1.7.7: Compressions- Isentropic- Salt Solutions
An extensive literature describes the isentropic compressibilities of salt solutions prompted by earlier studies by Passynski [1] described by Owen [2].
The isentropic compression of a given aqueous salt solution \(\mathrm{K}_{\mathrm{s}}(\mathrm{aq})\) is determined using the Newton-Laplace Equation in conjunction with speeds of sound and densities. An apparent molar compression of salt \(j \phi\left(K_{s} ; \text { def }\right)\) is calculated using equation (a).
\[\mathrm{K}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{s}} ; \operatorname{def}\right) \nonumber \]
Here \(\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)\) is the isentropic compression of the solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). For salt solutions, particularly aqueous salt solutions, the dependence of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) on the molality of the salt is generally examined in the light of equations describing the role of ion-ion interactions [3; see also reference 4]. For dilute salt solutions, equation (b) forms the basis for examining the dependence of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) on \(\left(\mathrm{m}_{\mathrm{j}}\right)^{1 / 2}\) where \(\mathrm{m}_{j}\) is the molality of the salt-\(j\).
\[\text { Then, } \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)=\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}+\mathrm{S}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
The form of the equation (b) has all the hallmarks of a pattern required by the \(\mathrm{DHLL}\). In practice \(\mathrm{S}_{\mathrm{KS}}\) cannot be calculated because the required isentropic dependence of the relative permittivity of the solvent on pressure is generally not known. However a plot is obtained using equation (b) yielding an estimate of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\).
For a large range of 1:1 salts \(\phi\left(K_{S_{j}} ; \operatorname{def}\right)^{\infty}\) is negative, a pattern attributed to electrostriction of neighbouring solvent molecules by electric charges on the ions [3-10]. \(\phi\left(\mathrm{K}_{\mathrm{S}_{j}} ; \text { def }\right)^{\infty}\) is more negative for solutions in \(\mathrm{D}_{2}\mathrm{O}\) than in \(\mathrm{H}_{2}\mathrm{O}\) as a consequence of more intense electrostriction in \(\mathrm{D}_{2}\mathrm{O}\) [5]. Further on the basis of the Desnoyers-Philip Equation, the difference \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) is small but not negligible, amounting to approx. 10%. For alkylammonium ions in aqueous solutions \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) decreases with increase in the hydrophobic character, matching a general increase in \(\phi\left(V_{j}\right)^{\infty}\) [11]. Group and ionic contributions to \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) have been estimated [10,11]. Indeed \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) is approximately a linear function of \(\left(m_{j} / m^{0}\right)^{1 / 2}\) for a wide range of aqueous and non-aqueous salt solutions [11-13]; e.g. salts in \(\mathrm{DMSO}\) [14] and in propylene carbonate [15]. \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) for copper(I) and sodium perchlorates in cyanobenzene, pyridine and cyanomethane show almost no dependence on salt molality [16].
A problem is further complicated by the fact that the \(\mathrm{DHLL}\) for \(\phi\left(K_{\mathrm{S} j} ; \text { def }\right)\) is itself a complicated function of salt molality [17], \(\mathrm{S}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) being however the leading term. A problem is encountered with the differential dependence of the molar volume of the solvent \(\mathrm{V}_{1}^{*}(\ell)\) on pressure at constant \(\mathrm{S}(\mathrm{s} \ln )\) describing how the volume of the solution would depend on pressure if it were held at the same entropy of the solution. Thus [18]
\[\begin{aligned}
-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{s}(\mathrm{aq})} &=\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \\
&+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{1}^{*}(\ell) \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\}
\end{aligned} \nonumber \]
By definition,
\[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{S} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
Then,
\[\begin{aligned}
&-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right) \\
&\quad+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{1}^{*}(\ell) \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\}
\end{aligned} \nonumber \]
Consequently the difference between \(-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{s}(\mathrm{aq})}\) and \(\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)\) is determined by the property \(\Delta \phi\), defined in equation (f).
\[\Delta \phi=\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\} \nonumber \]
However \(\Delta \phi / \mathrm{m}_{\mathrm{j}}\) is indeterminate at infinite dilution . But using L’Hospital‘s rule,
\[\operatorname{Limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \Delta \phi / \mathrm{m}_{\mathrm{j}}= \left[\left[\rho_{1}^{*}(\ell) \, \alpha_{p 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right] \,\left\{\frac{\phi\left(E_{p j}\right)^{\infty}}{\alpha_{p 1}^{*}(\ell)}\right]-\left[\frac{\phi\left(C_{p j}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\right]\right\} \nonumber \]
Despite the thermodynamic polish given to the analysis of isentropic compressions, the problem of contrasting conditions ‘at constant \(\mathrm{S}(\mathrm{aq})\)’ and ‘at constant \(\mathrm{S}_{1}^{*}(\ell)\)’ underlies the analysis.
Footnotes
[1] A. Passynski, Acta Physicochim. URSS, 1938, 8 ,385.
[2] B. B. Owen and P. L. Kronick, J. Am. Chem.Soc.,1961, 65 ,84.
[3] F. J. Millero, in Activity Coefficients of Electrolyte Solutions, ed. R. M. Pytkowicz, CRC Press, Boca Raton, Fl,1979 , chapter 13.
[4] F. J. Millero, F. Vinokurova, M. Fernandez and J. P. Hershey, J. Solution Chem.,1987, 16 ,269.
[5] J. G. Mathieson and B. E. Conway, J. Chem. Soc Faraday Trans.1, 1974, 70 ,752.
[6] B. B. Owen and H. L. Simons, J. Am. Chem. Soc.,1957, 61 ,479
[7] Transition metal chlorides(aq); A. Lo Surdo and F. J. Millero, J. Phys. Chem.,1980, 84 ,710.
[8] Nitroammino cobalt(III) complexes(aq); F. Kawaizumi, K. Matsumoto and H. Nomura, J. Phys. Chem., 1983, 87 ,3161.
[9] Sodium nitrate(aq) and sodium thiosulfate(aq); N. Rohman, and S. Mahiuddin, J. Chem. Soc. Faraday Trans.,1997, 93 ,2053.
[10] Bipyridine and phenanthroline complexes of Fe(II), Cu(II), Ni(II) and Cu(II) chlorides(aq); F. Kawaizumi, H. Nomura and F. Nakao, J. Solution Chem.,1987, 16 ,133
[11] R. Buwalda, J. B. F. N. Engberts, H. Høiland and M. J. Blandamer, J. Phys. Org. Chem., 1998, 11 , 59.
[12] E. Ayranci and B. E. Conway, J. Chem. Soc. Faraday Trans.1, 1983, 79 ,1357.
[13] G. Peron, G. Trudeau and J.E.Desnoyers, Can J.Chem.,1987, 65 ,1402.
[14] J. I. Lankford, W. T. Holladay and C. M. Criss, J. Solution Chem.,1984, 13 ,699.
[15] J. I. Lankford and C. M. Criss, J. Solution Chem.,1987, 16 ,753.
[16] D. S. Gill, P. Singh, J. Singh, P. Singh, G. Senanayake and G. T. Hefter, J. Chem. Soc., Faraday Trans., 1995, 91 , 2789.
[17] J. C. R. Reis and M. A. P. Segurado, Phys.Chem.Chem.Phys.,1999, 1 ,1501.
[18] M. J. Blandamer, J. Chem. Soc. Faraday Trans.,1998, 94 ,1057.