Rate constant diffusion control

Now, if Y were as transient as R , one would have approximately Ayr = k, = AlY, which gives smax = 1. This leads to the statistical product distribution [Pr] [Pc] [Py] = 1 2 1. However, for a persistent Y with a small self-termination constant ktY A AtR one obtains Smax l.14 For instance, if the rate constant for the self-termination of Y were AlY = 103 M 1 s 1 and the other rate constants diffusion controlled, Ac = ktR % 109 M 1 s 1 in nonviscous liquids, the selectivity would become Smax 1000, and this means 99.9% cross-reaction. Hence, the selectivity increases with decreasing self-termination rate constant of the persistent species however, this species need not be infinitely long-lived. Further, one notes that [Y]/[R] = Kr/Ky)112 implies [Y] [R] for ktR AlY, that is, the large excess of the persistent radical. Moreover, for kty kc ktR the self-termination terms on the... 

Bimolecular reactions involving radicals typically possess very low activation energies and very high intrinsic rate constants. Diffusion-controlled encounter can occur even in low viscosity solvents. Radicals are known to be involved in some classes of homogeneous catalytic reactions (47). 

It is worth noting that Scheme 15.6 is equivalent to Scheme 15.7, with fej = fea[M], where is the bimolecular association rate constant (diffusion controlled in most excimer formation reactions) and [M] is the concentration of monomer in the ground state k i is the dissociation rate constant, which is usually denoted k ). 

A semiquantitative procedure used to estimate the lifetimes of carbocations and oxocarbenium ions by using diffusion-controlled trapping of the cations by nucleophiles . Ions of intermediate stability react with azide ions at a constant, diffusion-controlled rate and react with water by an activated process. The ratio of the products obtained from the azide path and the water path is dependent on the electronic characteristics of the cation. 

The theory for cyclic voltammetry was developed by Nicholson and Shain [80]. The mid-peak potential of the anodic and cathodic peak potentials obtained under our experimental conditions defines an electrolyte-dependent formal electrode potential for the [Fe(CN)g] /[Fe(CN)g]" couple E°, whose meaning is close to the genuine thermodynamic, electrolyte-independent, electrode potential E° [79, 80]. For electrochemically reversible systems, the value of7i° (= ( pc- - pa)/2) remains constant upon varying the potential scan rate, while the peak potential separation provides information on the number of electrons involved in the electrochemical process (Epa - pc) = 59/n mV at 298 K [79, 80]. Another interesting relationship is provided by the variation of peak current on the potential scan rate for diffusion-controlled processes, tp becomes proportional to the square root of the potential scan rate, while in the case of reactants confined to the electrode surface, ip is proportional to V [79]. 

Equation (1.9) is also one of the most rapid chemical reactions. The second-order rate constant is one of the largest on record, 1.4 x 10" dm3 mol - 1 s - 1 at 25 °C. The reaction rate is diffusion controlled, i.e. the rate depends on the rate of diffusion of the reactants towards each other rather than their chemical characteristics, and there is a reaction every time the reactants meet. 

Note that the size of the particles drops out of the final expression for kr therefore the expression is equally valid for small molecules or colloidal particles so long as the various assumptions of the model apply. This constant describes the rate of diffusion-controlled reactions between molecules of the same size. In Example 13.2 we examine the numerical magnitude of the rate for the process we have been discussing. 

The rates for many of the e aq reactions in Table II are very fast, exceeding 1010M-1 sec.-1, and therefore, may be limited by the rates of diffusion-controlled encounters. The equation from which the diffusion-limited rate constants may be calculated for ionic species is due to Debye... 

First-order release obeying Fick s first law of diffusion the rate constant a controls the release kinetics, and the dimensionless solubility-dose ratio determines the final fraction of dose dissolved [90]. 

The activation rate constant a controlling diffusion within the wells is expressed by the following equation ]84] ... 

If the growth rate is diffusion controlled, q = where A is a constant. Now suppose Lheaerosolenteringthe chamber ismonodisperse thatis,no = uo),where5Cu—no)... 

This relaxation time is given by r = l/ fef[M] + fcb[A.], kf and kb denoting the forward and backward rate constants of the electron transfer. Thus, r is at the most 1 jis because either forward or backward electron transfer is exothermic and then its rate is diffusion controlled or only slightly slower... 

The effect of ionic environment on the rate of diffusion controlled e"aq reactions is revealed when one compares the experimental rate constants with the calculated values. In Table I the highly charged bis-pentacyano cobaltic peroxide (I) is much more reactive than expected for a pentavalent anion (1.11). It has been claimed (19) that polyvalent anions exhibit a lower effective charge in their kinetic behavior than expected from their structural formulae. We have checked the salt effect on the reaction of I + e m and compared it with the NOjf e m reaction. The results presented in Table IV and Figure 1 show that nitrate ions possess a normal salt effect, a result previously obtained by competition kinetics (15, 17). On the other hand, the salt effect of the I + e"aq reaction shows that this bis-pentacyano cobaltic peroxide ion has an... 

Fig. 2-13 An illustration of the spherically symmetric model used for deriving the rates of diffusion controlled reactions. The total flux through the surfaces of spheres concentric with B (—) is constant in the steady state. The concentration of A increases with increasing r for the association reaction and decreases with increasing r for the dissociation reaction.

The rate of diffusion controlled reaction is typically given by the Smoluchowski/Stokes-Einstein (S/SE) expression (see Brownian Dynamics), in which the effect of the solvent on the rate constant k appears as an inverse dependence on the bulk viscosity r), i.e., k oc (1// ). A number of experimental studies of radical recombination reactions in SCFs have found that these reactions exhibit no unusual behavior in SCFs. That is, if the variation in the bulk viscosity of the SCF solvent with temperature and pressure is taken into accounL the observed reaction rates are well described by S/SE theory. However, these studies were conducted at densities greater than the critical density, and, in fact, the data is inconclusive very near to the critical density. Additionally, Randolph and Carlier have examined a case in which the observed diffusion controlled, free radical spin exchange rates are up to three times faster than predicted by S/SE theory, with the deviations becoming most pronounced near the critical point. This deviation was attributed to some sort of solvent-solute clustering effect. It is presently unclear why this system is observed to behave differently from those which were observed to follow S/SE behavior. Possible candidates are differences in thermodynamic conditions or molecular interactions, or even misinterpretation of the data arising from other possible processes not considered. 

The rate constants of the reactions of metal complexes and e (aq) do not correlate with the redox potential of the metal ion, and neither with the affinity of the free ion for the electron. It is therefore inferred that the reaction of e (aq) with a metal complex does not involve a direct metal-ligand bond. It appears that the electron can quickly tunnel out of the solvent cage into an empty, in energy terms, most suitable d orbital. In such a case, the reduction rates with e (aq) approach diffusion control. It was assumed in earlier works that all these rates are diffusion-controlled since the corresponding energies of activation amounted to only 14.6 kJmol ( 3.5 kcalmor ), but subsequent more precise measurements have shown that the reactions with e (aq) are still too slow to be diffusion-controlled. ... 

In the literature of fast reactions we encounter reports as recently as 1959 of second-order rate constants in aqueous solution as large as sec. Hindsight is, of course, better than foresight, but greater attention should have been paid to Debye s theoretical equation (1942) for the specific rate of diffusion-controlled reactions in liquid solution. For the case of two ions recombining, the theoretical maximum specific rate is ... 

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