Sphere Reactions

The removal of potassium cations from the reaction sphere can be accomplished by their binding with 18-crown-6-ether (Scheme 2.16). 

An example of numerical solutions of Eq. (29) for the attractive Coulomb potential, subject to the initial and the boundary conditions (30)-(32), is presented in Fig. 4 [26]. The figure shows how the initially uniform electron concentration gradually decreases in the vicinity of the cation until the steady state is established. The rate of reaction is determined by the diffusive flow of electrons from large distances toward the reaction sphere. 

It is convenient to analyze the general equations (3.2.7) and (3.2.11) in terms of the simplest model of the recombination event called clear-cut radius or black reaction sphere - equation (3.1.1) ... 

Therefore, making use of this simple model one can solve and qualitatively analyze different peculiarities of the monomolecular recombination. Partially reflecting boundary conditions (grey recombination sphere) are studied in [50, 68], For the effects of reaction sphere anisotropy, see Chapter 5. 

Here the first term arises from the diffusive approach of reactants A into trapping spheres around B s it is nothing but the standard expression (8.2.14). The second term arises due to the direct production of particles A inside the reaction spheres (the forbidden for A s fraction of the system s volume). Unlike the Lotka-Volterra model, the reaction rate is defined by an approximate expression (due to use of the Kirkwood superposition approximation), therefore first equations (8.3.9) and (8.3.10) of a set are also approximate. 

Equation (8.3.14) is not an asymptotically exact result for the black sphere model due to the superposition approximation used. When deriving (8.3.14), we neglected in (8.3.11) small terms containing functionals I[Z], i.e., those terms which came due to Kirkwood s approximation. However, the study of the immobile particle accumulation under permanent source (Chapter 7) has demonstrated that direct use of the superposition approximation does not reproduce the exact expression for the volume fraction covered by the reaction spheres around B s. The error arises due to the incorrect estimate of the order of three-point density p2,i for a large parameter op at some relative distances ( f — f[ < tq, [r 2 - r[ > ro) the superposition approximation is correct, p2,i oc ct 1, however, it gives a wrong order of magnitude fn, oc Oq2 instead of the exact p2,i oc <7q 1 (if n — r[ < ro, fi — f[ < ro). It was... 

The removal of potassium cations from the reaction sphere can be accomplished by their binding with 18-crown-6-ether (Scheme 2-13). The removal of potassium cations makes the results of the liquid-phase and electrode reactions similar. In the presence of the crown ether, the liquid-phase process also leads to the azodianion. The azodianion was indeed identified via benzidine after protonation and rearrangement (Scheme 2-14). 

In general, a sensitizer transforms into its excited state, passes an excited electron (or hole) to a substrate, and then remains in the reaction sphere. Thus, electron back-transfer is characteristic of the whole process. During its lifetime, a substrate ion radical is often able to undergo some chemical transformation, especially when the transformation proceeds within a solvent cage. Examples are given at the end of Section 5.2.1. 

The biphasic kinetics curve for the reactions of polymers is very typical and is found frequently in the polymer literature. Daglen and Tyler showed62 that equation 27 gave excellent fit to these systems as well, which suggests the presence of reaction spheres is common in the mechanism of solid-state polymer photodegradation. 

If Rq > black sphere radius. This is the radius of a reaction sphere outside which neither excitation is yet quenched while inside the sphere all of them are already deactivated. In reality, of course, the border between outskirts and interior of the sphere is not as sharp, but Rq fixes this boundary and specifies the stationary (Markovian) rate of quenching (3.41). 

The GCK approximation includes the reaction layer adjacent to the contact in the reaction sphere and thus magnifies its external radius to the size of R considered as a fitting parameter. Using it instead of a, we obtain 4%RD instead of kn and transform Eq. (3.22) into the following expression ... 

The excitations that survive after static (instantaneous) quenching should have no surrounding acceptors, inside the reaction sphere of a certain volume v. If there are N acceptors in the volume V, then the probability of finding neither of them inside the selected spherical volume (in the limit of infinitely large sample) is... 

The most primitive but popular exponential model (EM) implies that the recombination occurs within the transparent reaction sphere where the ions are born [Fig. 3.22(a)]. The backward electron transfer to the ground state proceeds there with the uniform rate k et, but some ions escape recombination leaving the sphere, due to encounter diffusion that finally separate them. EM ascribes to this process the rate... 

Figure 3.22. The schematic representation of (a) the exponential model (EM) and (b) contact approximation. In EM the reaction sphere of radius a is transparent for particles, which leave it by a single jump with the rate ksep. In contact calculations, the same sphere surrounds an excluded volume and recombination takes place only at its surface, or more precisely in a narrow spherical layer around it.

Against all odds, this concept constitutes the formal basis of the exponential model. The EM kinetic equations are written for the survival probabilities of ions in the reaction sphere f2jn and out of it f>oul ... 

Contrary to this approximation, the exponential model, considered in Section V.A, does not assume recombination to be contact, but suggests that it takes place with a uniform backward transfer rate k-et within the reaction sphere of the volume v = 47ict3/3. As a result, Eq. (3.419) is replaced by the following one ... 

The recombination rate in such a microspace can be estimated according to Goselle et al. (1979). When two reactants, one of which is much smaller and more diffusive (H+) than the other (0-), are locked in a space with internal diameter R, we can regard the heavier reactant (0-) as practically immobile target in the center of a reaction sphere. The proton can diffuse through the reaction space, but wherever it penetrates the Coulomb cage of the proton emitter, protonation takes place. The rate constant of the reaction is thus controlled by two radii, the Debye radius (RD), where electrostatic interaction dominates, and the radius of the reaction sphere, out of which the proton cannot escape. 

These results can be subjected to more rigorous analysis. The rate constant of protonation of the excited anion of hydroxypyrene trisulfonate is k = 5 x 10loM-1 sec-1 (Weller, 1958). Thus, the effective concentration of H+ in the reaction sphere (R) is... 

For use of ionic melts as electrolytes in high-temperature chemical power sources it is necessary to know the metal-oxide s solubilities in these media, since the oxides may be formed as a result of reactions of electrochemically active substances with oxide ions which exist in the pure halide melts. This interaction is especially undesirable for the case of rechargeable high-temperature chemical power sources, since it results in irreversible removal of electrochemically active particles from the reaction sphere. 

Figure 7.5 Effectiveness factor for various values of CAb/Km (Michaelis-Menten-type reaction, sphere).

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