1.7.12: Compressions- Isentropic- Binary Liquid Mixtures

A given liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of liquid substance \(j\). The closed system is at equilibrium, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The volume of the system is defined by the following equation.

\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]

Similarly the entropy of the system is defined by equation (b).

\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]

The volume of the system is given by equation (c).

\[\mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\]

Here \(\mathrm{V}_{1}\) and \(\mathrm{V}_{j}\) are the partial molar volumes of the two substances in the system; \(\mathrm{V}\), \(\mathrm{V}_{1}\) and \(\mathrm{V}_{j}\) depend on the composition of the system. Similarly the entropy of the system is given by equation (d).

\[\mathrm{S}=\mathrm{n}_{1} \, \mathrm{S}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}\]

The molar volume of the system \(\mathrm{V}_{\mathrm{m}}\) is given by the ratio \(\mathrm{V} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)\).

\[\text { Hence, } \quad \mathrm{V}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{V}_{1}+\mathrm{x}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\]

\[\text { Similarly } \quad S_{m}=x_{1} \, S_{1}+x_{j} \, S_{j}\]

The system under examination is a binary liquid mixture such that the thermodynamic properties of the mixture are ideal.

\[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{*}(\ell)\]

\[\mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{S}_{1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}^{*}(\ell)+\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})\]

In other words the two reference states are the pure substances at the initially fixed \(\mathrm{T}\) and \(\mathrm{p}\). In equation (h) the term \(\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\) describes the ideal molar entropy of mixing, which is a function of composition only. The liquid mixture is perturbed to a local equilibrium state by a change in pressure along a path for which the entropy remains at that given by equation (h). At a specified mole fraction \(\mathrm{x}_{j}\) the change in volume is characterised by the isentropic compression, \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) which is \(\mathrm{K}_{\mathrm{m}}\left(\text { at constant } \mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\right)\) defined in equation (i).

\[\mathrm{K}_{\mathrm{Sm}}\left(\operatorname{mix} ; \mathrm{x}_{\mathrm{j}} ; \mathrm{id}\right)=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}\]

Hence,

\[\begin{aligned}
&\mathrm{K}_{\mathrm{Sm}}\left(\mathrm{mix} ; \mathrm{x}_{\mathrm{j}} ; \mathrm{id}\right)= \\
&\quad-\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{dd}), x(\mathrm{j})}-\mathrm{x}_{\mathrm{j}} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{id}), x(j)}
\end{aligned}\]

We note that the isentropic condition on the partial differentials in equation (j) refers to the entropy of an ideal mixture at mole fraction \(\mathrm{x}_{j}\). There is merit in relating these partial differentials to the isentropic compressions of the pure liquid substances 1 and \(j\) at the same \(\mathrm{T}\) and \(\mathrm{p}\), which are defined in equations (k) and (l).

\[\mathrm{K}_{\mathrm{si}}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(1)^{*}}\]

\[\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(j)^{*}}\]

The required relationship is obtained using a calculus operation. For substance 1 the different isentropic conditions are related by equation (m).

\[\begin{aligned}
&\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{jd}) \times(\mathrm{j})}= \\
&\left.\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(1)^{*}}+\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; ; \mathrm{d}) \times(j)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\ell)}\right)_{\mathrm{p}}\right)^{*}
\end{aligned}\]

In the latter equation we identify \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(1)^{*}} \text { and }\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\ell)}\right)_{\mathrm{p}}\) with, respectively, \(-\mathrm{K}_{\mathrm{Sl}}^{*}(\ell)\) and \(\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}\) which are thermodynamic properties of water(\(\ell\)). Using the same calculus operation, the remaining partial differential is related to an isothermal property in equation (n).

\[\begin{aligned}
&\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} \text {;id }) \times(\mathrm{j})}= \\
&\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{id}) \times(\mathrm{j})} \,\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}
\end{aligned}\]

But \(\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\), \(\left(\frac{\partial T}{\partial p}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{jd}) \mathrm{x}(\mathrm{j})}=\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{A} ; \mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{A} ; \mathrm{mix} ; \mathrm{id})}\), and \(\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\frac{\mathrm{C}_{\mathrm{pm}}^{*}(\ell)}{\mathrm{T}}\),

We combine these results with equation (n) to re-express equation (m) as equation (o).

\[\begin{aligned}
&\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{id}) \mathrm{x}(\mathrm{j})}= \\
&-\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}
\end{aligned}\]

Similarly, for the substance j we obtain equation (p).

\[\begin{aligned}
&\left(\frac{\partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{dd}) \times(\mathrm{j})}= \\
&-\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}+\mathrm{T} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell) \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}
\end{aligned}\]

Equations (o) and (p) can be used to recast equation (j) in the form of equation (q).

\[\begin{aligned}
&\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{Sl} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+ \\
&\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right. \\
&\left.-\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{pl}}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right] \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\}
\end{aligned}\]

\(\mathrm{K}_{\mathrm{Sm}}(\mathrm{A} ; \text { mix; id })\) is the ideal molar isentropic compression, which is commonly denoted by id \(\mathrm{K}_{\mathrm{Sm}}^{\mathrm{id}}\). The sum, \(\left[x_{1} \, E_{p 1}^{*}(\ell)+x_{j} \, E_{p j}^{*}(\ell)\right]\) is the ideal molar isobaric thermal expansion \(E_{p m}^{\text {id }}\) for the binary liquid mixture, here denoted as \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\), and analogously for the ideal molar isobaric heat capacity \(\mathrm{C}_{\mathrm{pm}}^{\mathrm{id}}= \mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\). Thus,

\[\begin{aligned}
&\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{s} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+ \\
&\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\}
\end{aligned}\]

The last term of equation (r) expresses a mixing property. In general terms,

\[\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \text { id })=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{sl}}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})\]

From equations (r) and (s) we obtain the following expression.

\[\begin{aligned}
&\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})= \\
&\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})}\right\} \\
&\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\mathrm{A} ; \operatorname{mix} ; \mathrm{id})=\mathrm{T}\left\{\mathrm{x}_{1}\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\right. \\
&\left.+\mathrm{x}_{\mathrm{j}}\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{pj}}^{*}(\ell)-\left[\mathrm{E}_{\mathrm{pm}}(\text { mix } ; \mathrm{id})\right]^{2} / \mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right\}
\end{aligned}\]

This is an important, albeit frequently neglected term, in the calculation of isentropic compressions of thermodynamically ideal liquid mixtures. In general ideal molar mixing values are non-zero for non-Gibbsian properties, the origin of which has been discussed [1].

We recall the definition of an apparent molar property. Hence,

\[\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{S} j}\right)(\mathrm{mix} ; \mathrm{id})\]

Here \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})\) is the apparent molar isentropic compression of chemical substance \(j\) in the ideal liquid mixture. Combination of equations (r) and (u) yields equation (v) [2].

\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}}\right)(\operatorname{mix} ; \mathrm{id})= \\
&\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\} / \mathrm{x}_{\mathrm{j}}
\end{aligned}\]

The limiting values for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)\) at \(\mathrm{x}_{j} = 1\) and \(\mathrm{x}_{j} = 0\) are of particular interest. For the pure liquid substance \(j\), equation (w) is readily obtained.

\[\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})=\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)\]

The latter is the expected property. However, the infinite dilution limit of equation (v) is not immediately obvious. In fact, both the numerator and denominator in the last term approach zero as we approach infinite dilution (\(\mathrm{x}_{j} = 0\)). What emerges is equation (x) [3], which is an example of the unusual formalism for non-Lewisian properties [4].

\[\begin{aligned}
&\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})= \\
&\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\mathrm{T} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2}
\end{aligned}\]

Thus for ideal liquid mixtures \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) is given by a combination of properties for both pure liquid components. In other words, the chemical nature of component 1 affects the non-Lewisian properties of its mate j in the ideal mixture. This is in contrast with apparent molar Lewisian properties, such as \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\), for which the values in ideal mixtures are the same as in the pure liquid state of substance \(j\).

An extensive literature describes isentropic compressions of binary liquid mixtures [5]

Footnotes

[1] G. Douhéret, M. I. Davis, J. C. R. Reis and M. J. Blandamer, ChemPhysChem, 2001, 2, 148.

[2] M. I. Davis, G. Douhéret, J. C. R. Reis and M. J. Blandamer, Phys. Chem. Chem. Phys., 2001, 3, 4555.

[3] It is instructive to show how the limiting value for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})\) at \(\mathrm{x}_{j} = 0\) (and hence \(\mathrm{x}_{1} = 1\)) is obtained from equation (v). In this limit the last term (without \(\mathrm{T}\)) of equation (v) becomes,

\[\begin{aligned}
&\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right)\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})}\right\} / 0\\
&=0 / 0
\end{aligned}\]

We apply L'Hospital's rule which asserts that this limit is equal to the ratio of limits (bb) and (cc).

\[\begin{aligned}
&\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \\
&\mathrm{d}\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right\} / \mathrm{dx} \mathrm{x}_{\mathrm{j}}
\end{aligned}\]

\[\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{dx}_{\mathrm{j}} / \mathrm{dx}_{\mathrm{j}}\]

The latter limit is unity. The former is obtained from the following differential.

\[\begin{aligned}
&\mathrm{d}\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right\} / \mathrm{dx} \mathrm{x}_{\mathrm{j}} \\
&=-\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)} \\
&-\left\{2 \, \frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]}{\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right]}\right\} \,\left[-\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right] \\
&+\left\{\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}\right\} \,\left[-\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right]
\end{aligned}\]

The limiting value of this differential is,

\[\operatorname{limit}\left(x_{1} \rightarrow 0\right) \frac{\left.\mathrm{d}_{\{} \ldots\right\}}{\mathrm{dx}_{\mathrm{j}}}=\mathrm{C}_{\mathrm{pj}}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} j}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2}\]

This is the value for the limit of equation (aa), which was used to obtain equation (x) above.

[4] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douhéret, Phys. Chem. Chem. Phys., 2001, 3, 1465.

[5]

1. methanol + alcohols; H. Ogawa and S. Murakami, J. Solution Chem.,1987, 16, 135.
2. γ-caprolactam + alcohols; S. K. Mehta, R. K. Chauhan and R. K. Dewan, J. Chem. Soc. Faraday Trans.,1996,92,4463.
3. ethanol + water; G. Onori, J. Chem. Phys.,1988, 89,4325.
4. alkoxyethanol + water; G.Douheret, A. Pal and M. I. Davis, J. Chem.Thermodyn.,1990,22,99.
5. 2-butoxyethanol(aq); 2-butanone(aq) Y. Koga, K..Tamura and S. Murakami, J. Solution Chem.,1995,24,1125.
6. pyrrolidin-2-one + alkanols; S. K. Mehta, R. K. Chauhan and R. F. Dewan, J. Chem. Soc. Faraday Trans.,1996,92,1167.
7. isomeric 2-butoxyethanols+water; G. Douheret, M. I. Davis, J. C. R. Reis, I. J. Fjellanger, M. B. Vaage and H. Hoiland, Phys. Chem. Chem. Phys., 2002,4,6034.
8. benzene + cyclohexane; G. C.Benson and O. Kiyohara, J. Chem. Thermodyn.,1979,11,1061.
9. water +_ ethanol; G. C. Benson + M. K. Kumaran, J. Chem. Thermodyn., 1983,15,799.