1.10.16: Gibbs Energies- Salt Solutions- Born Equation

The Born Equation [1] is based on a BBB model, “brass balls in a bathtub” [2]. The solvent is treated as a dielectric continuum characterised at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) by its relative permittivity \(\varepsilon_{r}\). The ions are treated as hard non-polarizable spheres, having radius \(r_{j}\). The Born Equation describes the difference in thermodynamic properties of a mole of i-ions forming a perfect gas and a mole of j-ions in an ideal solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The calculation is not straightforward [3-5]. What emerges is the difference in Helmholtz energies (at constant \(\mathrm{T}\) and \(\mathrm{V}\)) for a mole of \(j\) ions in incompressible liquid phases. The calculated quantities refer to the energies associated with the electric fields over the limits \(r_{j} \leq r \leq \infty\). In these terms the Born Equation describes the electrical part of the change in chemical potential on transferring an ion from the gas phase, permittivity \(\varepsilon_{0}\), to a solvent, relative (electric) permittivity \(\varepsilon_{r}\). In effect the Born Equation yields parameters characterizing the difference between the properties of one mole of \(j\) ions in ideal systems having equal concentrations at fixed \(\mathrm{T}\) and \(\mathrm{p}\) [6].

\[=-N_{A} \,\left(z_{j} \, e\right)^{2} \,\left[1-\left(1 / \varepsilon_{r}\right)\right] / 8 \, \pi \, r_{j} \, \varepsilon_{0}\]

Similarly for transfer of one mole of \(j\) ions from an ideal solution in solvent \(\mathrm{s}_{1}\) to an ideal solution in solvent \(\mathrm{s}_{2}\), the transfer chemical potential is given by the Born Equation assuming \(r_{j}\) is independent of solvent.

\[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \,\left[\left(1 / \varepsilon_{\mathrm{r}}(\mathrm{s} 2)\right)-\left(1 / \varepsilon_{\mathrm{r}}(\mathrm{s} 1)\right)\right] / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\]

Many attempts have been made to modify the Born Equation in order to attain agreement between theory and measured thermodynamic ionic properties, particularly in the case of aqueous salt solutions. A common concern is the extent to which near-neighbor water molecules form an electrostricted layer [7] around ions in solution, namely a layer of solvent molecules having dielectric properties which differ from those of the pure solvent at the same temperature and pressure [8 - 11]. A common concern in this subject is the definition of the ionic radius for a given ion [12 - 16]. There is no agreement concerning a set of ‘absolute’ ionic radii. As Conway pointed out in 1966, ‘ …theories .. based on the Born equation seem to have reached an asymptotic level of usefulness..’ [17].

Nevertheless correlations involving thermodynamic properties of salt solutions play an important role.

Standard partial molar entropies for alkali metal halides in various solvents are linear functions of the corresponding entropies in aqueous solution [18]. A similar correlation is reported for ionic entropies in mixed aqueous solvents and the corresponding entropies in aqueous solutions [19]. With reference to enthalpies, the analysis also suffers from the fact that the ionic radius is sensitive to temperature [20].

Footnotes

[1] M. Born, Z. Phys., 1920, 1, 45.

[2] H. S. Frank quoted by H. L. Friedman, J. Electrochem. Soc., 1977, 124, 421c.

[3] H. S. Frank, J. Chem. Phys., 1955, 23, 2023.

[4] J. E. Desnoyers and C. Jolicoeur, Modern Aspects of Electrochem, ed. B. Conway and J.O’M. Bockris.1969, 5,1.

[5] J. E. Desnoyers, R. E. Verrall and B. E. Conway, J. Chem. Phys., 1965,43, 243.

[6]

\[\begin{aligned}
& Delta(\mathrm{pfg} \rightarrow \mathrm{s} \ln ) \mu_{\mathrm{j}}=\left[\mathrm{mol}^{-1}\right] \,[\mathrm{C}]^{2} \,\{[1]-[1]\} /[1] \,[1] \,[\mathrm{m}] \,\left[\mathrm{Fm}^{-1}\right]= \\
&{[\mathrm{mol}]^{-1} \,\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] /[\mathrm{m}] \,\left[\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-3}\right]=\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{2} \mathrm{~s}^{-2}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]}
\end{aligned}\]

[7] M. H. Abraham, E. Matteoli and J. Liszi, J. Chem. Soc. Faraday Trans.I, 1983, 79,2781; and references therein.

[8] D. R. Rosseinsky, Chem. Rev.,1965,65,467; and references therein.

[9] A. A. Rashin and B. Hornig, J. Phys. Chem., 1985, 89,5588.

[10] T. Abe, Bull. Chem. Soc. Jpn.,1991, 64,3035.

[11] S. Goldman and R.G.Bates, J.Am.Chem.Soc.,1972,94,1476.

[12] L. Pauling, Nature of the Chemical Bond, Cornell University Press, Ithaca, 3rd edn., 1960, chapter 8.

[13] M. Bucher and T. L. Porter, J. Phys. Chem.,1986,90,3406; and references therein.

[14] M. Salomon, J.Phys.Chem.,1970,74,2519.

[15] K. H. Stern and E. S. Amis, Chem. Rev.,1959,59,1.

[16] Y. Marcus, Chem. Rev.,1988,88,1475.

[17] B. E. Conway, Annu. Rev. Phys. Chem.,1966,17,481.

[18] C. M. Criss, R. P. Held and E. Luksha, J.Phys.Chem.,1968,72,2970.

[19] F. Franks and D. S .Reid, J. Phys.Chem.,1969,73,3152.

[20] B. Roux, H.-A. Yu and M. Karplus, J. Phys. Chem.,1990, 94,4683.