Encounter pairs

If, on the other hand, the encounter pair were an oriented structure, positional selectivity could be retained for a different reason and in a different quantitative sense. Thus, a monosubstituted benzene derivative in which the substituent was sufficiently powerfully activating would react with the electrophile to give three different encounter pairs two of these would more readily proceed to the substitution products than to the starting materials, whilst the third might more readily break up than go to products. In the limit the first two would be giving substitution at the encounter rate and, in the absence of steric effects, products in the statistical ratio whilst the third would not. If we consider particular cases, there is nothing in the rather inadequate data available to discourage the view that, for example, in the cases of toluene or phenol, which in sulphuric acid are nitrated at or near the encounter rate, the... 

A careful study of isomer distributions might thus provide information about the structure of the encounter pair. 

One interesting proposal is that the encounter pair is a radical pair N02ArH + formed by an electron transfer (SET), which would explain why the... 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

If the interaction between the donor and acceptor in the encounter pair is strong (Scheme 4.3), this encounter pair (DA) is called an exciplex (see Section 4.4). 

Any weakly attractive, short-lived complex that is typically formed as an intermediate in a reaction mechanism. When there are only two such molecular entities engaged in the formation of a particular encounter complex, that complex is often called an encounter pair. lUPAC (1979) Pure and Appl. Chem. 51, 1725. 

There are at least three possible mechanisms for the spontaneous breakdown of hemiorthoesters, hemiacetals, and related species. Firstly, there may be a rapid and reversible ionization equilibrium followed by hydronium-ion catalysed breakdown of the anion (9) (Gravitz and Jencks, 1974). A necessary condition for this mechanism to be valid is that k2 calculated from kHi0 and Ka should fall below the diffusion controlled limit of c. 10loM 1s 1. The second mechanism (10) is similar to this but involves formation of the anion and hydronium ion in an encounter pair which react to give products faster than the diffuse apart (Capon and Ghosh, 1981). With this mechanism therefore the ionization equilibrium is not established and the rate constant for... 

One interesting proposal86 is that the encounter pair is a radical pair N02 ArH formed by an electron transfer (SET), which would explain why the electrophile, once in the encounter complex, can acquire the selectivity that the free N02+ lacked (it is not proposed that a radical pair is present in all aromatic substitutions only in those that do not obey the selectivity relationship). The radical pair subsequently collapses to the arenium ion. There is evidence87 both for and against this proposal.88... 

Reverting to eqn. (1), the overall rate coefficient for the formation of products, providing that the concentration of the encounter pair (AB) is small and assuming a steady state, is kAka/(k-d + ka). When the reaction of the encounter pair to form products becomes fast compared with the diffusive forward and backward processes, the rate of formation of products is determined by the rate at which A and B can diffuse together to form the encounter pair (AB). The opposite situation is where the rate of reaction of the encounter pair is very slow compared with the backward diffusion (ka < fe d), and the products form at a rate... 

This is the inner boundary condition. It has two serious flaws. The reaction between A and B may not occur at a rate very much faster than the reactants can approach one another. As was discussed in Sect. 3.1, this can lead to an appreciable probability of formation of the species (AB), which can be better described as an encounter pair. This difficulty was neatly handled by Collins and Kimball [4] and is discussed in Sect. 4 and Chap. 8 Sect. 2.4. The other flaw is the specification of one definite distance at which reaction occurs, the encounter distance. Even if the reaction proceeds with similar rates when the separation distance varies by 0.1nm (the largest likely variation of bond distance), this will be a small variation compared with the encounter distance, which is typically >0.5nm. Means to circumvent this difficulty are discussed in Chap. 8 Sect. 2.4 and Chap. 9 Sect. 4. 

It is convenient to label the relative slowness of encounter pair reaction as due to an activated process and to remark that the chemical reaction (proton, electron or energy transfer, bond fission or formation) can be activation-limited. This is an unsatisfactory nomenclature for several reasons. Diffusion of molecules in solution not only involves a random walk, but oscillations of the molecules in solvent cages. Between each solvent cage in which the molecule oscillates, a transformation from one state to another occurs by passage over an activation barrier. Indeed, diffusion is activated (see Sect. 6.9), with a typical activation energy 8—12 kJ mol-1. By contrast, the chemical reaction of a pair of radicals is often not activated (Pilling [35]), or rather the entropy of activation... 

When the activation process is comparable with or slower than the rate of approach of reactants to form encounter pairs, it is no longer satisfactory to say that the reactants can not co-exist within a distance R of one another. Because the rate of reaction, /eact, of the activation process is finite, so too is the lifetime (and hence concentration) of encounter pairs non-zero. The inner boundary condition, which describes reaction of A and B together in the diffusion analysis, is unsatisfactory. Collins and Kimball [4] suggested an alternative boundary condition and the remainder of this section analyses their work following Noyes [5]. Firstly, the boundary condition is developed and then included in the diffusion equation analysis to obtain the density distribution. Finally, the rate coefficient is obtained. 

The width of the encounter pair reactivity zone, 672, is to be considered small. There is no reason for this choice, save convenience. Probably rather larger widths would be more appropriate following work on gas-phase collision kinetics or long-range transfer processes (Chap. 4). In such circumstances, the partially reflecting boundary condition is no longer suitable and other techniques have to be used (see Chap. 8 Sect. 2.4 and Chap. 9 Sect. 4). 

The second term on the right-hand side contains the dependence on feact. In the limit of very fast encounter pair reaction, eqn. (24) reduces to eqn. (21). 

To find the second-order rate coefficient for the reaction of A and B subject to the encounter pair reacting with a rate coefficient feact, the method developed in Sect. 3.7 can be used. Using eqn. (19), the rate coefficient, k(t), can be defined in terms of the diffusive current of B towards the central A reactant. But the partially reflecting boundary condition (22) equates this to the rate of reaction of encounter pairs. The observed rate coefficient is equal to the rate at which the species A and B could react were diffusion infinitely rapid, feact, times the probability that A and B are close enough together to react, p(R). 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

In Sect. 2, a few experimental results were mentioned which strongly indicate that some molecular reactions are limited by the rate of approach of reactants to form an encounter pair. There have been many hundreds of studies of the rates of reaction in solution. Some studies are discussed in books and reviews by Grunwald et al. [19], Pilling [35], Sutin [15], Rice and Pilling [39], Hart and Anbar [17], Birks [6] and Lorand [22]. As it is not the purpose of this article to consider all these studies but to select some of the more recent, detailed and interesting studies to compare theory and experiment, the reader is encouraged to consult these articles as well as this review. 

In the preceding sections of this chapter, a theory of diffusion-limited chemical reactions has been described for two cases, (a) where the reaction of the encounter pair is much faster than its formation and (b) where these rates are of comparable magnitude. Some experimental evidence for both these cases has been described. During this discussion, a number of other difficulties in the interpretation of diffusion-limited reactions were indicated. This section details the complications and when they may be expected. The following chapters serve to amplify these comments. Chapter 8 provides a resume and conclusion as well as recommendations for future areas of both experimental and theoretical study. 

This discussion highlights the difficulty of deciding at what separation A and B form an encounter pair and then whether this reacts or separates. Noyes [5] and Wilemski and Fixman [51] have taken the encounter distance to be that separation which, if reduced slightly, will lead to reaction. Where these authors disagree is that Noyes [5] only allows reaction to occur in a very narrow range of separation distances about R (which is the usual assumption) and Wilemski and Fixman [51] assume that any separation distance less than the encounter distance, R, can lead to reaction between A and B and that A and B can diffuse through each other till their centres of mass coincide (Chap. 9, Sect. 4). Neither assumption is good, but the differences in predicted rate coefficients are so small that an experimental test of these theories could not be definitive. 

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