The time-dependent rate coefficient

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found 

Both Hong and Noolandi [72] and Rice et al. [73] inverted Laplace transforms of order s 1/2 for small s to get the term in f 1/2. The nature of higher-order time dependence was not discussed. For the totally absorbing boundary condition (47), with Rice et al. showed 

Finally, several attempts have been made to solve the Debye—Smoluchowski equation in the time domain using approximate techniques based on uniformly small perturbations (Montroll [74], Abell and Mozumder 

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We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

The solution of this problem yields the time-dependent rate coefficient [94]... 

The time-dependent rate coefficient for the bulk ion recombination may be obtained as the total current of particles flowing through the reaction surface dA. r = R, divided... 

While no spectroscopic evidence of a ground-state complex between anthracene and carbon tetrachloride, naphthalene or 1,2-benzanthracene and carbon tetrabromide has been found, Nemzek and Ware [7] were unable to explain their steady-state fluorescence quenching measurements with the parameters deduced from the determination of the time-dependent rate coefficients unless a ground-state complex was present. This cannot be regarded as a satisfactory and consistent analysis because the time-dependent rate coefficient would be modified by the presence of the initial distribution of quencher and fluorophor in the ground state. 

Rice [147] has noted the similarity of form between the time-dependent rate coefficients from the Smoluchowski, eqn. (19), Debye— Smoluchowski, eqn. (53), diffusion and Forster transfer equations,... 

For times shorter than this, the rate coefficient may be expected to be time-dependent. For instance, with A 108 s 1, L 0.14 nm and D 10"18 m2 s 1, Reff 1 nm and > 0.3 s. This would require a long-lived phosphorescent state, such as of phenanthrene (r0 4 s), for time dependence to be observed. Since the residual time dependence of eqn. (96) contains no dependence on the radial co-ordinate, r, the time-dependent rate coefficient, eqns. (73)—(75), is simply... 

Little attention has so far been paid to studying exchange energy-transfer processes in media so viscous that a steady-state is no longer established. Butler and Pilling [200] specifically sought experimental evidence for time-dependent rate coefficients of the form of eqn. (98). They chose to study triplet phenanthrene in methanol—water mixtures and used cupric chloride as the acceptor since it is readily soluble and a very efficient quencher of triplet phenanthrene. To observe even the t 1/2 dependence of the time-dependent rate coefficient, concentrations [A] > 10-2 are required that is with Re 1 nm and [A] > 10 mmol... 

One further conceptual point in the favour of the inclusion of a noninfinite rate of encounter reaction into the theory of diffusion-limited reactions follows from the behaviour of the time-dependent rate coefficient at short times. The time-dependent Smoluchowski rate coefficient depends on time as r1/2 at short times. Indeed, immediately after formation, the rate of reaction of reactants is essentially infinite. This is because all those reactants within a separation 5R of the encounter... 

There is broad agreement between these results and those deduced by Emeis and Fehder [456a] from their molecular dynamics calculation. Finally, for reaction of neutral species with each other, Rice [484] found that the time-dependent rate coefficient was of the form... 

The time-dependent rate coefficient was first evaluated by Felderhof [460] to be... 

Fig. 53. Comparison of the time-dependent rate coefficients for the Smoluchowski...

Figure 2 (left panel) shows the energy profile for a two-level system weakly coupled to the reaction coordinate. Both the ground and excited state surfaces have two minima separated by a high barrier at (Rq) = The right panel of this figure compares the time dependent rate coefficients for quantum (QRB) and classical (CRB) treatments of the reaction coordinate for a moderately low temperature (/3 = 2). At t = 0, the CRB result for the time-dependent transmission coefficient, (t) = where is determined from a... 

The plateau value of the time-dependent rate coefficient k t) = kAB t) + kBA t), which is the sum of the forward and reverse rate constants, determines the overall chemical relaxation time, tchem = for the proton transfer. The nonadiabatic time-dependent rate coefficient k t) is shown in Figure 10.2. The rate constant extracted from the plateau value of this plot is k = 0.163 ps Up to six nonadiabatic transitions were required to obtain converged results. An examination of the trajectories in the ensemble revealed that the major nonadiabatic correction to the rate comes from two quantum transitions ground state -> coherent state ground state. This picture of how nonadiabatic transitions influence the reaction rate is quite different from that in standard surface-hopping methods. 

Another use of Brownian dynamics is to compute the time-dependent rate coefficient k t) of diffusion-controlled reactions. In this case particles start on a boundary near the active site and undergo Brownian motion until they either react or their lifetimes exceed some preset cut-off. The starting positions on this boundary are assigned according to the distribution /c(ro)exp[— f/(ro)]. In this case /c(ro) is the space-dependent intrinsic bimolecular rate constant, is k T), and t/(ro) is the potential of mean force between the two particles. 

The time-dependent rate coefficient for a system with comparable time constants for reaction (from different vibrational states) and energy relaxation (within these states) is described by a master equation (Troe and Wagner, 1967 Pritchard, 1975). The important aspects are illustrated by a two level system for which the mechanism of reaction and energy transfer is... 

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