Gibbs energy of an ion

The Gibbs energy of an ion changes on transfer from one solvent to another primarily because the electrostatic interaction between ions and the medium changes as a result of the varying dielecric constant of the solvent. This can be expressed roughly by the Born equation (see Eq. 1.2.7),... 

If there were no electrostatic interactions the behavior of ions would be ideal and the Gibbs energy of an ion of type i would be given by the relationship ... 

The Figure 13 illustrates the effect of image forces on the electrostatic Gibbs energy of an ion at the oil-water interface. 

The Gibbs energy of an electroneutral system is independent of the electrostatic potential. In fact, when substituting into Eq. (3.7) the electrochemical potentials of the ions contained in the system and allowing for the electroneutrality condition, we can readily see that the sum of aU terms jZjF f is zero. The same is true for any electroneutral subsystem consisting of the two sorts of ion and (particularly when these are produced by dissociation of a molecule of the original compound k into x+ cations and x anions), for which... 

This dependence is fundamental for electrochemistry, but its key role for liquid-liquid interfaces was first recognized by Koryta [1-5,35]. The standard transfer energy of an ion from the aqueous phase to the nonaqueous phase, AGf J, denoted in abbreviated form by the symbol A"G is the difference of standard chemical potential of standard chemical potentials of the ions, i.e., of the standard Gibbs energies of solvation in both phases. 

Gibbs transfer energy of an ion i from phase a to p AG g Gibbs energy for ion-solvent interaction in phase a A log P partition coefficient difference between two solvent systems A 0 Galvani potential difference between a and p phases Ag(pi/2 half-wave potential... 

The relation between the standard molar Gibbs energy of hydrogen ion and that of hydronium ion is obtained by two methods that differ only in the assumption used. The result is the same. In the first method we choose to define the acid species either as hydrogen ion or as hydronium ion, and we do not consider an equilibrium between the two species and the solvent. In this case the chemical potential of the hydrogen ion is related to that of the hydronium ion in aqueous solution by... 

At a microscopic level, solutes in polar solvents undergo strong solvation. For example, the Born model predicts that the Gibbs free energy of an ion with charge q (in Coulombs) and radius r will be changed in the solvent compared with the gas phase by an amount... 

The treatment starts from a generalized virial expansion for Gxs, the total Gibbs energy of the solution minus the Gibbs energy of an ideal solution of the same composition. Although the virial coefficients are not individually measurable, measurable combinations of the virial coefficients have been identified - —/. Differentiation of Gxs with respect to the amount of water allows computation of the activity of the water and differentiation with respect to the amount of an ion yields the activity coefficient of that ion. 

Within the framework of the same dielectric continuum model for the solvent, the Gibbs free energy of solvation of an ion of radius and charge may be estimated by calculating the electrostatic work done when hypothetically charging a sphere at constant radius from q = 0 q = This yields the Bom equation [13]... 

The standard Gibbs energy of electrolyte transfer is then obtained as the difference AG° x ° = AG° ° - AG° x. To estabfish the absolute scale of the standard Gibbs energies of ion transfer or ion transfer potentials, an extrathermodynamic hypothesis must be introduced. For example, for the salt tetraphenylarsonium tetraphenyl-borate (TPAs TPB ) it is assumed that the standard Gibbs energies of transfer of its ions are equal. 

By varying the scan rate, this equation allows then the evaluation of the diffusion coefficient of the transferring ion. With the determination of the formal transfer potential of an ion and thus of its Gibbs energy of transfer by application of Eq. (10), this is the most important application of cyclic voltammetry. 

The difference between the electronic energies of the final and initial states must include the energy of ionization of the ion B(z-1)+ in vacuo (where its ionization potential is complemented by the entropy term TA5/), the interaction energy of the ions Bz+ and B(z-1)+ with the surroundings, i.e. the solvation Gibbs energies, and finally the energy of an electron at the Fermi level in the electrode. These quantities can be expressed most simply... 

The Gibbs energy AG°d depends on the electrode potential . This dependence will be different for anions, cations, and neutral species. The simplest possible case is the adsorption and total discharge of an ion according to the equation ... 

Application of an overpotential r] changes the Gibbs energy for the ion transfer by an amount ze r/, where z is the charge number of the... 

Figure 9.2 Gibbs energy for the transfer of an ion from the solution to the electrode surface.

There are other ways of estimating inner potential differences. Gi rault and Schiffrin [4] assume that the difference in the inner potential is negligible at the pzc, because the interface consists of an extended layer where both solvents mix, so that any dipole potentials will be small. The resulting scale of Gibbs energies of transfer agrees reasonably well with the TPAs+/TPB scale, if the small difference in the radii of these ions is accounted for. 

---