Diffusion, coefficient controlled bimolecular reactions

The bulk polymerization of acrylonitrile in this range of temperatures exhibits kinetic features very similar to those observed with acrylic acid (cf. Table I). The very low over-all activation energies (11.3 and 12.5 Kj.mole-l) found in both systems suggest a high temperature coefficient for the termination step such as would be expected for a diffusion controlled bimolecular reaction involving two polymeric radicals. It follows that for these systems, in which radicals disappear rapidly and where the post-polymerization is strongly reduced, the concepts of nonsteady-state and of occluded polymer chains can hardly explain the observed auto-acceleration. Hence the auto-acceleration of acrylonitrile which persists above 60°C and exhibits the same "autoacceleration index" as at lower temperatures has to be accounted for by another cause. 

Although the extinction coefficient of the benzyl radical was not known, product analysis gave an approximate total yield of benzyl radicals, from whence it could be deduced that e318 a 1,100 l.mole-1.cm-1 and the rate coefficients for the second order processes were 4xl07 and 2x 108 l.mole-1.sec 1 respectively. These rate coefficients are almost equal to those for the diffusion-controlled bimolecular reaction. 

The rate of MV formation was also dependent on pH. The bimolecular rate constant, as calculated from the first order rate constant of the MV build-up and the concentration of colloidal particles, was substantially smaller than expected for a diffusion controlled reaction Eq. (10). The electrochemical rate constant k Eq. (9) which largely determines the rate of reaction was calculated using a diffusion coefficient of of 10 cm s A plot of log k vs. pH is shown in Fig. 24. 

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

Because of the limitations imposed by activity coefficients and specific interactions, a precise quantitative check of experimental data against the collision formula presented here is not possible. However, the frequency factors of bimolecular reactions which are diffusion-controlled (i.e., those which occur on nearly every collision) such as free radical recombinations,... 

Let us examine now the set of equations controlling the creation and annihilation of neutral A and B particles in the Euclidean space which have equal diffusion coefficients D = Dq = D [93]. It has the form of equations (2.2.20) to (2.2.21). Here K is a reaction rate of bimolecular recombination in particular, it can be equal to K = SirDro. Also, and... 

Thus, the kinetics of diffusion-controlled bimolecular electron-transfer reactions in the micellar interiors differ from that in the homogeneous solution. Numerous data have shown that Eq. 9 reproduces the dynamics of electron-transfer reactions within micelle interiors [80]. Diffusion coefficients (D) estimated from Eqs. 8 and 9 are very similar to those obtained by independent measurements. For example, Eq. 8 gave ku = 7.5 X 10 s for electron transfer from excited pyrene to CH2I2 in SDS micelles [79b]. One estimates from Eq. 8, with = 20 A and ai = 1.5 (calculated assuming d = 7 A), a value of Z) = 1.3 x 10 cm s, nearly identical with the experimentally determined value of Z) = lO " cm s [45]. 

The introductory remarks about unimolecular reactions apply equivalently to bimolecular reactions in condensed phase. An essential additional phenomenon is the effect the solvent has on the rate of approach of reactants and the lifetime of the collision complex. In a dense fluid the rate of approach evidently is determined by the mutual diffusion coefficient of reactants under the given physical conditions. Once reactants have met, they are temporarily trapped in a solvent cage until they either diffusively separate again or react. It is common to refer to the pair of reactants trapped in the solvent cage as an encounter complex. If the unimolecular reaction of this encounter complex is much faster than diffusive separation i.e., if the effective reaction barrier is sufficiently small or negligible, the rate of the overall bimolecular reaction is diffusion controlled. 

Reactions that are bimolecular can be affected by the viscosity of the medium [9]. The translational motions of flexible polymeric chains are accompanied by concomitant segmental rearrangements. Whether this applies to a particular reaction, however, is hard to tell. For instance, two dynamic processes affect reactions, like termination rates, in chain-growth polymerizations. If the termination processes are controlled by translational motion, the rates of the reactions might be expected to vary with the translational diffusion coefficients of the polymers. Termination reactions, however, are not controlled by diffusions of entire molecules, but only by segmental diffusions within the coiled chains [10]. The reactive ends assume positions where they are exposed to mutual interaction and are not affected by the viscosity of the medium. 

Just as in the gas-phase, thermodynamics tells only part of the story in respect of reactions in solution kinetics also plays its part. An important additional consideration is that, in solution, if a bimolecular reaction is intrinsically fast as, for example, in acid-base neutralisation, the rate-determining process can be the diffusion of the reactants through the solvent before they encounter one another. If the reaction occurs every time the reactants (say, A and B) meet and they are assumed to be spheres with radii Ta and rg, it can be shown that the rate coefficient ( d) for the diffusion-controlled reaction is given by ... 

In aqueous solution is of the order oflO M" s for molecules and slightly more or less for ions of opposite or the same charge. This value constitutes the limit for rate coefficients of bimolecular processes, which are then diffusion-controlled and have an activation energy of 3 to 5 kcal mole Many proton-transfer reactions are of this type. 

Election transfer reactions have been well studied in ionic liquids, with some very interesting results. They are such fast reactions that bimolecular election transfers are almost always diffusion controlled. The diffusion-limited rate coefficient is related to the bulk viscosity of the pure solvent by Equation 10.4 ... 

A major milestone in the history of polymer science was the macromolecular hypothesis by Staudinger [1]. The molecular structure of polymers started to emerge and nowadays, almost 80 years later, a knowledge base of respectable size has been built by the contributions of thousands of researchers. Nevertheless, there are still many aspects of free-radical polymerizations that are not fully understood. The bimolecular free-radical termination reaction is one such example. The first scientific papers dealing in some detail with the kinetics of this reaction, can be traced back to the 40 s when the gel-effect was discovered [2-4]. From subsequent research it became apparent that this reaction has a very low activation energy and is diffusion controlled under almost all circumstances. A major consequence of this diffusion-controlled nature is that the termination rate coefficient kt) is governed by the mobility of macroradicals in solution and is thus dependent upon all parameters that can exert an effect on the mobility of these coils. Consequently, kt is a highly system-specific rate coefficient and benchmark values for this coefficient do not exist. 

The number of models that has been presented in literature to account for the complicated diffusion control of bimolecular termination reactions is overwhelming. The majority of these efforts are based on the Smoluchowski model [202], which was originally developed for colloidal systems. The Smoluchowski model describes the rate coefficient for a diffusion-controlled reaction and is based on the appearance of concentration gradient in a homogeneous mixture. This gradient is formed as a result of the chemical reaction between species, radicals in this case, which causes a local depletion of the reactants. Consequently, a net flux of reactants is induced towards those loci which are poor in radicals. Von Smoluchowski was the first to use this treatment for diffusion-controlled reactions. ... 

For the kinetic relationship, we calculated the molecular weight dependence empirically from diffusion-controlled rate coefficient data for two ends of the same chain to meet, or ends from different chains to diffuse to each other. As described above, Winnik and coworkers [33, 34] used pyrene fluorescence to determine over a molecular weight range from 3,900 to 27,000, with the relationship log = 11.97 -1.52 log(M v). The rate coefficient, ki, is equal to the chain-length dependent termination in free-radical bimolecular termination between two chains of equal length i, represented by the rate coefficient k , since 2 for this termination reaction is much greater than k i (Scheme lb). We have previously found in dilute solutions (i.e., below c ) the following empirical relationship [35, 36] ... 

Another use of Brownian dynamics is to compute the time-dependent rate coefficient k t) of diffusion-controlled reactions. In this case particles start on a boundary near the active site and undergo Brownian motion until they either react or their lifetimes exceed some preset cut-off. The starting positions on this boundary are assigned according to the distribution /c(ro)exp[— f/(ro)]. In this case /c(ro) is the space-dependent intrinsic bimolecular rate constant, is k T), and t/(ro) is the potential of mean force between the two particles. 

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