Rate constant diffusion controlled

The upper limit on a rate constant for a reaction is imposed by the rate at which Ae reactants can diffuse together. This limit can be of significance when a particular mechanism would require a rate constant beyond the 

If two reactants A and B with radii and r, respectively, diffuse together and react at an interaction distance (r + r ), then theories developed from Brownian motion predict that the diffusion-controlled second-order rate constant is given by 

N is Avogadro s number, and are the diffusion coefficients for A and B, respectively, and Zb are their respective charges, e is the electron charge (1.602xlff C), e is the dielectric constant of the solvent, Cq is the vacuum permittivity (47t o= 1.113x10 ) and is Boltzmann s constant (1.381xl0 J K ). If one or both of the species are neutral, then U=0 and the right-hand term in brackets in Eq. (1.86) equals 1. 

Diffusion coefficients can be approximated from the Stokes-Einstein equation, D = k TI(m r, where k = 1.381x10 erg K , r is the solvent viscosity and r is the radius of the species, so that 

One can estimate k jj from Eq. (1.87) without knowing the diffusion coefficients. Some values for various reactant sizes and charge products in water (ti = 0.00894 poise, e = 78.3), are given in Table 1.1. It is apparent from these data that diffusion-controlled rate constants in water can be expected to be in the range of 10 to 10 M s . For a solvent with e = 20, kjj(y is 5 times larger for Za b and smaller for Za b  

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With this expression, kjkn can be obtained by the measurement of one set of [RI ], [R2 ] values, at full light intensity only. As to kii itself, which is needed to evaluate kc, one must either do a separate experiment by time-resolved EPR spectroscopy (see Chapter 11) or, with less accuracy and reliability, one can assign it the value for the diffusion-controlled rate constant in that solvent. 

This value, often approximated as 1010 L mol"1 s 1, is referred to as the diffusion-controlled rate constant. It is rather insensitive to the chemical species that participate in the reaction. A larger molecule diffuses more slowly than a smaller one, but that effect is roughly compensated by a higher probability of encounter given its larger radius. 

For entities of molecular size the numerical coefficient in the denominator of Eq. (9-12) should be reduced from 6 to 4 or less,6 which would also alter the expression for the diffusion-controlled rate constant from the traditional form given in Eq. (9-13) ... 

Values of diffusion-controlled rate constants at 298 K calculated for different solvents according to Eq. (9-13)... 

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. 

Influence" of ionic charges on second-order diffusion-controlled rate constants in aqueous solution at 25 °C... 

In liquids, collisional energy transfer takes place by multistep diffusion (the rate determining step) followed by an exchange interaction when the pair is very close. The bimolecular-diffusion-controlled rate constant is obtained from Smoluchowski s theory the result, including the time-dependent part, may be written as... 

Pressure effects The diffusion through liquids is governed by the number of defects or atomic-sized holes in the liquid. A high external pressme can reduce the concentration of holes and slow diffusion. Therefore, in a liquid, a diffusion-controlled rate constant also depends on the pressure. 

The product cystine is presumably formed in the recombination of two thiyl radicals. This free-radical model is suitable for formal treatment of the kinetic data however, it does not account for all possible reactions of the RS radical (68). The rate constants for the reactions of this species with RS-, 02 and Cu L, (n = 2, 3) are comparable, and on the order of 109-10loM-1s-1 (70-72). Because all of these reaction partners are present in relatively high and competitive concentrations, the recombination of the thiyl radical must be a relatively minor reaction compared to the other reaction paths even though it has a diffusion controlled rate constant. It follows that the RS radical is most likely involved in a series of side reactions producing various intermediates. In order to comply with the noted chemoselectivity, at some point these transient species should produce a common intermediate leading to the formation of cystine. 

At higher concentrations in solution, the photodimerization of tS has been studied by means of picosecond electronic absorption spectroscopy. The 5i state of tS in benzene at 22°C is quenched with a diffusion-controlled rate constant of 2.03 X lO M s to give a new reactive intermediate exhibiting an absorption maximum at 480 nm. This new species decays unimolecularly with a rate constant of (2.40 0.37) X 10 s. It has tentatively been assigned to either the excimer or a biradicaloid species located at the pericyclic minimum. 

Using this pXa value and assuming that protonation of dG(-H) occurs with a diffusion-controlled rate constant (2x10 ° s ) [48], we estimate... 

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 the reaction with HO you can assume that both compounds react with a nearly diffusion controlled rate constant (see Fig. 16.3). 

It may be concluded from the known rate coefficients for the hydrolysis of acetic anhydride, for example, that the hypothetical k0H value is larger than a diffusion-controlled rate constant. If Xd is taken as about 107 for acetic anhydride, then from k0n = h2o where kHiQ is the second-order water... 

This diffusion controlled rate constant depends only on the viscosity rj of the solvent and on its temperature T the size of the molecules does not appear in it, and this can be something of a surprise at first sight. The reason for this is that although larger particles move slower, their encounter cross-section is that much larger, and in a simple model these two effects cancel out. 

Thus by plotting l//emitted vs. the concentration of A, we obtain a straight line having a slope of A flT°//ab3 and an intercept of 1//Bb8. From these quantities we can calculate kqr°. If kq can be assumed to be diffusion-controlled (see above), a knowledge of the diffusion-controlled rate constant in the solvent used allows calculation of t°. (Alternatively, direct measurement or theoretical calculation of r° allows calculation of kq.)... 

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. 

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... 

If HA is a stronger acid than the ammonium function of 2, the rate constant for proton transfer to 2, kuA, will be for diffusion and the observed rate constant will be independent of the acidity of HA. On the other hand, if HA is a weaker acid than the ammonium function of 2, the proton transfer from general acids, HA, to the nitrogen of 1 in Scheme 11.8 will be given by k K /K, where K A is the acid dissociation constant of HA and kA is the diffusion-controlled rate constant. The observed rate will nowbe dependent upon the acidity of the catalyst HA as described by a Bronsted correlation with slope equal to —1. 

When U(r) / 0, the approximate expression for k t) behaves correctly at t —> oo, and the diffusion-controlled rate constant becomes... 

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