Effective reaction radius

The signs + and are for repulsion and attraction, respectively.) L is called Onsager radius [73] at which the Coulomb interaction energy equals the thermal energy kBT. Finally, for the Coulomb attraction (see more details in Chapter 4), the effective reaction radius is [67]... 

When L rB, the effective reaction radius coincides practically with the Onsager radius,... 

Similarly to the black sphere model, equation (4.1.63), the effective reaction radius i eff could be defined through Kq — 47rDi eff. Comparing the i eff obtained in such a way, with the results of Chapter 3, the conclusion suggests that they coincide, i.e., both definitions of the effective radii turn... 

Estimates of the effective reaction radius and reaction rate... 

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

The kinetics of the A + B - 0 bimolecular reaction between charged particles (reactants) is treated traditionally in terms of the law of mass action, Section 2.2. In the transient period the reaction rate K(t) depends on the initial particle distribution, but as f -> oo, it reaches the steady-state limit K(oo) = K() = 47rD/ieff, where D — Da + >b is a sum of diffusion coefficients, and /4fr is an effective reaction radius. In terms of the black sphere approximation (when AB pairs approaching to within certain critical distance ro instantly recombine) this radius is [74]... 

Bakale et al. [261] have studied the reaction of a wide range of dipolar reactants with the solvated electron in cyclohexane at 293 K and found that the effective reaction radius increases approximately proportionately to the dipole moment. They suggested that the correct reaction radius was such that U(r) = — k T in eqn. (120), i.e. about 0.8, whereas from eqn. 

Considering first, however, polar solvents where only the coulomb and screened coulomb potentials are signficant by comparison with thermal energies, it should be emphasized that theory is quite successful in explaining the experimental results. The Debye [68] expression for the rate of reaction has been discussed in Chap. 3, Sect. 1. With an Onsager distance 0.7—1.0 nm, the effective reaction radius is 0.9—1.2 nm or... 

Diffusion-controlled phenomena affecting the termination and propagation reactions, as well as the initiator efficiency (gel-, glass- and cage effect, respectively) are expressed in terms of a reaction-limited term and a diffusion-limited one (Keramopoulos and Kiparissides, 2(X)2). The latter depends on the diffusion coefficients of the corresponding species (i.e., polymer and monomer) and an effective reaction radius. 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

The reaction scheme of Schwarz, with the specific rates, is shown in Table 7.1. Comparison with later compilations (Anbar et al., 1973, 1975 Farhataziz and Ross, 1977) indicates that most of these rates are reasonable within the bounds of experimental error. Some of the rates are pH-dependent, and when both reactants are charged there is a pronounced ionic strength effect these have been corrected for by Schwarz. He further notes that the second-order rates are not accurate for times less than 1 ns if the reaction radius... 

Equation (9.2) shows the effect of the reaction radius rl on the escape probability, which, remarkably, is free of the diffusion coefficient. Normally rc r which reduces Eq. (9.2) to the celebrated Onsager formula 0 = exp(- r /rg) as given by Eq. (9.1). 

The results obtained in Ref 30 for partially diffusion-controlled recombination show that the field dependence of the recombination rate constant is affected by both the reaction radius R and the reactivity parameter p [cf. Eq. (33)]. Depending on their relative values, the rate constant can be increased or decreased by the electric field. The latter effect predominates at low values of p, where the reactants staying at the encounter distance are forced to separate by the electric field. 

In Eq. (20), k is the rate coefficient of the excited molecule decay without quencher k = r ). Using this equation in Fig. 5, we show the time dependence of the relative excited cyclohexane molecule concentration in a solution containing 0.05 mol dm CCI4. In order to show the effect of static quenching we chose a large reaction radius of a = 1.3 nm. It is... 

Therefore, equation (4.2.21) with the substitution of for R cannot describe correctly the process of the steady-state formation if the diffusion process is controlled by the strong tunnelling (x 3> 1). In other words, strong tunnelling could be described in terms of the effective recombination radius i eff analogous to the black sphere in the steady-state reaction stage only. 

Fig. 4.10. The ratio of effective hopping reaction radius to the diffusion one vs. hop length [81]. 1 - exact calculation. Different approximations curve 2 - equation (4.3.26), 3 - (4.3.22), 4 - (4.3.23), 5 - (4.3.27). Diffusion mechanism holds between the axis x = 0 and the vertical line, hopping on the right hand side of it.

As it was mentioned above, up to now only the dynamic interaction of dissimilar particles was treated regularly in terms of the standard approach of the chemical kinetics. However, our generalized approach discussed above allow us for the first time to compare effects of dynamic interactions between similar and dissimilar particles. Let us assume that particles A and B attract each other according to the law U v(r) = — Ar-3, which is characterized by the elastic reaction radius re = (/3A)1/3. The attraction potential for BB pairs is the same at r > ro but as earlier it is cut-off, as r ro. Finally, pairs AA do not interact dynamically. Let us consider now again the symmetric and asymmetric cases. In the standard approach the relative diffusion coefficient D /D and the potential 1/bb (r) do not affect the reaction kinetics besides at long times the reaction rate tends to the steady-state value of K(oo) oc re. 

Here Rq(D) is the diffusion-dependent effective quenching radius, which plays in the liquid-state kinetics the same role as the energy-dependent reaction cross sections in gas-phase kinetics. This is the single parameter that determines the universal form of ns(r) at large r following from Eq. (3.41) and boundary condition ns(oo,t) = 1 [7] ... 

Figure 3.8. (a) The linear viscosity dependence of the inverse ionization rate in the reaction studied in Ref. 98. Bullets—experimental points solid line—fit performed with the generalized Collins—Kimball model, (b) The effective quenching radius for the same reaction in the larger range of the viscosity variation. Bullets—experimental points solid fine—fit performed with the encounter theory for the exponential transfer rate. The diffusion coefficient D given in A2/ns was calculated from the Stokes—Einstein relationship corrected by Spemol and Wirtz [100]. 

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