The Born Model of Solvation

Since ionic compounds, when dissolving, produce a cationic and anionic pair, the thermodynamics of ionic solutions becomes complicated in that values for a single ion are unknowable. Conventional values are obtained by assigning ∆_fG°[H⁺]=0 for all temperatures and then calculating every other ion value with respect to that reference. This process works well in Hess' Law-type calculations as long as we don't assign meaning to individual ion values (particularly true for negative S_f° values).

Though ultimately less successful, we can gain a lot of insight into the solvation process by attempting to derive the thermodynamic values from scratch, that is, from first principles. In the Born model, we imagine a hard-sphere of radius r being charged up from 0 to a final value q. The difference between the work of charging this ion in the medium of choice and a vacuum will be taken as ∆_solvG_i° for ion i.

Recalling that the infinitesimal work of charging is dw=φdq and the electric potential is φ=q/4πεr, we obtain for 1 mole of ions, after integrating and substituting q=ze:

∆_solvG_i°=z²e²N_A/8πε_or(1/ε_r-1)

Since the dielectric constant is greater than 1 for all materials, the solvation of any ion is a spontaneous process in any solvent.

Focusing on 298K water as the solvent leads us to gibbs energy of hydration:

∆_hydG°=-68.6z²/r

A test of the Born model can be obtained by plotting experimental ∆_hydG° values versus z²/r for a bunch of ions and see whether or they not they track along a line of slope –68.6. As seen in class, most univalent ions generally agree with the Born results but polyvalent ions deviate appreciably, especially as the valence z gets large.

This is a good example of how physical chemistry works: Propose a model that is as simple as possible and see how much of the chemical behavior is captured. When it fails for some cases, go back and examine the assumptions (and tweak and repeat). In this model, the assumption of the aqueous solvent being structureless (that is, a continuum) is likely the culprit. Indeed, we saw that many of the high-valence ions seem to be tracking along a line in which the dielectric constant was approximately 2 (indicative of a case in which the mobility of water has been decreased to the point that it acts effectively as a nonpolar substance). An additional deviation from the Born model, not discussed much in class, arises from ions that are highly polarizable and would therefore not be adequately treated as a hard-sphere.

Many models exists treating the solvent as a molecular entity exist and experiments and simulations support the theory that an ion in solution produces a hydration sphere encompassing 4-8 water molecules (the actual number being a function of ionic size). Indeed, femtosecond laser studies provide hydration lifetimes for water molecules in the presence of a particular ion. Some ions with high charge densities have very long lifetimes, supporting the idea that water in those cases has been effectively locked immobilized.

Though the underlying explanation has not been fully developed, we can experimentally verify that some ions in solution increase the viscosity when compared to water. These kosmotropic ["structure-making"] ions include Li⁺, Na⁺ and F^-. Analogously, viscosity is decreased by chaotropic ["structure-breaking"] ions like Rb⁺, Cs⁺ and NO₃^-.

On Wednesday, a brief return to colligative properties (to adjust them for ionic solutions) before tackling chemical potentials of ionic solutions, activity and the Debye-Huckel theory.