Diffusion controlled reactions encounters

After the jump, the particle is taken to have reacted with a given probability if its distance from another particle is within the reaction radius. For fully diffusion-controlled reactions, this probability is unity for partially diffusion-controlled reactions, this reaction probability has to be consistent with the specific rate by a defined procedure. The probability that the particle may have reacted while executing the jump is approximated for binary encounters by a Brownian bridge—that is, it is assumed to be given by exp[—(x — a)(y — a)/D St], where a is the reaction radius, x andy are the interparticle separations before and after the jump, and D is the mutual diffusion coefficient of the reactants. After all... 

Green and Pimblott (1989) have extended the IRT model to partially diffusion-controlled reactions between neutrals. They derive an analytical expression that involves an additional parameter, namely the reaction velocity at encounter. For reactions between charged species, W generally cannot be given analytically but must be obtained numerically. Furthermore, numerical inversion to get t then... 

Alkyl radicals react in solution very rapidly. The rate of their disappearance is limited by the frequency of their encounters. This situation is known as microscopic diffusion control or encounter control, when the measured rate is almost exactly equal to the rate of diffusion [230]. The rate of diffusion-controlled reaction of free radical disappearance is the following (the stoichiometric coefficient of reaction is two [233]) ... 

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... 

For a fuUy diffusion-controlled (or encounter-controlled) reaction. 

Diffusion-Controlled Reactions. Chemical reactions without Transition States (or energy barriers), the rates of which are determined by the speed in which molecules encounter each other and how likely these encounters are to lead to reaction. 

Let us first consider a very fast reaction between uncharged nonpolar reactants in solution. In this case, the rate is controlled by the number of encounters. Once A and B diffuse into the same solvent cage, they will react hence the rate of these diffusion-controlled reactions is determined by how fast A and B diffuse together in solution. 

Starting with Fick s first law, one can calculate for a solution of two reactants A and B the frequency of A-B encounters, which is in effect the reaction rate constant for diffusion-controlled reactions. This is given by the following, in units of L mol 1 s-1 ... 

Similar considerations apply to the role of spin equilibria in electron transfer reactions. For many years spin state restrictions were invoked to account for the slow electron exchange between diamagnetic, low-spin cobalt(III) and paramagnetic, high-spin cobalt(II) complexes. This explanation is now clearly incorrect. The rates of spin state interconversions are too rapid to be competitive with bimolecular encounters, except at the limit of diffusion-controlled reactions with molar concentrations of reagents. In other words, a spin equilibrium with a... 

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. 

The classical treatment of such processes derives from the consideration of the coagulation of colloids (Smoluchowski, 1917), but many accounts have been given of how the same approach can be used for diffusion-controlled reactions (Noyes, 1961 North, 1964 Moelwyn-Hughes, 1971). The starting point is the assumption of a random distribution of the two reactants (here given the symbols X and B) in the solution. Then, if B is capable of reacting on encounter with a number of molecules of X, it follows that such reactions deplete the concentration of X in the neighbourhood of B and therefore set up a... 

One remaining problem is why this reaction path should so closely simulate the characteristics of a diffusion-controlled reaction between the free amine and the nitronium ion. This probably arises in part because the concentration of encounter pairs involving the nitronium ion and the hydrogen-bonded free amine follows the concentration of free amine as calculated from the appropriate acidity function. Also, as outlined in Section 2, the rate coefficient for reaction within the encounter pair may be near its limiting value. However, in this example, it may be unrealistic to consider the nitronium ion and free amine as separate species within the encounter pair. This point is considered further in Section 6. 

The values of the rate coefficients in Table 10 approach that expected for a diffusion-controlled reaction (cf. k = 3.4 x 109 mol-1 s-1 dm3 for 3,5,N,N-tetramethylaniline). However, it is worth noting that most of these results refer to hydrogen-ion concentrations >0.5 mol dm 3 and that the basicity of the amines is such that protonation probably occurs on encounter. Thus, the half-life of the free amine is likely to be <10-10 s and, under these conditions, the... 

Classification of reactions. Before considering the individual reactions it is helpful to look at the factors that determine the balance between the diffusion-controlled and pre-association mechanisms. Consider first a diffusion-controlled reaction according to Scheme 3 in which the rate of formation of the encounter pair B.X is rate-determining. These conditions require that k l > k2 R] so that the reaction rate is given by (45).21 The corresponding rate of... 

The value of the equilibrium constant for an encounter is certainly of prime importance in the discussion of interchange pathways of complex formation. This was first suggested, in fact by Werner [4] as early as in 1912. Most of the work on ligand substitution in complexes is based on the assumption that encounter equilibria could be calculated from the ion-pairing equation of Fuoss [5] which was derived in turn from a consideration of diffusion-controlled reactions by Eigen [6]. At zero ionic strength, the encounter equilibrium constant, Kp is given as... 

The motion of molecules in a liquid has a significant effect on the kinetics of chemical reactions in solution. Molecules must diffuse together before they can react, so their diffusion constants affect the rate of reaction. If the intrinsic reaction rate of two molecules that come into contact is fast enough (that is, if almost every encounter leads to reaction), then diffusion is the rate-limiting step. Such diffusion-controlled reactions have a maximum bimolecular rate constant on the order of 10 ° L mol s in aqueous solution for the reaction of two neutral species. If the two species have opposite charges, the reaction rate can be even higher. One of the fastest known reactions in aqueous solution is the neutralization of hydronium ion (H30 ) by hydroxide ion (OH ) ... 

However, our concern is with the cationic surface which promotes a rapid exchange of an electron from dimethylaniline to pyrene, and thereafter maintains a long-lived ion which can react with further solutes added to the system. Hie concept of the experiment is, that dimethylaniline transfers the electron rapidly to pyrene via a diffusion controlled reaction, which occurs by movement of the reactants on the surface of the micelle until they encounter each other. Electron transfer then occurs, and the back reaction of the two ions is prevented by the surface of the micelle, which holds the reactants in an unsuitable configuration for back reaction to occur. However, the repulsive positive force of the micelle on the dimethylaniline cation rapidly drives it away from the micelle, and effective and efficient charge separation is achieved, with a quantum yield Q of unity for the process of charge separation. 

There is some question as to whether such a system actually transfers electrons via an encounter, or whether the electron is actually ejected from DMA to pyrene over a distance, and before encounter occurs. This latter statement is shown not to be correct (5) as the kinetics follow that of a diffusion controlled reaction... 

In a diffusion-controlled reaction the steps (Scheme 1, Chapter 1) subsequent to the formation of the encounter complex become faster than the diffusion rate constant ( 1 ns ). 

The value of 4TrNr i Sr/lOMs typically 2 x 10 M s , whereas the value of Z used in the classical formalism is 10" M s. This value of Z seems low, because the rate constants for a diffusion-controlled reaction of two uncharged reactants are ca. 1 X 10 M s, and there are ca. 10 collisions in the solvent cage per encounter. On this basis, Z should be closer to 10 M s". ... 

Thus, if for any reason the average lifetime of the growing radical increases, then there will be an increase in the polymer chain length. An often-encountered example of this is the Trommsdorff or gel effect that occurs in the polymerization of solutions of high monomer concentration when the viscosity, rj, increases and, after a certain extent of conversion, there is a rapid acceleration in the rate of polymerization. This is interpreted as an indication of the decrease in the rate of termination as this reaction becomes diffusion-controlled. A feature of diffusion-controlled reactions is that the rate coefficient, is not chemically controlled but depends on the rate at which the terminating radicals can collide. This is most simply given by the diffusion-controlled rate coefficient, k, in the Debye equation ... 

The frequency with which two reactive species encounter one another in solution represents an upper bound on the bimolecular reaction rate. When this encounter frequency is rate limiting, the reaction is said to be diffusion controlled. Diffusion controlled reactions play an important role in a number of areas of chemistry, including nucleation, polymer and colloid growth, ionic and free radical reactions, DNA recognition and binding, and enzyme catalysis. 

Electronically excited states are highly reactive and so are many of their primary photoproducts. Therefore, intermolecular processes that occur on every encounter between reactant molecules in solution, also called diffusion-controlled reactions, are fairly common in photochemistry. 

Having confirmed that the concept of reduction of dimensionality can play an important role in determining the efficiency of diffusion-controlled reactions in both symmetrical and asymmetrical compartmentalized systems, one may ask How does a substrate know, upon first encounter with the boundary, to move along the interior surface of a cellular unit, to react (eventually) with (say) a membrane-bound enzyme While substrate-specific, surface binding or association forces can conspire to keep the substrate in immediate vicinity of the boundary, once the latter has been encountered for the first time, certain (chemically) nonspecific, statistical factors are also likely to play a role in reduction of dimensionality. 

---