1.7.5: Compressions- Isentropic- Solutions- Partial and Apparent Molar
Isentropic properties of aqueous solutions are defined in a manner analogous to that used to define isothermal compressions and isothermal compressibilities. The assertion is made that a system (e.g. an aqueous solution) can be perturbed along a pathway where the affinity for spontaneous change is zero by a small change in pressure \(\delta \mathrm{p}\), to a neighbouring state having the same entropy. The (equilibrium) isentropic compression is defined by equation (a).
\[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=-[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{p}]_{\mathrm{S}(\mathrm{aq}), A=0} \nonumber \]
The constraint on this partial differential refers to 'at constant \(\mathrm{S}(\mathrm{aq})\)'. The definition of \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) uses non-Gibbsian independent variables. In other words, isentropic parameters do not arise naturally from the formalism which expresses the Gibbs energy in terms of independent variables in the case of, for example, a simple solution, \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}, \mathrm{n}_{\mathrm{j}}\right]\) [1]. The isothermal compression of a solution \(\mathrm{K}_{\mathrm{T}}(\mathrm{aq})\) and partial molar isothermal compressions of both solvent \(\mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})\) and solute \(\mathrm{K}_{\mathrm{T} j}(\mathrm{aq})\) are defined using Gibbsian independent variables. Unfortunately the corresponding equations cannot be simply carried over to the isentropic property \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\). The volume of a solution is expressed in terms of the amounts of solvent \(\mathrm{n}_{1}\) and solute \(\mathrm{n}_{j}\).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
The latter equation is differentiated with respect to pressure at constant entropy of the solution \(\mathrm{S}(\mathrm{aq})\). The latter condition includes the condition that the system remains at equilibrium where the affinity for spontaneous change is zero. We emphasize a point. The entropy which remains constant is that of the solution.
\[\mathrm{K}_{\mathrm{s}}(\mathrm{aq})=-\mathrm{n}_{1} \,\left[\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq}) ; \mathrm{A}=0}-\mathrm{n}_{\mathrm{j}} \,\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq}) ; \mathrm{A}=0} \nonumber \]
\(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) is an extensive property of the aqueous solution. \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) may also be re-expressed using Euler’s theorem as a function of the composition of the solution.
\[\mathrm{K}_{\mathrm{s}}(\mathrm{aq})=\mathrm{n}_{1} \,\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}+\mathrm{n}_{\mathrm{j}} \,\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})} \nonumber \]
Because \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) is defined using non-Gibbsian independent variables, two important inequalities follow.
\[-\left[\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} \neq\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})} \nonumber \]
\[-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} \neq\left[\partial \mathrm{K}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
\(\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}\) and \(\left[\partial \mathrm{K}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}\) are respectively the partial molar properties of the solvent and solute. Because partial molar properties should describe the effects of a change in composition on the properties of a solution, we write equation (d) for an aqueous solution in the following form.
\[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{S} 1}(\mathrm{aq} ; \text { def })+\mathrm{n}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def }) \nonumber \]
\[\text { Hence, } \quad \mathrm{K}_{\mathrm{sj}}(\mathrm{aq} ; \text { def }) \neq-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} \nonumber \]
In view of the latter inequality \(\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})\) is a non-Lewisian partial molar property [2]. We could define a molar isentropic compression of solute \(j\) as (minus) the isentropic differential dependence of partial molar volume on pressure. This alternative definition is consistent with equation (g) expressing a summation rule analogous to that used for partial molar properties. However some other thermodynamic relationships involving partial molar properties would not be valid in this case. Therefore, \(-\left[\partial V_{j}(a q) / \partial p\right]_{S(a q)}\) is a semi-partial molar property. A similar problem is encountered in defining an apparent molar compression for solute \(j\), \(\phi\left(\mathrm{K}_{\mathrm{Sj}_{\mathrm{j}}}\right)\) in a solution where the solute has apparent molar volume \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\); cf. equation (h) [3,4]. We might assert that \(\phi\left(\mathrm{K}_{\mathrm{Sj}_{\mathrm{j}}}\right)\) is related to the isentropic differential dependence of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) on pressure, \(-\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}\). Alternatively, using as a guide the apparent molar properties \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) and \(\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)\), we could define \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)\) using equation (i).
\[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\mathrm{l})+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{S} \mathrm{j}} ; \text { def }\right) \nonumber \]
\(\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})\) as given by equation (d) and \(\phi\left(K_{S j} ; \text { def }\right)\) are linked; equation (j).
\[\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \operatorname{def})=\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)+\mathrm{n}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})} \nonumber \]
Equation (j) is of the general form encountered for other apparent and partial molar properties. This form is also valid in the case of partial and apparent molar isobaric expansions, isothermal compressions and isobaric heat capacities. On the other hand, the semi-partial molar isentropic compression defined by \(-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{s}(\mathrm{aq})}\) and the semi-apparent molar isentropic compression defined by \(-\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}\) are related. The isentropic pressure dependence of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) is given by equation (k).
\[\begin{aligned}
&-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}= \\
&-\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}-\mathrm{n}_{\mathrm{j}} \,\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})} / \partial \mathrm{p}\right\}_{\mathrm{s}(\mathrm{aq})}
\end{aligned} \nonumber \]
However,
\[\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} / \partial \mathrm{p}\right\}_{\mathrm{S}(\mathrm{aq})} \neq\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} / \partial \mathrm{n}_{\mathrm{j}}\right\}_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
Hence, the analogue of equation (j) does not hold for these 'semi' properties. The inequalities (e) and (f) highlight the essence of non-Lewisian properties. Their origin is a combination of properties defined in terms of Gibbsian and non-Gibbsian independent variables as in equations (e) and (f). This combination is also the reason for the inequality (l). We stress that the isentropic condition in equations (e) and (f) refers to the entropy \(\mathrm{S}(\mathrm{aq})\) of the solution defined as is the volume \(\mathrm{V}(\mathrm{aq})\) by the Gibbsian independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\). But this is not the entropy \(\mathrm{S}_{1}^{*}(\ell)\) of the pure solvent having volume \(\mathrm{V}_{1}^{*}(\ell)\). \(\mathrm{S}(\mathrm{aq})\) at fixed composition is not simply related to \(\mathrm{S}_{1}^{*}(\ell)\) as, for example, linear functions of temperature and pressure.
The isentropic condition is involved in the definitions of isentropic compression, \(\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)\) and isentropic compressibility \(\kappa_{\mathrm{S} 1}^{*}(\ell)\) of the solvent.
\[\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=-\left[\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right] \text { at constant } \mathrm{S}_{1}^{*}(\ell) \nonumber \]
\[\begin{aligned}
&\kappa_{\mathrm{Sl}}^{*}(\ell)=\mathrm{K}_{\mathrm{Sl}}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\\
&=-\left[\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right] / \mathrm{V}_{1}^{*}(\ell) \text { at constant } \mathrm{S}_{1}^{*}(\ell)
\end{aligned} \nonumber \]
The different isentropic conditions in equation (a) and in equations (m) and (n) signal a complexity in the isentropic differentiation of equation (o) with respect to pressure [5,6].
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Footnotes
[1] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret, Chem. Phys. Phys. Chem., 2001, 3 ,1465.
[2] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,2,1982, 78 ,1565.
[3] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc. Rev.,2001, 30 ,8.
[4] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998, 94 ,2385.
[5] M. J. Blandamer, J. Chem. Soc. Faraday Trans.,1998, 94 ,1057.
[6] M. J. Blandamer, Chem. Soc. Rev.,1998, 27 ,73.