Modified encounter theory

Although small, this is a principal disadvantage of the simplest integral theory. The near-contact density of the products nonlinear in c is lost in the lowest-order approximation to this parameter. However, the nonzero contribution to this region is provided by a modified encounter theory outlined in Section XII. The chief merit of MET is that the argument of the Laplace transformation of n r,t) in (3.311) is shifted from 1 /td to 1/xd + ck. As a result, in the limit xD = oo we have instead of (3.313) [133] ... 

The modified encounter theory is the only one that gives dilferent forms for S(j) in the trap (Table VIII) and target (Table IX) limits. In the case of the... 

This behavior, inherent to the IET description of either reversible or irreversible transfer, can be eliminated using modified integral encounter theory (MET) [41,44], or an improved superposition approximation [51,126],... 

When this is used in V(R), the resulting transition state rate is equiva-lent to that of free-energy variational transition state theory [5]. Application of the encounter theory shows [30] that the onset of diffusion control occurs at bath densities of several hundred atmospheres. However at these pressures dielectric screening and "solvation effects cannot be neglected and the theory would have to be modified. 

Reactions in solution proceed in a similar manner, by elementary steps, to those in the gas phase. Many of the concepts, such as reaction coordinates and energy barriers, are the same. The two theories for elementary reactions have also been extended to liquid-phase reactions. The TST naturally extends to the liquid phase, since the transition state is treated as a thermodynamic entity. Features not present in gas-phase reactions, such as solvent effects and activity coefficients of ionic species in polar media, are treated as for stable species. Molecules in a liquid are in an almost constant state of collision so that the collision-based rate theories require modification to be used quantitatively. The energy distributions in the jostling motion in a liquid are similar to those in gas-phase collisions, but any reaction trajectory is modified by interaction with neighboring molecules. Furthermore, the frequency with which reaction partners approach each other is governed by diffusion rather than by random collisions, and, once together, multiple encounters between a reactant pair occur in this molecular traffic jam. This can modify the rate constants for individual reaction steps significantly. Thus, several aspects of reaction in a condensed phase differ from those in the gas phase ... 

If the rate of reaction of encounter pairs is comparably fast to the rate of formation of encounter pairs, Collins and Kimball [4] suggested that the slowness of the chemical reaction rate could be incorporated into the theory of diffusion-limited reaction rates by modifying the Smoluchowski [3] boundary condition, eqn. (5), to the partially reflecting boundary... 

The binary-encounter-dipole (BED) model of Kim and Rudd [31] couples the modified form of Mott cross section [32] with the Bom-Bethe theory [27]. BED requires the differential continuum oscillator strength (DOS) which is rather difficult to obtain. The simplest approximate version of BED is the binary-encounter-Bethe (BEB) [31] model, which does not need the knowledge of DOS for calculating the EISICS. 

In dealing with orthogonal polynomials we have encountered Hankel determinants and modified Hankel determinants, which were an important tool in discussing power series and continued fractions. Thus we may expect a close analogy between the theory of orthogonal polynomials and that of continued fractions. This expectation is corroborated by the following theorem. 

Interparticle forces are a determinant factor for most properties of dispersions, including rheological behavior. They are produced by the molecular forces on the surfaces of the particles, due to their nature or to adsorbed molecules, that modify the interface. These are electrical forces arising from charges on the particles and London-van der Waals attraction forces. The role of these forces on suspension stability has been extensively study and is known as the DLVO theory. In addition, sterical forces encountered on dispersions stabilized with nonionic species also exert an important influence on rheological behavior. The nature of these forces will not be considered since they are matters of discussion in Chapters 1-4. However, from a rheological point of view it is impwtant to understand how these factors modify the flow characteristics of dispersions. 

Commonly, gels are obtained by a hydrolysis step followed by a condensation reaction, like in the case of silica and most of the aerogels of catalytic interest. The destabilization of a sol into a gel, which mainly occurs by modifying the pH of the dispersion (see the DLVO theory) can also be performed. The first situation, which is by far the most encountered, is now described. Generally, an alkoxide in solution... 

Simple collision theory can, in fact, be modified and extended to reactions in solution. In solutions, which contain solvated molecules and ions rather than simple molecules or atoms, interactions are known as encounters rather than collisions. It would be expected that encounter rates should be smaller than collision frequencies because the solvent molecules reduce the collision rate between reactants. However, encounters may be more likely than collisions where molecules are trapped in a temporary cage of solvent molecules (Figure 6.20). 

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