvent around a solute molecule where the organization differs from that in the bulk solvent at some distance from a given solute molecule \(j\). For example, the limiting partial molar volume of solute \(j\) can be understood as the sum of two terms, \(\mathrm{V}\)(intrinsic) and \(\mathrm{V}\)(cosphere). Then \(\mathrm{V}\)(cosphere) is an indicator of the role of solvent-solute interaction, hydration in aqueous solution. Thus,
\[\left.V_{j}^{\infty}(a q)=V_{j} \text { (int rinsic }\right)+V_{j}(\cos p h e r e) \nonumber \]
Hence,
\[\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}} \text { (int rinsic) }+\mathrm{K}_{\mathrm{T}_{j}}(\cos \text {phere}) \nonumber \]
The argument is advanced that \(\mathrm{K}_{\mathrm{T}j}\)(intrinsic) for simple ions such as halide ions and alkali metal ions is zero. \(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\) is an indicator of the hydration of a given solute in aqueous solution. \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) is obtained from the dependence of \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) on, for example, concentration \(\mathrm{c}_{j}\) using equation(l) for neutral solutes and equation (m) for salts, the latter being based on the DHLL.
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{a}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{b}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)^{1 / 2} \nonumber \]
One might have expected an extensive scientific literature reporting \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) for a wide range of solutes. Unfortunately measurement of isothermal compressions of liquids is difficult at least to the precision required for the estimation of \(\mathrm{K}_{\mathrm{Tj}}^{\infty}(\mathrm{aq})\). Indeed direct measurement of the volume change of a liquid when compressed at constant temperature is difficult because the isothermal condition is difficult to satisfy. Two procedures have been adopted to over come this problem. In both cases isentropic compressibilities calculated from densities and speeds of sound have been used.
In one set of procedures, isentropic compressibilities, densities and isobaric heat capacities are used to calculate isothermal compressions for a given solution, molality \(\mathrm{m}_{j}\). For example, Bernal and Van Hook [5] use the Desnoyers-Philip Equation to evaluate \(\phi\left(K_{T_{j}}\right)^{\infty}\) for glucose, sucrose and fructose [5] in aqueous solutions at \(348 \mathrm{~K}\). An alternative procedure equates \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\) with the experimentally accessible limiting apparent isentropic compression, ∞ φ(K ) Sj . In another approach, the starting point is equation (a) which is differentiated with respect to pressure at constant temperature to yield equation (n).
\[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{Tl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq}) \nonumber \]
Equation (n) is divided by volume \(\mathrm{V}(\mathrm{aq})\). Hence
\[\begin{aligned}
\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{1}(\mathrm{aq}) \\
&+\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{j}}(\mathrm{aq})
\end{aligned} \nonumber \]
We use \(\phi_{1}\left[=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \text { and } \phi_{\mathrm{j}}\left[=\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right]\) to express volume fractions.
\[\kappa_{\mathrm{T}}(\mathrm{aq})=\phi_{1} \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})+\phi_{\mathrm{j}} \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{Tj}}(\mathrm{aq}) \nonumber \]
The latter equation is not tremendously helpful. Although \(\kappa_{\mathrm{T}}(\mathrm{aq})\) can be measured, the right hand side involves six terms about which we have no information ‘a priori’ and which depend on the composition of the solution.
Footnotes
[1] F. T. Gucker, Chem. Rev.,1933, 14 ,127.
[2] F. T. Gucker, J. Am. Chem. Soc.,1933, 55 ,2709.
[3] Units; \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right] ; \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\); \(\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]\)
For the solution \(\kappa_{\mathrm{T}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{Pa}^{-1}\right]\)
For the solvent \(\kappa_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\left[\mathrm{Pa}^{-1}\right]\)
Isothermal compressions have units of ‘volume per unit of pressure’ whereas compressibilities have units of ‘reciprocal pressure’. \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is an apparent molar isothermal compression on the grounds that the units of this quantity are \(\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\). Some reports use the term ‘apparent molar isothermal compressibility’ which should be avoided because in the present context this term corresponds to a different property; see J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998, 94 ,2385.
[4] R.W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
[5] P. D. Bernal and W. A. Van Hook, J. Chem. Thermodyn., 1986, 18 ,955.