Born equations

The polarization energy may be calculated with the help of a Born model, if we know the dielectric constant c. A charged sphere with radius R gives rise to a field F(r) outside the sphere  

The difference in field energy between a sphere in vacuum = 1) and a sphere in a dielectric medium with dielectric constant index c is then 

Such a situation has been thoroughly reported in solid state physics books on dielectrics subjected to external electric fields [3] and may be applied directly here to aqueous solutions. Consider, for example, a pair consisting of a cation and an anion (each of charge - - q and — q) that are freshly formed in vacuum. Suppose their centers are separated by a distance r. Then, Coulomb s law of electrostatics gives the change in electrostatic energy of attraction as 

e (= 1) is the dielectric constant in vacuum. If these ions are in an aqueous solution, because of the rearrangement of the polar molecules around these ions, one needs to assign a new dielectric constant (82) that is characteristic of this new medium. The net 

Pictorial representation of formation of chemically bonded phosphate ceramic. 

---

The generalised Born equation has been incorporated into both molecular mechan calculations (by Still and co-workers [Still et al. 1990 Qiu et al. 1997]) and semi-empiri quantum mechanics calculations (by Cramer and Truhlar, in an ongoing series of mod called SMI, SM2, SM3, etc. [Cramer and Truhlar 1992 Chambers et al. 1996]). In th( Ireafirients, the two terms in Equation (11.61) are combined into a single expression of 1 following form ... 

When one chooses a radical as the reference system, the stability of a carbocation or carbanion can be defined by the free-energy change for discharging the ion in vacuum, and the change can be approximately described by the classical Born equation (3) (Bom, 1920), provided that the ion is represented by a conducting sphere on which the charge is located. 

The Born equation thus derived is based on very simple assumptions that the ion is a sphere and that the solvents are homogeneous dielectrics. In practice, however, ions have certain chemical characters, and solvents consist of molecules of given sizes, which show various chemical properties. In the simple Born model, such chemical properties of ions as well as solvents are not taken into account. Such defects of the simple Born model have been well known for at least 60 years and some attempts have been made to modify this model. On the other hand, there has been another approach that focuses on short-range interactions of an ion with solvent molecules. 

In the Born equation, the ion solvent interaction energy is determined only by one physical parameter of the solvent, i.e., the dielectric constant. However, since actual ion-solvent interactions include specific interactions such as the charge-transfer interaction or hydrogen bonds, it is natural to think that the Born equation should be insufficient. It is well known that the difference in the behavior of an ion in different solvents is not often elucidated in terms of the dielectric constant. 

As seen in Table 2, the order of the magnitude of rjj for alkali metal ions is the reverse of that of the magnitude of r. This means that a more hydrophilic ion has a larger rji. However, this fact does contradict the expectation from Bornian electrostatic theories. As can be seen in the Born equation [Eq. (2)], it is expected that the larger the radius an ion has, the more positive the value the ion has, that is, the more hydrophobic it... 

The Born equation for the Gibbs energy of solvation is thus... 

Although the Born equation is a rough approximation, it is often used for comparison of the solvation effects of various solvents. The simplification involved in the Born theory is based primarily on the assumption that the permittivity of the solvent is the same in the immediate vicinity of the ion as in the pure solvent, and the work required to compress the solvent around the ion is neglected. 

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),... 

Ionic AHelectrostatiC Obtained with The Born Equation In The Form of Eq. (42) Compared To Experimental AHhydration, In kcal/mole"... 

A second, more detailed approach was then investigated. From Scheme 1 it is clear that in order to calculate the solubility product, KB, it is necessary to know both the free energy of solution, AG9sol(A+,X ) and the lattice energy, Uc. Abraham s recent developments [9] of the Born Equation [10] make it possible to calculate AG9sol to within 10 kj mol1 from the ionic radii a+, a ), but it is not possible to calculate Uc... 

These points indicate that the continuum theory expression of the free energy of activation, which is based on the Born solvation equation, has no relevance to the process of activation of ions in solution. The activation of ions in solution should involve the interaction energy with the solvent molecules, which depends on the structure of the ions, the solvent, and their orientation, and not on the Born charging energy in solvents of high dielectric constant (e.g., water). Consequently, the continuum theory of activation, which depends on the Born equation,fails to correlate (see Fig. 1) with experimental results. Inverse correlations were also found between the experimental values of the rate constant for an ET reaction in solvents having different dielectric constants with those computed from the continuum theory expression. Continuum theory also fails to explain the well-known Tafel linearity of current density at a metal electrode. ... 

Born equation Jphys chemJ An equation for determining the free energy of solvation of an ion in terms of the Avogadro number, the ionic valency, the ion s electronic charge, the dielectric constant of the electrolytic, and the ionic radius. born i kwa zhan J... 

Extraction processes that proceed according to the model of ion-pair extraction are described by a formalism different from that presented in section 16.4.2, and are based on partition of single ions and their association in the organic phase [76] (see also section 2.6). The Born equation has been widely used to describe the transfer of an ion of the charge q and radius r from vacuum to the liquid (water) of the dielectric constant e ... 

The solvation energy described by the Born equation is essentially electrostatic in nature. Born equations 8.116 and 8.120 are in fact similar to the Born-Lande equation (1.67) used to define the electrostatic potential in a crystal (see section 1.12.1). In hght of this analogy, the effective electrostatic radius of an ion in solution r j assumes the same significance as the equilibrium distance in the Born-Lande equation. We may thus expect a close analogy between the crystal radius of an ion and the effective electrostatic radius of the same ion in solution. 

In the Coulomb term, the electrostatic and solvation effects depending on the polarity of the media are summarized. For solvent-separated ion pairs (solvation energy calculated by the Born equation), it is given by... 

This crude analysis is based on the behavior postulated by the Born equation. However, ion-pair formation equilibrium constants have been observed to deviate ma edly from that behavior (22/ -222)1 Oakenful, and Fenwick (222) found a maximum in the ion-pair formation constants of several alkylamines with carboxylic acids when determined at various methanol-water solvent compositions as shown by their data in Fig. 54. The results demonstrate that in this system the stability constant decreases with increasing organic solvent concentration above a.critical value which yields maximum stability. The authors suggested that this was due to a weakening of hydrophobic interactions between the ion-pair forming species by increased alcohol concentrations. In practice the effect of added organic solvent has been either to decrease the retention factor or to have virtually no effect. 

Also, the dielectric constant of a solvent near an ion may differ from that of the bnlk hqnid. Althongh the Born equation is only an approximation, it does indicate that free energies of solvation of ions will be larger as the solvent dielectric constant increases. 

---