Encounter distance

Number of chains Box edge (A) CPU-time (min) Maximal collision Minimal encountered distance (A)... 

When the probability of finding a quencher molecule within the encounter distance with a molecule M is less than 1, this situation is relevant to static... 

When this probability is equal to 1 (uniform concentration), the reaction is of pseudo-first order. This is the case, for example, in photoinduced proton transfer in aqueous solutions from an excited acid M (=AH ) (see Section 4.5) M is always within the encounter distance with a water molecule acting as a proton acceptor, and thus proton transfer occurs effectively according to a unimolecular process. This is also the case of photoinduced electron transfer in aniline or its derivatives as solvents an excited acceptor is always in the vicinity of an aniline molecule as an electron donor. In both cases, the excited-state reaction occurs under non-diffusive conditions and is of pseudo-first order. 

The results obtained in Ref 30 for partially diffusion-controlled recombination show that the field dependence of the recombination rate constant is affected by both the reaction radius R and the reactivity parameter p [cf. Eq. (33)]. Depending on their relative values, the rate constant can be increased or decreased by the electric field. The latter effect predominates at low values of p, where the reactants staying at the encounter distance are forced to separate by the electric field. 

Energetic considerations based on the separation of solvated ions at the encounter distance a show that solvated ion-pair formation from 1M is sufficiently exothermic in polar solvents to effectively prevent the production of excited singlet states 1M by the reverse process. Table XVIII lists values for free energies AGIM of ion-pair formation in acetonitrile estimated24 from the oxidation and reduction potentials, D/D+ and EA-tA, of donor and acceptor using the relationship... 

When the distance between each A reactant is very large compared with that between each pair of B reactants, at a point about midway between a pair of A reactants, the concentration of B reactants is unlikely to be significantly affected by the presence of the A reactants. Smoluchowski suggested that such B reactants are effectively an infinite distance from the A reactants under discussion. By effectively an infinite distance is meant perhaps 1000 times the molecular diameter or encounter distance R. In this region, the concentration of B reactants at any time during the reaction is very close to the initial concentration, i.e. [B](1000iZ) [B]0 for all time (t > 0). From the definition of the density distribution, eqn. (2), this boundary condition as r - °° is... 

Because reaction between A and B has been presumed to be effectively instantaneous compared with the rate of migration of the reactants, there is no probability of observing A and B when they are close enough to react. Smoluchowski suggested that at a separation distance, r, between A and B equal to the encounter distance, R, the reactants very rapidly react to form products. When A and B are separated by distances larger than this, no bonds can be formed or broken, nor can any energy or an electron be transferred. It is only when the separation distance is equal to the encounter distance that reaction does occur (and the encounter... 

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. 

Fig. 3. The Smoluchowski rate coefficient, eqn. (18), for a diffusion-limited reaction with the mutual diffusion coefficient Z) = 10-9m2s 1 and the encounter distance R = 0.5 nm.

Fig. 4. Green s function, eqn. (21), for a diffusion-limited reaction of A with B. The mutual diffusion coefficient is D = 10-9m2s-1, the encounter distance R = 0.5 nm and initial separation distance is 2 nm. As time proceeds, Green s function progressively diminishes and broadens.------, t = 0 ----, f = 0.1 ns - - - -, t = 1 ns ., t = 10 ns.

Thus the partially reflecting boundary condition reduces the effective encounter distance by a factor of fcact (4nRD + fcact) 1 for both the steady-state and transient terms in the rate coefficient. 

In Chaps. 3 and 4, estimates of encounter distances and mutual diffusion coefficients from similar experiments to those of Buxton et al. [18] are discussed. The complications to the analysis of diffusion-controlled rate processes in solution when the reactants interact strongly with one another or the reaction can occur over distances much larger than typical encounter distances do not lead to markedly different time-dependent rate coefficient expressions to the Smoluchowski form. Indeed, replacing R in eqn. (29) by an effective encounter distance, Reff, allows the compactness of the Smoluchowski rate coefficient to be extended to other situations. Means of estimating Reff are discussed in Chaps. 3, 4, 5 (Sect. 4.3), 8 (Sect. 2.6) and 9 (Sects. 4 and 6). 

Nemzek and Ware [7] evaluated the mutual diffusion coefficient, D, and encounter distance, J2eff. D was found to be in close agreement with values of molecular diffusion coefficients from independent measurements. The encounter distance was 0.90nm in 1,2-propanediol. There were small... 

Assuming that the true encounter distance is about 1.20 nm (the exact value is not very important), feact > 4 x 109 dm3 mol-1 s-1. The quenching of 1,2-benzanthracene and naphthalene fluorescence by carbon tetrabromide can also be followed by illuminating a solution with light of constant intensity and measuring the fluorescence intensity, /, for a number-of solutions containing different quencher concentrations. These steady-state experiments of Nemzek and Ware [7] are discussed in Sect. 5.5. 

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. 

Since the study of diffusion-limited reactions in solution seeks to discover more about the nature of the reaction path, the nature of the encounter pair, the energetics of the reaction and possibly the rate of reaction of the encounter pair, ftact, it is to be recommended that experimentalists actively seek to measure the diffusion coefficients of the reactants (or similar species), as well as any other parameters which may have an important bearing on the rate coefficient. By so doing, some of the uncertainty in estimating encounter distance may be removed and inconsistencies between diffusion coefficients measured independently and those obtained from an analyses of rate coefficient time dependence may provide valuable insight into the nature of the diffusion process at short distances. 

Estimates of the effective encounter distance, f eff, and diffusion coefficient, D, from quenching of triplet phenanthrene by cupric ions in methanol—water mixtures (Butler and Pilling [200])... 

Hart and Anbar [17] pointed out that effective rate coefficients of solvated electron reactions with many strong oxidants were larger than those implied by the encounter distances for the solvated electron and oxidant by a factor of 1.5—2.0 times. Some of these effective encounter distances are listed in Table 5, together with others from recent work. For... 

Effective encounter distances for reaction of solvated electrons with electron scavengers at room temperature compared with crystallographic encounter distances Unless otherwise noted, the solvent is water (containing an inert electrolyte in some cases). Corrections for ionic interactions according to eqn. (106) were applied and reaction rate coefficient were extrapolated to zero ionic strength (Chap. 3, Sect. 1.6 and 1.7). Many of these studies have been mentioned in Chap. 3, Sect. 2... 

Electron scavenger Crystallographic encounter distance, Ra (nm) Effective encounter distance, i eff (nm)... 

Crystallographic encounter distance 0.3 nm + bond lengths + van der Waals radius. bRef. 17. cRef. 116. dRef. 120. eRef. 164. 

Effective encounter distances for reactions of solvated electrons with electron scavengers at low temperatures, compared with crystallographic encounter distances, from a fit between experiment and eqn. (105)... 

Crystallographic encounter distance 0.3 nm + bond lengths + van der Waals radius. 

The flux J is direct radially, that is angular diffusion of one radical around another leads to no loss of radicals. The current is the integral of the flux over a surface S (= 4irr2i, where r is a unit radial vector) perpendicular to the direction of the flux. If reaction occurs when the radicals are separated by an encounter distance R, then the surface of reactions is the spherical... 

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