Diffusion-controlled reactions theory

The diffusion kinetics were studied, at 220 to 270K, by analyzing the photo-induced dissociation of an etchant-generated D-C complex. Under suitably strong illumination, the annihilation rate of the complex was proportional to the P density. This indicated that the rate-determining step was the diffusion of D to P atoms. By invoking diffusion-controlled reaction theory, it was deduced that the diffusion was described by ... 

Other particular theories are confined to diffusion-controlled reactions (109), to the so called cooperative processes (113), in which the reactivity depends on the previous state, or to resistance of semiconductors (102), while those operating with hydrogen bridges (131), steric factors (132), or electrostatic effects (133, 175) are capable of being generalized less or more. 

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... 

Effects of intermolecular photophysical processes on fluorescence emission Box 4.1 Theories of diffusion-controlled reactions... 

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... 

This expression is a fundamental result of the theory of bulk ion recombination and has been extensively used in interpreting experimental results of diffusion controlled reactions. The Debye-Smoluchowski expression can also be written in terms of the mobility,... 

Over molecular length scales, the diffusion distances become very short (< 1 nm) so that only very rapid events can be influenced by these short diffusion times. Necessarily, this limits the number of systems to only relatively few, where the rate at which the reactants can approach one another is slow or comparable with the rate at which the reactants react chemically with each other. Some typical systems which have been studied are discussed in Sect. 2. The Smoluchowski [3] theory of reactions in solution, which occur at a rate limited solely by how fast the reactants can approach each other (sufficiently closely to react chemically almost instantaneously) is discussed in Sect. 3. If the chemical reaction is not so rapid, the observed rate of reaction may be influenced by both the rate of approach and the rate of subsequent chemical reaction. Collins and Kimball [4], and later Noyes [5], have extended the Smoluchowski theory (1917) to consider this situation (Sect. 4). In light of these quantitative theoretical models of diffusion-limited rate processes, some of the more recent and careful experiments on diffusion-controlled reactions in solution are considered briefly in Sect. 5. As the Smoluchowski theory... 

In the remainder of this section, the general equations and theory appropriate for an incorporation of these long-range transfer effects into the diffusion-controlled reaction process are discussed. Later in this chapter, the specific cases of interest outlined above are developed and their relation to experimental work commented upon. 

This volume is concerned with providing a modern account of the theory of rates of diffusion-controlled reactions in solution. A brief elementary discussion of this area appeared in Volume 2 of this series, which was published in 1969. Since then, the subject has undergone substantial development to the point where we consider it timely that a complete volume devoted to the field appears. Unlike previous volumes of Comprehensive Chemical Kinetics, Volume 25 has been written entirely by one author, Dr. Rice, and his view of the objectives and scope of the book are summarised in Chapter 1. 

In the theory of diffusion-controlled reactions it is called the radiative boundary condition . This quaint name originated from H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford 1947) p. 13. 

The authors of this book started working on chemical kinetics more than 10 years ago focusing on investigations of particular radiation - induced processes in solids and liquids. Condensed matter physics, however, treats point (radiation) defects as active particles whose individual characteristics define kinetics of possible processes and radiation properties of materials. A study of an ensemble of such particles (defects), especially if they are created in large concentrations under irradiation for a long time, has lead us to many-particle problems, common in statistical physics. However, the standard theory of diffusion-controlled reactions as developed by Smoluchowski... 

The theory of irreversible diffusion-controlled reactions is discussed in Chapter 6 the effects of particle Coulomb and elastic interactions are analyzed in detail. The many-particle effects (which in principle cannot be explained in terms of the linear theory) are demonstrated. Special attention is paid to the pattern formation and similar particle aggregation in systems of interacting and noninteracting particles. 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

A wide range of condensed matter properties including viscosity, ionic conductivity and mass transport belong to the class of thermally activated processes and are treated in terms of diffusion. Its theory seems to be quite well developed now [1-5] and was applied successfully to the study of radiation defects [6-8], dilute alloys and processes in highly defective solids [9-11]. Mobile particles or defects in solids inavoidably interact and thus participate in a series of diffusion-controlled reactions [12-18]. Three basic bimolecular reactions in solids and liquids are dissimilar particle (defect) recombination (annihilation), A + B —> 0 energy transfer from donors A to unsaturable sinks B, A + B —> B and exciton annihilation, A + A —> 0. 

Diffusion-Controlled Reactions. The specific rates of many of the reactions of elq exceed 10 Af-1 sec.-1, and it has been shown that many of these rates are diffusion controlled (92, 113). The parameters used in these calculations, which were carried out according to Debye s theory (41), were a diffusion coefficient of 10-4 sec.-1 (78, 113) and an effective radius of 2.5-3.0 A. (77). The energies of activation observed in e aq reactions are also of the order encountered in diffusion-controlled processes (121). A very recent experimental determination of the diffusion coefficient of e aq by electrical conductivity yielded the value 4.7 0.7 X 10 -5 cm.2 sec.-1 (65). This new value would imply a larger effective cross-section for e aq and would increase the number of diffusion-controlled reactions. A quantitative examination of the rate data for diffusion-controlled processes (47) compared with that of eaq reactions reveals however that most of the latter reactions with specific rates of < 1010 Af-1 sec.-1 are not diffusion controlled. 

Summing up, the two-phase model is physically consistent and may be applied for designing industrial systems, as demonstrated in recent studies [10, 11], Modeling the diffusion-controlled reactions in the polymer-rich phase becomes the most critical issue. The use of free-volume theory proposed by Xie et al. [6] has found a large consensus. We recall that the free volume designates the fraction of the free space between the molecules available for diffusion. Expressions of the rate constants for the initiation efficiency, dissociation and propagation are presented in Table 13.3, together with the equations of the free-volume model. 

Eigen (1964) found that a plot of ApR against the rate constant for proton transfer between acetylacetone and a series of bases gave a curved plot. It should be noted, however, that Eigen s explanation for curvature is quite different from the one based on Marcus theory and the reactivity-selectivity principle. The curvature discussed by Eigen is attributed to a change from a rate-determining proton transfer to a diffusion controlled reaction which is independent of the catalyst p. 

The rates of proton transfer reactions cover a wide spectrum, from exasperatingly slow to diffusion controlled. Any theory which can rationalize this range has obvious merit. Such a rationalization is in fact accomplished, to a large degree, by Br nsted and Pedersen s (1923) relationship between rate (kinetic acidity) and p/sTa (thermodynamic acidity). The relationship, known as the Br0nsted equation, has the form (8) where B is the catalytic rate constant. The... 

When the growing radicals are small, their termination constant can be calculated by means of the theory of diffusion-controlled reactions [12]. The mean termination rate constant is then given by the relation... 

The Arrhenius A factors for the propagation reactions are low and of the order one would expect from any of the transition-state theories for a bimolecular reaction between two large molecules (Table XII.2). The activation energies Ep for propagation are also low and of the order observed for similar addition reactions in the gas phase of radicals to a double bond. The values of At are in the range to be expected for diffusion-controlled reactions (Sec. XV.2) except for vinyl chloride, which must certainly be in error. As pointed out earlier in discussing diffusion-controlled reactions, it is expected that the activation energies will be of the order of a... 

The boundary condition that Cp is zero cannot be strictly valid, as shown by Noyes [18] using as an example the recombination of iodine atoms in carbon tetrachloride solution, a quintessential diffusion-controlled reaction. The rate coefficient for collisions calculated from the simple kinetic theory of gases, 9.15 x 10-11 cm3 molecule-1 sec-1, is equated to k, which gives a value of 1.35 for the term (1 + 4irpDk l). More elaborate treatments of the rate of a diffusion-controlled reaction, as reviewed by Noyes [18], use more realistic boundary conditions, but this simple treatment gives answers certainly correct to within an order of magnitude. 

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