Born model, assumptions

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. 

An alternative strategy is to synthesize a molecular wave function, on chemical intuition, and progressively modify this function until it solves the molecular wave equation. However, chemical intuition fails to generate molecular wave functions of the required spherical symmetry, as molecules are assumed to have non-spherical three-dimensional structures. The impasse is broken by invoking the Born-Oppenheimer assumption that separates the motion of electrons and nuclei. At this point the strategy ceases to be ab initio and reduces to semi-empirical quantum-mechanical simulation. The assumed three-dimensional nuclear framework is no longer quantum-mechanically defined. The advantage of this model over molecular mechanics is that the electron distribution is defined quantum-mechanically. It has been used to simulate the H2 molecule. 

Although polyhalide anions are most frequently studied in aqueous solution, they are in fact more stable in less polar solutions. For example, D(l2-I )=17 kJ/mol in aqueous solution, and 47 kJ/mol in acetone. While halide anions are spherical, the trihalide anions are oblate. The Born model is still reasonably accurate for these systems if a spherical ion approximation is used with either an experimental value of the radius or with the assumption that the volume of X is simply three times the volume of X . However, the model does not work for the pentahalide ions IJ and Br. Thus, application of the Born model to larger anions is an oversimplification. As more data become available, it may be possible to use more sophisticated models of solvation to correlate gas- and solution-phase data. 

In contrast to this, the adsorption of ions obeying GC theory is termed as nonspecific adsorption. It is important to emphasize that the distinction between specific and nonspecific adsorption is based on an arbitrarily chosen model assumption. It should, however, be borne in mind that it is impossible, by means of thermodynamic arguments, to determine the extent of the contribution of the specific adsorption to the overall surface excess concentration and to the charge density. 

The Born model is only a rough approximation. Improvements of the method take into account a local permittivity e and effective ionic radii fl= a -I- 5 , where Si is the distance between an ion and an adjacent solvent dipole. More elaborate models include in the calculation the energy of formation of a spherical cavity in the pure solvent into which an ion and its solvation shells can be transferred from the vacuum. Further interactions that can be taken into account result from ion-quadrupole, ion-induced dipole, dipole-dipole, dispersion, and repulsion forces. For nonaqueous electrolyte solutions most of the molecular and structural data needed for this calculation of the solvation energy are unknown, and ab initio calculations have not so far been very successful. Actual information on ion solvation in nonaqueous solutions is based almost exclusively on semiempirical methods and/or the extrathermodynamic assumptions quoted in Section II.C. 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

It should also be mentioned that a theoretical model using an empirical LEPS potential energy surface has successfully been used to reproduce the vibrational population distribution of the products of this surface reaction.40 This approach confines itself to the assumptions of the Born-Oppenheimer approximation and underscores one of the major questions remaining in this field do we just need better Born Oppenheimer potential surfaces or do we need a different theoretical approach ... 

The Langmuir model first assumes the adsorption sites are energetically identical. Actually, this assumption is not borne out when adsorption occurs predominantly by physisorption. The spread of A/Tads values between the various sites can be as high as 2 kJmol-1, which is often a significant fraction of the overall enthalpy of adsorption when physisorption is the sole mode of adsorption. By contrast, energetic discrepancies between sites can be ignored when adsorption occurs by chemisorption. 

The bias observed between experimental measurements and Kieffer s model predictions is due to the relative paucity of experimental data concerning cutoff frequencies of acoustic branches, and also to the assumption that the frequencies of the lower optical branches are constant with K and equivalent to those detected by Raman and IR spectra (corresponding only to vibrational modes at K = 0). Indeed, several of these vibrational modes, and often the most important ones, are inactive under Raman and IR radiation (Gramaccioli, personal communication). The limits of the Kieffer model and other hybrid models with respect to nonempirical computational procedures based on the equation of motion of the Born-Von Karman approach have been discussed by Ghose et al. (1992). 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

As fine particles arise from many sources, it would be desirable to replace the fine particle mass concentration in the equations by the concentration of an element borne by the fine particles from coal combustion and no other source. The best candidate for such an element is Se (2.4.17). If coal-fired power plants were the only significant source of Se (probably a good assumption in many areas), one could measure emission rates of SO2, SO4 and Se from the source and their concentrations at a downwind location and plug the values into the equations and solve them to obtain the conversion and deposition rates averaged over the travel time of the plume. The model is a useful first step towards the use of... 

The nature of the solvent is represented in the Born equation only by the static dielectric permittivity e, which includes all types of polarization. The Born assumption on the structureless nature of the solvent may be only approximately fulfilled in the case when the ion is cdhsiderably larger than the solvent molecules. The problem of ion size and the applicability of the dielectric model has been discussed by Evans et al. [23]. 

Since Eq. (28) was obtained under assumptions similar to those used by Born, the calculation of AGq suffers from the same limitations as the Born solvation model. The dielectric continuum model is valid for electron transfer in a structureless dielectric medium with a reactant approximated by a hard conducting sphere. It is obeyed when the specific solute-solvent interactions are negligible. 

Comparing the leading order term in (15.75) with the step value in (15.74), we see that the inverse formula provides the correct estimate of a step, b, with the accuracy determined by the leading term in the representation (15.75). However, Bleistein inversion is based on a Born approximation formula which is valid only to leading-order in 6cg. Thus the theoretical inversion result for this simple model fits well the basic assumption of the Bleistein method. 

The IPAH model incorporated a number of factors that can modify the toxicity of the sediment-borne PAHs. Equilibrium partitioning was used to estimate the concentration of each PAH in the pore water of the sediment. The assumption was that the pore water material is the fraction that is bioavail-able. QSAR was also used to estimate the interstitial water concentration based on the octanol-water partition coefficient of several PAHs. Amphipods were used as the test organism to represent environmental toxicity. A toxic unit (TU) approach was used and the toxicity is assumed to be additive. The assumption of additivity is justified since each of the PAHs has a similar mode of action. Finally, a concentration-response model was formulated using existing toxicity data to estimate the probability of toxicity. 

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