Diffusion-Controlled Rapid Reactions

The fastest reactions in solution are limited in their reaction rates by mass transfer. For most bimolecular reactions this limitation arises because the two reactants must diffuse together in order to form a reactant pair. Therefore it is interesting to estimate the magnitude of the rate constant for a diffusion-con-trolled reaction and to compare it with experimentally determined rate constants for very fast reactions. The treatment presented here applies to bimolecular reactions involving either molecules or ions. 

It is convenient to re-express the electrostatic term in terms of the electrostatic energy C/eA, where 

The problem is now solved, assuming that diffusion is a spherically symmetrical process and that it occurs at a steady state. Furthermore, the reactants 

The fact that both A and B move in reality is now dealt with by using Z a + -Dg as the mutual diffusion coefScient this effectively accounts for the motion of the coordinate system which is now defined with respect to B. Thus, 

The number of molecules moving through the surface of a sphere of radius r can now be calculated  

---

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. 

When chemisorption takes place, the rate may be diffusion-controlled or reaction-controlled. The former mode Is expected when all arriving molecules are rapidly scavenged by the reaction. Reaction-controlled adsorption has a kinetics typical for chemical processes, with an activation energy and an Arrhenius type of temperature dependence. 

Finally a word on rates of acid-base reactions. All the protons in water are undergoing rapid migration from one oxygen atom to another, and the lifetime of an individual HsO+ ion in water is only approximately 10 13 sec. The rate of reaction of HsO+ with a base such as OH" in water is very fast but also diffusion-controlled.13 Reaction occurs when the solvated ions diffuse to within a critical separation, whereupon the proton is transferred by concerted shifts across one or more solvent molecules hydrogen-bonded to the base. 

At one place in the apparatus, where the pressure is 1 std atm abs and the temperature 200 C. the analysis of the bulk gas is 33.33% NHj (A), 16.67% N2 (B), and 50.00% (Q by volume. The circumstances are such that NH3 diffuses from the bulk-gas stream to the catafyst surface, and the products of the reaction diffuse back, as if by molecular diffusion through a gas film in laminar flow 1 mm thick. Estimate the local rate of cracking, kg NH3/(m catalyst surface) > s, which mi t be considered to occur if the reaction is diffusion-controlled (chemical reaction rate very rapid) with the concentration of NH3 at the catalyst surface equal to zero. 

From Equation 7.20, N 03 corresponds to K2 70. Assuming (=200nm is the closest attainable tip/substrate separation and a typical value of D=10" cm s", the upper limit on the range of measurable rate constants is estimated as k2 s (8.75 x 10 °cm s )a /ca. The absolute value of then depends on the value of a and Ca, with a large value of the former parameter and small value of the latter, favoring the measurement of rapid kinetics. Even for moderate values of a=2.5 pm and Ca= 10" mol dm", diffusion-controlled solution reactions are potentially accessible from TG/SC measurements. 

The experimental value is still more rapid, apparently because of the electrostatic interaction. Thus reaction (a) is diffusion controlled. For reaction (b),... 

The first-order reaction of hydrogen with Ni3C at 443 K is relatively more rapid than the decomposition [669], indicating facile hydrogenation of the residual carbon at the reactant surface and the possibility of diffusion control is mentioned. 

In the pulse radiolysis studies on the reaction of MV with TiOj, the sol contained propanol-2 or formate and methyl viologen, MV Ionizing radiation produces reducing organic radicals, i.e. (CH3)2COH or C02 , respectively, and these radicals react rapidly with MV to form MV. The reaction of MV with the colloidal particles was then followed by recording the 600 nm absorption of MV . The rate of reaction was found to be slower than predicted for a diffusion controlled reaction. 

In an EC mechanism the ratio of the forward and backward reaction rates is decisive for k/ d in , the chemical follow-up reaction has no influence here, so that for a sufficiently rapid electron transfer step the limiting current remains diffusion controlled.)... 

Approximation refers to the bringing together of the substrate molecules and reactive functionalities of the enzyme active site into the required proximity and orientation for rapid reaction. Consider the reaction of two molecules, A and B, to form a covalent product A-B. For this reaction to occur in solution, the two molecules would need to encounter each other through diffusion-controlled collisions. The rate of collision is dependent on the temperature of the solution and molar concentrations of reactants. The physiological conditions that support human life, however, do not allow for significant variations in temperature or molarity of substrates. For a collision to lead to bond formation, the two molecules would need to encounter one another in a precise orientation to effect the molecular orbitial distortions necessary for transition state attainment. The chemical reaction would also require... 

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]) ... 

Diffusion of particles in the polymer matrix occurs much more slowly than in liquids. Since the rate constant of a diffusionally controlled bimolecular reaction depends on the viscosity, the rate constants of such reactions depend on the molecular mobility of a polymer matrix (see monographs [1-4]). These rapid reactions occur in the polymer matrix much more slowly than in the liquid. For example, recombination and disproportionation reactions of free radicals occur rapidly, and their rate is limited by the rate of the reactant encounter. The reaction with sufficient activation energy is not limited by diffusion. Hence, one can expect that the rate constant of such a reaction will be the same in the liquid and solid polymer matrix. Indeed, the process of a bimolecular reaction in the liquid or solid phase occurs in accordance with the following general scheme [4,5] ... 

The simple reaction scheme outlined in Scheme 2 easily accommodates the results obtained in the investigation of XA. The basic scheme is just the same as it is for BA except that the relative energies of the singlet and triplet states of the carbene are reversed. All of the evidence points to a singlet ground state for XA. This includes the absence of an epr signal, the nearly diffusion-controlled formation of ether in direct and sensitized experiments, the stereospecific cyclopropanation, and the absence of a rapid reaction of the carbene with 02. 

Some rare reactions occur at a rate that appears to be insensitive to temperature. Such reactions are extremely rapid, and are termed diffusion-controlled reactions. 

There are some processes occurring in solutions, e.g. quenching of the fluorescence in solution, certain heterogeneous reactions etc., in which the diffusion is the rate controlling. These reactions occur very rapidly, e.g. ionic recombinations. 

Electronic excitation energy in a crystal is in many cases highly mobile It may diffuse very rapidly through many thousands of molecules and eventually be trapped at some appropriate defect site. If, then, photoreaction occurs at this site, the stereochemistry of the reaction pathway will be determined by the symmetry of this site, and not by the symmetry of the bulk crystal. Nevertheless, the bulk symmetry is found empirically to be the determining factor in most cases studied (topochemical control). 

The rate constants for reaction of Bu3SnH with the primary a-alkoxy radical 24 and the secondary ce-alkoxy radical 29 are in reasonably good agreement. However, one would not expect the primary radical to react less rapidly than the secondary radical. The kinetic ESR method used to calibrate 24 involved a competition method wherein the cyclization reactions competed with diffusion-controlled radical termination reactions, and diffusional rate constants were determined to obtain the absolute rate constants for the clock reactions.88 The LFP calibrations of radical clocks... 

---