Sphere diffusion with fast reaction

F. DIFFUSION IN A SPHERE WITH FAST REACTION - SINGULAR PERTURBATION THEORY ... 

F. Diffusion in a Sphere with Fast Reaction - Singular Perturbation Theory ... 

It has been shown in previous studies by several fast reaction techniques that the complexation reaction between a metal ion and a bidentate ligand involves several discrete steps (5). As shown in figure 2, step 1 represents rapid (diffusion controlled) formation of an outer-sphere complex and step 2 the formation of a monodentate complex for which the forward rate constant can be identified with that for water exchange (kex) measured by n.m.r. methods. Since PADA is a bidentate ligand, a further ring closure step (3) takes place in which the final product is formed. The rate of ring closure is generally assumed to be rapid compared with the rate of dissociation of the monodentate complex (i.e. kj k 2) ... 

For spherical particles, diffusion-controlled reaction rates are given by Equation (18.25). Diffusion control implies that reactions are fast. Other reactions involve additional rate-limiting steps that occur after the reacting molecules have come into contact with the sphere. Diffusion control defines an upper limit on the speed of reactions. Any other kind of process must be slower than the diffusion-controlled process because the reaction must take additional time to complete after contact. Association rates are often expressed in terms of a rate coefficient ka defined by 1(a) = -kaC, where... 

Figure 2.3 Left, reduction models. In the shrinking core or contracting sphere model the rate of reduction is initially fast and decreases progressively due to diffusion limitations. The nucleation model applies when the initial reaction of the oxide with molecular hydrogen is difficult. Once metal nuclei are available for the dissociation of hydrogen, reduction proceeds at a higher rate until the system comes into the shrinking core regime. Right the reduction rate depends on the concentration of unreduced sample (1-a) as f(a) see Expressions (2-5) and (2-6).

Inner sphere oxidation-reduction reactions, which cannot be faster than ligand substitution reactions, are also unlikely to occur within the excited state lifetime. On the contrary, outer-sphere electron-transfer reactions, which only involve the transfer of one electron without any bond making or bond breaking processes, can be very fast (even diffusion controlled) and can certainly occur within the excited state lifetime of many transition metal complexes. In agreement with these expectations, no example of inner-sphere excited state electron-transfer reaction has yet been reported, whereas a great number of outer-sphere excited-state electron-transfer reactions have been shown to occur, as we well see later. 

Transition metal complexes functioning as redox catalysts are perhaps the most important components of an ATRP system. (It is, however, possible that some catalytic systems reported for ATRP may lead not only to formation of free radical polymer chains but also to ionic and/or coordination polymerization.) As mentioned previously, the transition metal center of the catalyst should undergo an electron transfer reaction coupled with halogen abstraction and accompanied by expansion of the coordination sphere. In addition, to induce a controlled polymerization process, the oxidized transition metal should rapidly deactivate the propagating polymer chains to form dormant species (Fig. 11.16). The ideal catalyst for ATRP should be highly selective for atom transfer, should not participate in other reactions, and should deactivate extremely fast with diffusion-controlled rate constants. Finther, it should have easily tunable activation rate constants to meet sped c requirements for ATRP monomers. For example, very active catalysts with equilibrium constants K > 10 for styrenes and acrylates are not suitable for methacrylates. 

Just as in the gas-phase, thermodynamics tells only part of the story in respect of reactions in solution kinetics also plays its part. An important additional consideration is that, in solution, if a bimolecular reaction is intrinsically fast as, for example, in acid-base neutralisation, the rate-determining process can be the diffusion of the reactants through the solvent before they encounter one another. If the reaction occurs every time the reactants (say, A and B) meet and they are assumed to be spheres with radii Ta and rg, it can be shown that the rate coefficient ( d) for the diffusion-controlled reaction is given by ... 

---