The Bom Equation

In the preceding text, particularly in Section 2.15.11, use is made of Bom s equation, a famous classical equation first deduced in 1920. This equation is generally 

from electrostatics, the work done, W, when there is a change of charge Aq 

To find the work ITdone in a real finite buildup of charge, one has to overcome a problem—that y/ itself depends on the degree of charge—and hence express y/ in terms of q. 

It is easy to show that for a conducting sphere, the value of yr is given by 

With this (and the assumptions) as background, one may write for the work to build up a charge q on the sphere  

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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]... 

A. rather complex procedure is used to determine the Born radii a values of which. calculated for each atom in the molecule that carries a charge or a partial charge. T Born radius of an afom (more correctly considered to be an effective Born radii corresponds to the radius that would return the electrostatic energy of the system accordi to the Bom equation if all other atoms in the molecule were uncharged (i.e. if the other ato only acted to define the dielectric boundary between the solute and the solvent). In Sti force field implementation, atomic radii from the OPLS force field are assigned to ec... 

The quantitative theory of ionic reactions, within the limitations of a continuum model of the solvent, is based on the Bom equation for the electrostatic free energy of transfer of an ion from a medium of e = 1 to the solvent of dielectric constant... 

FIG. 1 Modification of the Bom equation by correcting ionic radii. Solid curves for crystal radii, dotted for corrected radii (for cations, Ar = 0.85A for anions, Ar = 0.1A). (From Ref. 14. Copyright 1939 the American Institute of Physics.)... 

A straightforward approach to ionic solvation, particularly for monatomic ions, is by means of the Bom equation,17 Eq. (32), which was introduced in Section III.2.i ... 

The Bom equation does not take into account the mutually polarizing effect that the solute and solvent have upon each other. This can be done, within the framework of continuum solvent... 

For ionic as for molecular solutes (Section III.3), some studies have applied the discrete molecular model to the solvent in the immediate environment of the solute, and treated the remainder as a continuum. This can in principle help to deal with the problem of inner-shell structure as well as that of long-range effects. Thus Straatsma and Berendsen used the Bom equation to correct simulation-obtained free energies of hydration for six monatomic ions.174 This helped in some instances but not in others. 

In section 2.3.3, the Bom equation was introduced in discussing the electrostatic interactions when an ion is transferred from a vacuum to a solvent of dielectric constant e. The Bom equation can be used also to estimate the electrostatic interaction between an anion and a cation in a solution. The equation in this case has the form ... 

In general the results of recent investigations support the thesis that the forces operative in ionic crystals are those, described above, that underlie the Bom equation for the crystal energy and we may feel justified in investigating the further consequences of this postulate. The question of the sizes of ions is studied from this point of view in the following section. 

In the ionic model, binding energies represent the nett effects of attraction between charges on the anions and central cation, repulsion between the anions and all electrons on the cation, and repulsion between the nuclei. The lattice energy, U0, of a binary ionic solid, such as periclase, may be expressed by the Bom equation, one form of which is... 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

Partial molar entropies of ions can, for example, be calculated assuming S (H+) = 0. Alternatively, because K+ and Cl ions are isoelectronic and have similar radii, the ionic properties of these ions in solution can be equated, e.g. analysis of B-viscosity coefficients (Gurney, 1953). In other cases, a particular theoretical treatment which relates solvation parameters to ionic radii indicates how the subdivision could be made. For example, the Bom equation requires that AGf (ion) be proportional to the reciprocal of the ionic radius (Friedman and Krishnan, 1973b). However, this approach involves new problems associated with the definition of ionic radius (Stem and Amis, 1959). In another approach to this problem, the properties of a series of salts in solution are plotted in such a way that the value for a common ion is obtained as the intercept. For example, when the partial molar volumes of some alkylammonium iodides, V (R4N+I ) in water (Millero, 1971) are plotted against the relative molecular mass of the cation, M+, the intercept at M + = 0 is equated to Ve (I-) (Conway et al., 1966). This procedure has been used to... 

The expression given in Eq. (10) for the work assumes that p = 0, where p is the ionic strength of the medium. AG is the free-energy of the equilibrated excited-state (AG AE00), rD and rA are the molecular radii of the donor and acceptor molecules, e5 is the static dielectric constant or permittivity of the solvent, and z is the charge on each ion. ss is related to the response of the permanent dipoles of the surrounding solvent molecules to an external electrical field. Equation (9), the Bom equation, measures the difference in solvation energy between radical ions in vacuo and solution. 

As the non-electrostatic part of AG j cannot be calculated in as simple a manner as AG° several extrapolation procedures using the Bom equation in a modified form have been used. 

Plots of the diverse AG functions versus r often fail to be linear, complicaitiiiig the long range extrapolation, and if the plot is linear the slope differs from the theoretical slope deduced from the Bom equation. This discrepancy was eliminated by adding constant increments 6+ and 6 to the crystal radii, as was shown by Latimer et al. and Strehlow et al. 

We know from S.P.T. calculations that the cavity term will be negative for the transfer from water to water + acetone mixtures (Table IV). According to the Bom Equation AGt° (EL) is positive. Insofar as the separate consideration of a specific and and an electric term is not arbitrary we may note that in order to have AGt° (SPEC) < 0 (preferen-... 

Thus, the Bom Equation predicts a decrease in Gibbs energy of 0.53 kcal mol 1 for the transfer of HBr from water to H20/NMA, oc2 = 0.5, if the radii of the hydrated hydrogen and bromide ions are taken to be 2.8 and 1.96 A, respectively. That the observed decrease is more than three times this value may have a variety of explanations among them are deficiencies in the Bom calculation, increased basicity of the medium through addition of the amide component, and solvent-solvent interactions of an undetermined nature. 

The shape of the cavity affects the difficulty in solving the Poisson equation. If the cavity is a sphere, then an analytical solution is obtained. If the solute is an ion with charge q, we get the Bom equation... 

An alternative to solving the Poisson equation is an approximation based on the Bom equation (Eq. (1.66)). The generalized Bom (GB) equation gives the electronic energy... 

There are a number of fundamental difficulties with the Bom equation, which is still presented in this text because of the prominent part it plays in most theories (it accounts for around one-third of the hydration energy calculated for simple ions). 

Although differing in quantitative aspects, other ab initio calculations using a different model of solvation, based upon the Bom equation, verify that interaction with a medium... 

Transfer activity coefficient The electrostatic, or nonspecific, contribution to the transfer activity coefficient can be obtained by estimating the free-energy change involved in transferring a sphere of radius r and charge Zfi from one solvent to another of different dielectric constant. According to the Bom equation, " when 1 mole... 

Calculations such as in Example 4-1 are but a first estimate and do not allow for spedfic solute-solvent interactions. The calculations cannot be entirely correct because on the molecular level a solvent is not a dielectric continuum the effective dielectric constant near the intense field of an ion is decreased. Furthermore, the equation assumes that ions are spherical, nonpolarizable entities with the charge located at the center. Latimer, Pitzer, and Slanski modified the Bom equation by... 

A comparison of anion solvation by methanol, a protic solvent, and dimethylformamide, a dipolar aprotic solvent, is instructive. The electrostatic contribution, d/i , to the Gibbs free energy of solvation per mole of an ion is sometimes estimated quite successfully (Stokes, 1964) from the Bom model, in which a charged sphere of radius r is transferred from vacuum to a medium of uniform dielectric constant, c. The Bom equation (17) suggests that an anion should be similarly solvated in methanol and in DMF, because these solvents have effectively the same dielectric constant (33-36). The Born equation makes no allowance for chemical interactions, such as hydrogen-bonding and mutual... 

When looking at the basic properties of these amines in aqueous solution we have to consider solvation effects as well because they play a crucial role in stability of these cations in water. The Bom equation (equation 4.9) shows that solvation energy is inversely proportional to the radius of the cations. This means that the lowest solvation energy is going to be observed for the largest cation of the group, that is (CHj)jNH, and the highest value for the ammonium cation (the smallest of the four). 

It has long been recognized that the Bom equation cannot account for differences in the solvation energy of ions of the same size. Various modifications of the Bom equation have been proposed in order to account for the effect of partial dielectric saturation in the near vicinity of ions and or to allow for differences in the size of ions in solution and in a crystal lattice (14—18). The modified equations cannot explain differences in the solvating power of different aprotic solvents as will be shown later. 

The best procedure is probably to use the Bom equation with the lower limit being the distance from the origin to the start of the region corresponding to bulk water. This allows (ii) and (iii) to be assessed separately. 

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