Diffusion control rate equations

The boundary conditions indicate that electrolysis of the species of interest occurs at a diffusion-controlled rate (Equation 13.7) and the adsorbate is inert on the insulating sheath surrounding the electrode (Equation 13.8). Additional conditions define zero radial flux at the axis of symmetry (Equation 13.9) and the recovery of the bulk concentration of the species beyond the radial edge of the tip/substrate domain (Equation 13.10). This latter assumption is particularly valid for... 

Pulse radiolysis results (74) have led other workers to conclude that adsorbed OH radicals (surface trapped holes) are the principal oxidants, whereas free hydroxyl radicals probably play a minor role, if any. Because the OH radical reacts with HO2 at a diffusion controlled rate, the reverse reaction, that is desorption of OH to the solution, seems highly unlikely. The surface trapped hole, as defined by equation 18, accounts for most of the observations which had previously led to the suggestion of OH radical oxidation. The formation of H2O2 and the observations of hydroxylated intermediate products could all occur via... 

The temperature dependence of a diffusion-controlled rate constant is very small. Actually, it is just the temperature coefficient of the diffusion coefficient, as we see from the von Smoluchowski equation. Typically, Ea for diffusion is about 8-14 kJ mol"1 (2-4 kcal mol-1) in solvents of ordinary viscosity. 

The sulfur compounds RSH and disulfides are highly reactive, but RS and RSR are not. Nitro compounds usually react at diffusion-controlled rates. Aromatic compounds also fit into the Hammett (1940) equation when log k is plotted against free-energy change due to polar effects log(k/kg) = ap. Anbar... 

The cyclohexadienyl radicals decay by second-order kinetics, as proven by the absorption decay, with almost diffusion-controlled rate (2k = 2.8 x 109 M 1 s 1). The cyclohexyl radicals 3 and 4 decay both in pseudo-first-order bimolecular reaction with the 1,4-cyclohexadiene to give the cyclohexadienyl radical 5 and cyclohexene (or its hydroxy derivative) (equation 15) and in a second order bimolecular reaction of two radicals. The cyclohexene (or its hydroxy derivative) can be formed also in a reaction of radical 3 or... 

K = k T. When energy-transfer experiments were used to deduce the x values, the quenching was assumed to proceed with a diffusion-controlled rate, and the transfer was supposed to take place when the acceptor and donor molecules were in contact (no static quenching). The transfer rate coefficients were calculated using the usual diffusional equations [24,25,59] (see Sec. 3.1). 

Solution First we evaluate kr, using Equation (32). It is convenient to use cgs units for this calculation therefore we write kr = 4 (1.38 10-16) (293)/(3 )(0.010) = 0.54 10-11 cm3 s 1. Recall that the coefficient of viscosity has units (mass length-1 time-1), so the cgs unit, the poise, is the same as (g cm -1 s -1). As a second-order rate constant, kr has units (concentration -1 time -1), so we recognize that the value calculated for kr gives this quantity per particle, or kr = 0.54 10-11 cm3 particle-1 s-1. Note that multiplication by Avogadro s number of particles per mole and dividing by 103 cm3 per liter gives kr = 3.25 109 liter mole-1 s-1 for the more familiar diffusion-controlled rate constant. 

For a reaction in solution occurring at a diffusion-controlled rate, Arc 109 I mol-1 s-1, and if Air is negligible and AtSc 9x 108s then for [Bl = 0.1 M, the efficiency of singlet state reaction is (equation 7.14)... 

D. The chronoamperometric results can also be used to ascertain the number of electrons involved in the formation of benzonitrile from p-chloro-benzonitrile. In order to translate the chronoamperometric data into a meaningful n value, a compound is selected that has a diffusion coefficient very similar to that of p-chlorobenzonitrile and that gives a stable, known product upon electroreduction. Tolunitrile, which satisfies these criteria, is known to be reduced to its radical anion at a diffusion-controlled rate. Since this one-electron process gives a value of 168 pA s1/2- M x cm 2 for it1/2/CA, the corresponding value of 480 pA s1/2 A/ 1 cm-2 for the reduction of p-chlorobenzonitrile to benzonitrile anion radical must represent an overall three-electron process. When we subtract the one electron that is required to reduce benzonitrile to its radical anion from this total, we immediately conclude that two electrons are involved in cleavage of the carbon-chlorine bond in p-chlorobenzonitrile. A scheme that is consistent with these data is described by Equations 21.1 to 21.6. 

This effect of N08 ion is quantitatively consistent with a reaction mechanism (43) in which N08 interacts with an electronically excited water molecule before it undergoes collisional deactivation by a pseudo-unimolecular process (the NOs effect is temperature independent (45) and not proportional to T/tj (37)). Equation 1, according to this mechanism, yields a lifetime for H20 of 4 X 10 10 sec., based on a diffusion-controlled rate constant of 6 X 109 for reaction with N08 Dependence of Gh, on Solute Concentration. Another effect of NOa in aqueous solutions is a decrease in GH, with increase in N08 concentration (5, 25, 26, 38, 39). This decrease in Gh, is generally believed to result from reaction of N08 with reducing species before they combine to form H2. These effects of N08 on G(Ce+3) and Gh, raise the question as to whether or not they are both caused by reaction of N08 with the same intermediate. 

Simple silenes readily couple to yield the head-to-tail dimers, 1,3-disilacyclobuta-nes1,2,15,157. The dimerization is extremely facile and silenes bearing only small alkyl groups dimerize in an argon matrix even at 40 K, i.e., the dimerization proceeds at a diffusion controlled rate. Bulky substituents slow down the dimerization rate and allow the isolation of stable silenes. The head-to-tail dimerization (equation 98) is the predominant dimerization path for silenes, including those of the Auner-Jones 79,80 and Wiberg type72,73,78. 

In the case where AH = HsO"1", the experimental result is kG K = 104 M-2 s 1, which is a credible value for the third-order k in Equation 11.7. However, since Ka is known to be 2.8 x 108 M, k in Equation 11.8 needs to be 2.8 x 1012 M 1 s-1 to support the second mechanism in Scheme 11.5b with A = H20[5], Since this value is larger than the likely diffusion-controlled rate constant (7.4 x 109 M 1 s 1) [ 6 ], the second mechanism is ruled out. (The calculated rate constant is even larger than that of the fastest known... 

For example, anthracene>having a triplet energy of 42 kcal/mol quenches the excited state ( 50 kcal/mol) at an essentially diffusion controlled rate by electronic energy transfer. The excited state can also be quenched by electron transfer, equations (15) and (16) (56). Both of these processes are... 

The equation indicates that the diffusion controlled rate constant k for reaction of two ions is given by... 

After polymerization has started, the free volume of the reaction system will continuously decrease. The smaller the amount of free volume, the lower is the mobility of the long-chain molecules. At a certain point, the rate constant of termination becomes diffusion-controlled and autoacceleration takes place. The diffusion-controlled rate constant kt will become smaller than the original chemically controlled rate constant kt. To keep track of the diffusion-controlled kt, one can use the following equation ... 

As the Debye equation, eqn. (16), shows, diffusion-controlled rates of reaction depend only slightly on the charges on the diffusing particles, and this is borne out by the observation that the rate of reaction between ammonia and the hydroxonium ion is similar to the rate of reaction between the acetate and the hydroxonium ion. The decrease in rate due to the lack of ionic attraction is compensated for by the increase due to the slightly more favourable steric factor for ammonia. 

The second model (Fig. 20c) assumes that upon melting of reactant A, a layer of initial product forms on the solid reactant surface. The reaction proceeds by diffusion of reactant B through this layer, whose thickness is assumed to remain constant during the reaction (Aleksandrov et al., 1987 Aleksandrov and Korchagin, 1988). The final product crystallizes (C) in the volume of the melt after saturation. Based on this model, Kanury (1992) has developed a kinetic expression for the diffusion-controlled rate. Using this rate equation, an analytical expression for the combustion wave velocity has been reported (Cao and Varma, 1994)... 

On closer inspection, the combination rate constants are about 1/4 of the estimated diffusion-controlled rate constant. For acetonitrile, for example, fcjj - 2.9 X 10 L mol" s from the von Smoluchowski equation wiA a diffusion coefficient from a modified version of the Stokes-Einstein relation, D - fcT/4jiT r. Owing to the restriction to singlet state recombination, an experimental rate constant 1 /4 of is quite reasonable. On the other hand, for these heavy metals, the spin restriction may not apply, in which case one would argue that the geometrical and orientational requirements of these large species could well give recombination rates somewhat below the theoretical maximum. 

When spectroelectrochemistry is used as a tool in reaction kinetics, it is important to know accurately the rate of generation of reactive intermediates, that is, the accurate potential of the working electrode. This requirement becomes a particular problem when an OTE is the preferred electrode because of the ohmic drop in the electrode itself and the nonuniform current distributions often encountered. For the OTTLEs in particular, the accurate modeling of the diffusion in the cell also leads to rather complicated mathematical equations [346]. The most profitable way of operation is therefore to use a potential-step procedure where the potential is stepped to a value at which the heterogeneous electron transfer reaction proceeds at the diffusion-controlled rate. In transmission spectroscopy the absorbance, AB(t), of the initial electrode product B, in the absence of chemical follow-up reactions, is given by Eq. (99) [347,348], where b is the extinction coefficient of B. 

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