1.7.8: Compresssions- Isentropic- Aqueous Solution

A given aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\). The thermodynamic properties of this solution are ideal.

\[\text { Then, } \quad V_{m}(a q ; \text { id })=x_{1} \, V_{1}^{*}(\ell)+x_{j} \, \phi\left(V_{j}\right)^{\infty}\]

\[\text { Here } \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})=\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \mathrm{V}(\mathrm{aq} ; \mathrm{id})}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)}\]

The molar entropy of the ideal solution is given by equation (c).

\[\mathrm{S}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{S}_{1}^{*}(\ell)-\mathrm{x}_{1} \, \mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\]

\(\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\) is the partial molar entropy of solute \(j\) at the same \(\mathrm{T}\) and \(\mathrm{p}\). The solution is perturbed by a change in pressure and displaced to a neighbouring state having the same entropy, \(\mathrm{S}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})\).

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

From equation (a),

\[\mathrm{K}_{\mathrm{Sm}(\mathrm{a} ; ; \mathrm{dd})}=-\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}-\mathrm{x}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; ; \mathrm{dd}), \mathrm{x}(\mathrm{j})}\]

On these partial differentials, the isentropic condition is not the most convenient because it refers to the entropy of an ideal solution. Using the technique adopted for liquid mixtures, we obtain in equation (f), an expression for the unconventional isentropic compression of the solvent .

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

Or,

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

Except for the different ideal reference state, equation (g) for the solvent is formally identical to the corresponding equation for liquid mixtures. However, in this case we need to follow a different approach for chemical substance \(j\). The appropriate choice for isentropic conditions on solute properties is the entropy of the pure solvent at same \(\mathrm{T}\) and \(\mathrm{p}\). Hence,

\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}=\]

Or,

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

Or,

\[\begin{aligned}
&\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \times(\mathrm{j})}=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)} \\
&-\mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \,\left[\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right]
\end{aligned}\]

The isentropic pressure dependences of apparent and partial molar volumes are complicated functions. We are interested in obtaining an expression for \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})\) in terms of the limiting apparent or partial molar isentropic compression of solute \(j\), \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\). We use the following expression [1-3].

\[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)}=-\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left(\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right)\]

We combine the results in equations (e), (g), (j) and (k) to obtain an equation for \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})\); equation (l).

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

Finally, by noting that \(\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}\) after slight simplification we arrive at an expression for \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})\); equation (m).

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

The complexity of equation (m) for solutions can be attributed to a combination of the non-Gibbsian character of \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})\) with the non-Lewisian character of \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\). Clearly \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})\) and \(\mathrm{K}_{\mathrm{mix}}(\mathrm{aq} ; 1 \mathrm{~d})\) are not equal because the reference states for chemical substance \(j\) differ. We are interested in the apparent molar isentropic compression of solute \(j\) in ideal aqueous solutions \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})\), which is defined in equation (n) and expressed by equation (o).

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

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

Limiting values for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})\) are interesting. For the ideal solution at \(\mathrm{x}_{j} = 0\), which is the same state as the real solution at infinite dilution, we naturally obtain \(\phi\left(\mathrm{K}_{\mathrm{Sj}_{j}}\right)^{\infty}\) although using equation (o) for this purpose requires solving an indeterminate form. For the ideal solution at \(\mathrm{x}_{j} = 0\) we obtain equation (p), which yields equation (q) after major reorganisation.

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

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

The latter equation expresses the molar isentropic compression of solute \(j\) in a standard state of unit mole fraction in terms of properties for the pure solvent and for the solute at infinite dilution. The ideal aqueous solution may be described as a non-ideal liquid mixture. An excess property is defined by equation (r).

\[\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})-\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\text { mix} ; \mathrm{id})\]

\[\text { Or, } \phi\left(K_{S j}\right)^{E}=\left[K_{S m}(\mathrm{aq} ; i d)-K_{S m}(\operatorname{mix} ; i \mathrm{~d})\right] / x_{j}\]

A working equation for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}\) can be generated from equation (o). After little reorganisation, we obtain equation (t).

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

Interestingly, the first four terms on the right end side of equation (t) express the difference \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}-\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{*}(\ell)\). For solution chemists the important reference state is at infinite dilution. The limiting excess property \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\) is given by equation (u) [4].

\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{S} \mathrm{j}}\right)^{\mathrm{E}, \infty}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \\
&-\mathrm{T} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} j}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2}
\end{aligned}\]

Estimates of \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) using \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\) data for binary liquid mixtures often neglect the last term in equation (u).

Equation (u) works in two ways. A solution chemist will estimate \(\phi\left(K_{S j}\right)^{\infty}\) from data reporting \(\phi\left(K_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\). A chemist interested in the properties of liquid mixtures will estimate \(\phi\left(K_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\) from data reporting \(\phi\left(K_{S j}\right)^{\infty}\).

Footnotes

[1] J. C. R. Reis, J. Chem. Soc., Faraday Trans.2. 1982, 78, 1595.

[2] M. J. Blandamer, J. Chem. Soc., Faraday Trans., 1998, 94, 1057.

[3] J. C. R. Reis, J. Chem. Soc., Faraday Trans., 1998, 94, 2395.

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