Reaction radius

Of the ethers, rate constants for es reactions are available for tetrahydrofuran (THF). Since the neutralization reaction, THF+ + es, is very fast, only fast reactions with specific rates 10u-1012 M s"1 can be studied (see Matheson, 1975, Table XXXII). Bockrath and Dorfman (1973) compared the observed rate of the reaction es + Na+ in THF, 8 x 1011 M 1s 1, with that calculated from the Debye equation, <3 x 1011 M-1s-1. Although the reaction radius is not well known, the authors note on a spectroscopic basis that Na+ and es are strongly coupled in THF Thus, the reaction of a solute with (Na+, es) in THF is much slower, sometimes by an order of magnitude, than the corresponding reaction with es only. Reaction with pyrene is an example. 

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... 

After the jump, the particle is taken to have reacted with a given probability if its distance from another particle is within the reaction radius. For fully diffusion-controlled reactions, this probability is unity for partially diffusion-controlled reactions, this reaction probability has to be consistent with the specific rate by a defined procedure. The probability that the particle may have reacted while executing the jump is approximated for binary encounters by a Brownian bridge—that is, it is assumed to be given by exp[—(x — a)(y — a)/D St], where a is the reaction radius, x andy are the interparticle separations before and after the jump, and D is the mutual diffusion coefficient of the reactants. After all... 

The independent reaction time (1RT) model was introduced as a shortcut Monte Carlo simulation of pairwise reaction times without explicit reference to diffusive trajectories (Clifford et al, 1982b). At first, the initial positions of the reactive species (any number and kind) are simulated by convolving from a given (usually gaussian) distribution using random numbers. These are examined for immediate reaction—that is, whether any interparticle separation is within the respective reaction radius. If so, such particles are removed and the reactions are recorded as static reactions. 

The time approach relies entirely on independent diffusion-reaction time without reference to distances. The reaction product inherits the time sequence of one of the parents chosen at random however, its residual time to react with another species is scaled inversely relative to its mutual diffusion confident. A heuristic correction is also made for the change of reaction radius (Clifford et al, 1986). 

The position approach strives to get the positions of the reactive particles explicitly at the reaction time t obtained in the IRT model. While the nonreac-tive particles are allowed to diffuse freely, the diffusion of the reactive particles is conditioned on having a distance between them equal to the reaction radius at the reaction time. Thus, following a fairly complex procedure, the position of the reactive product can be simulated, and its distance from other radicals or products evaluated, to generate a new sequence of independent reaction times (Clifford et al, 1986). 

FIGURE 7.5 Same as in Figure 7.4, where reaction of products is included with equal reaction radius. MC and RRT simulations (time approach) for the product AAB. From Clifford et al. (1986), by permission of The Royal Society of Chemistry . 

Another virtue of the procedure is that it can explicitly take into account a partially diffusion-controlled recombination reaction in the form of Collins-Kimball radiation boundary condition—namely, j(R, t) = -m(R, t) where j(R, t) is the current density at the reaction radius and K is the reaction velocity k— < > implies a fully diffusion-controlled reaction. Thus, the time dependence of e-ion recombination in high-mobility liquids can also be calculated by the Hong-Noolandi treatment. 

MC simulation for multiple ion-pair case is straightforward in principle. A recombination, if necessary with a given probability, is assumed to have taken place when an e-ion pair is within the reaction radius. Simulation is continued until either only one pair is left or the uncombined pairs are so far apart from each other that they may be considered as isolated. At that point, isolated pair equations are used to give the ultimate kinetics and free-ion yield. 

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). 

FIGURE 9.1 Simplified derivation of Onsager s escape probability formula. In the stationary state a unit electron current at ro partitions as I toward the reaction radius and as Is toward the sink at infinity the latter is the escape probability. Reproduced from Mozumder (1969a), with the permission of John Wiley Sons, Inc. ... 

In the solvents ethane, propane, and hexane, a reaction radius, R, of 1.4 nm was found for attachment to SFg. This is close to the theoretical maximum radius for electron attachment in the gas phase [106]. 

Some reactants have rate constants higher than 3xl0 M sec examples are nitrobenzene and o-dinitrobenzene. These two compounds have large dipole moments of 4.1 and 6.1 Debye, respectively, and it has been shown [110] that the rate constants in cyclohexane increase with dipole moment because the reaction radius increases. That dependence is given by [117] ... 

The value of for attachment to SFg given in Fig. 8 is 2.1 x reaction has also been studied in other high mobility liquids including methane (/id = 400 cm /Vs), argon = 400 cm /Vs), and xenon = 2000 cm /Vs) [127-129], and the rate constant is nearly constant at 3 1 x lO sec h This has been explained by Warman [106] and others as due to the fact that the residence time, td, of an electron within a reaction radius, R, is short compared to the attachment time, ta. Thus rate constants would be expected to fall off with increasing mobility according to the equation ... 

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. 

Equation 2 indicates that if the distance r between the cation radical and the electron becomes less than the reaction radius R, they recombine. The typical shape of/(r,ro)> the initial distribution function, is expressed by the following exponential or Gaussian. 

Figure 14 (a) Time-dependent behavior of cation radicals in liquid -dodecane monitored at 790 nm. The dotted and the solid lines represent the experimental curve and the simulation curve, respectively. The parameters of the electron dilfusion coefficient (De) = 6.4 x 10 " cm /sec, the cation radical diffusion coefficient (D + ) = 6.0 x 10 cm /sec, the relative dielectric constant e = 2.01, the reaction radius R = 0.5 nm, and the exponential function as shown in Eq. (19) with ro = 6.6 nm were used, (b) Time-dependent distribution function obtained from fitting curve of (a), r indicates the distance between the cation radical and the electron. The solid line, dashed line, and dots represent the distribution of cation radical-electron distance at 0, 30, and 100 psec after irradiation, respectively. 

A Monte Carlo simulation with the parameter A/a, where A is the mean free path of electrons and a is the reaction radius [39,42,114]. 

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... 

In the usual treatments there is only one parameter to be fitted, i.e., the reaction radius. In practice three cases should be distinguished ... 

The only unsatisfactory aspect of this work is that it does not enable estimates of the reaction radius, R, or the mutual diffusion coefficient, D, to be made from experimentally measured rate coefficients. Nevertheless, the agreement between experiment and theory is very encouraging. Korth et al. have studied the recombination of cyano-substituted alkyl radicals and found a similar close relation between the measured rate coefficient, ft, and T/r) [45b], They presented evidence to suggest that the radicals recombine on an attractive potential energy surface. 

Transfer of an electron from a photoexcited donor to an acceptor has also been studied. Ballard and Mauzerall [219] photoexcited zinc octaethylporphyrin and found both the triplet—triplet rate coefficient and ion yields indicate a reaction radius of 2.0 0.1 nm, some 0.6 nm larger than twice the radius of this metal ligand. However, electron transfer from pyridine—ruthenium complexes does not appear to be facilitated by electron tunnelling (see Chap. 3, Sect. 2.1). 

Fig. 17, The correction factor to the Smoluchowski rate coefficient for reaction between an isotropic reactant, B, and an axially symmetric reactant, A, which has a cap of reactivity of the spherical surface, subtending a semi-angle of 77/9, 77/2, 577/6 at the sphere s centre. The reaction radius is R and the radius of B is rB. Translational and rotational diffusion coefficients are given by eqn. (118), for reactions with G = 1, i.e,...

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