Born equation parameter values

Table IV. Values of Born Equation Parameters, Cellulose Acetate (Eastman E-398)...

It should be dear that if the Born equation is not corrected in any way, there is no doubt as to its failure. According to eqn. 2.11.29 it follows that AG (ion) will always be positive for Sr > This is not believed to be true for many solvents as discussed below. Conversely for eqn. 2.11.29 predicts a negative value for AGf(ion), which is also questionable. Noyes chose to evaluate effective dielectric constants as his variable parameter to correct the Born equation. Using Pauling radii and correcting for non-electrostatic effects, he found that the effective dielectric constant, eff was different for each ion. Latimer et chose to vary the crystal radii by adding constant terms to and r such that 5 + 4=. The d corrections can be adjusted so that the... 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

The techniques collectively termed molecular mechanics (MM) employ an empirically derived set of equations to describe the energy of a molecule as a function of atomic position (the Born—Oppenheimer surface). The mathematical form is based on classical mechanics. This set of potential energy functions (usually termed the force field) contains adjustable parameters that are optimized to fit calculated values of experimental properties for a known set of molecules. The major assumption is, of course, that these parameters are transferable from one molecule to another. Computational efficiency and facile inclusion of solvent molecules are two of the advantages of the MM methods. 

The value of X for a typical polar solvent is approximately two. This equation was introduced earlier in the development of the MSA for ion-solvent interactions (section 3.5). It was seen that the MSA gives an improved description of ion solvation parameters with respect to the Born model. However, it fails to distinguish between the solvation of cations and anions of the same size. In other words, it fails to distinguish between the short-range chemical interactions which stabilize ions of differing charge. 

Munze (19) has used a Born-type equation to calculate stability constants of In(III) and An(III) complexes of carboxylates as well as other ligands which agreed well with experimental values. His approach was modified by allowing the dielectric constant to be a parameter (the "effective" dielectric constant, De) in an analysis of fluoride complexation by M(II), M(III) and M(IV) cations (20). A value of De = 57 was found satisfactory to calculate trivalent metal fluoride stability constants which agreed with experimental values for Ln(III), An(III) and group IIIB cations (except Al(III). Subsequently, the equation was used... 

The force-field method involves the other part of the Born-Oppenheimer approximation, that is the positioning of the nuclei. The electronic system is not considered explicitly, but its effects are of course taken into account indirectly. This method is often referred to as a classical approach, not because the equations and parameters are derived from classical mechanics, but rather because it is assumed that a set of equations exist which are of the form of the classical equations of motion. The problem from this point of view is one of establishing just which equations are necessary, and determining the numerical values for the constants which appear in the equations. In general there is no limit as to what functions may be chosen or what parameters arc to be used, except that the force-field must duplicate the experimental data. 

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