Pair Kinetic Equation

The pseudo-Liouville operator does couple these doublet fields to triplet fields such as 8 abs cds involving the solvent molecules. Thus one of the simplest forms for the pair kinetic equation can be obtained by explicitly including doublet and triplet fields in the generalized Langevin equation. This procedure yields a treatment of the effects of solvent dynamics on the motion of the reactive pair that is much more sophisticated than that given in the singlet kinetic equation discussed in the preceding... 

Analogous BGK models can be constructed at the pair kinetic equation level. The simplest model of this type would approximate the collision operator in (7.32) as... 

Since the derivation of the pair kinetic equation is similar to that given in Appendix C for the singlet kinetic equation, we only outline the calculation. We again restrict the calculation to the irreversible reaction the details of the full reversible reaction case are given in Ref. 53. 

The pair kinetic equation in Section VII.D follows directly from these results if the dynamic memory function " xbs.abs neglected, and the static structural correlations in (D.3) to (D.6) are approximated so that all binary collisions are calculated in the Enskog approximation. [This is the singly independent disconnected (SID) approximation, which is discussed in detail in Ref. 53.] We have also used the static hierarchy to obtain the final form involving the mean force, given in (7.32). This latter reduction involving the static hierarchy is carried out below in the context of a comparison of the singlet and doublet formulations. 

To examine the relation between the pair kinetic equation (7.32) and the corresponding propagator for the doublet field that enters into the singlet field equation derived in Appendix C, consider (C.12). The static memory kernel ab,ab defined in (C.l 1) may be written in a form closely related to that in (7.32) by using the static hierarchy. For a hard-sphere system, the static hierarchy takes the form" ... 

The operator on this correlation function, involving the doublet field 5ajab(12), may now be compared directly with the operator in the pair kinetic equation (7.32). There, of course, the possibility of soft forces between the solute species was also taken into account. The ring operator in (7.33) and (7.34) takes the place of < >ab,ab above. In the singlet kinetic equation that we used in Section VII.C, we ignored fl t... 

In particular if the reaction rate depends only on Cj, which is the case, for example, if the reaction is irreversible with mass,-action kinetics, then these reduce further to a pair of equations, namely... 

When this holds, the kinetic equations reduce to single exponentials. Chipperfield6 demonstrates that approximate adherence to Eq. (4-25) suffices to fit 20 absorbancetime pairs spaced at equal times over the first 75 percent of the reaction with correlation coefficients better than 0.999. 

Furthermore, kinetic analysis of the decay rate of anthracene cation radical, together with quantum yield measurements, establishes that the ion-radical pair in equation (76) is the critical reactive intermediate in osmylation reaction. Subsequent rapid ion-pair collapse then leads to the osmium adduct with a rate constant k 109 s 1 in competition with back electron-transfer, i.e.,... 

The kinetic equations. Having established from electrochemical considerations that the simultaneous presence of appreciable concentrations of free cations and ion-pairs can reasonably be expected in many of the systems of interest in the present context, the next task is to establish the shape of the curves relating p and q to c, and, hence, the curves to be expected for the dependence of rate on c under various conditions. These considerations are made much easier by the fact that the shapes of the curves are independent of the value of K. The curves of p against c and q against c (all in units of K) given by equation (14) are parabolas (curves A and B of Figure 1). 

The nucleophile in the S.v2 reactions between benzyldimethylphenylammonium nitrate and sodium para-substituted thiophenoxides in methanol at 20 °C (equation 42) can exist as a free thiophenoxide ion or as a solvent-separated ion-pair complex (equation 43)62,63. The secondary alpha deuterium and primary leaving group nitrogen kinetic isotope effects for these Sjv2 reactions were determined to learn how a substituent on the nucleophile affects the structure of the S.v2 transition state for the free ion and ion-pair reactions64. 

However, Waite s approach has several shortcomings (first discussed by Kotomin and Kuzovkov [14, 15]). First of all, it contradicts a universal principle of statistical description itself the particle distribution functions (in particular, many-particle densities) have to be defined independently of the kinetic process, but it is only the physical process which determines the actual form of kinetic equations which are aimed to describe the system s time development. This means that when considering the diffusion-controlled particle recombination (there is no source), the actual mechanism of how particles were created - whether or not correlated in geminate pairs - is not important these are concentrations and joint densities which uniquely determine the decay kinetics. Moreover, even the knowledge of the coordinates of all the particles involved in the reaction (which permits us to find an infinite hierarchy of correlation functions = 2,...,oo, and thus is... 

In fact, Waite s approach needs a kind of demon to mark (enumerate) pairs of particles at moments of their birth and then following the time development w(r,t) of all such geminate pairs - even when these pairs already completely have mixed at the bimolecular reaction stage What is said above demonstrates quite well how the violation of statistical principles in deriving kinetic equations can lead to unphysical paradoxes (discussed in detail in [14, 15]). 

Below we take into account the non-linear terms in the kinetic equations containing functionals J (coupling spatial correlations of similar and dissimilar particles) but neglect the perturbation of the pair potentials assuming that il(r, t) = l3U(r). This is justified in the diluted systems and for the moderate particle interaction which holds for low reactant densities and loose aggregates of similar particles. However, potentials of mean force have to be taken into account for strongly interacting particles (defects) and under particle accumulation when colloid formation often takes place [67],... 

Here p is a creation rate (dose rate) of stable pairs of defects, while the quantity 5 is a fraction of the reaction volume overlapped by recombination spheres, i.e., the ratio of the effective recombination volume to the entire volume of the crystal. The problem of constructing the kinetic equation of... 

Kinetic equations for the electron tunneling reactions in the case of non-pair distributions of reagents have been obtained [8-11]. Two methods have been used in the literature to obtain these equations. Both of them have been used earlier to describe the kinetics of energy transfer processes. These are the method of pair density and that of conditional concentrations. It has been shown [15] that these two different methods are, in fact, equivalent and lead to identical results. The detailed description of the pair density method can be found in refs. 3,5,7,13,16, and 33 and that of the method of conditional concentrations in refs. 5,8,15, and 17. 

Comparison of eqns. (3), (18), and (20) shows that both for the pair and non-pair reagent distributions the derivation of kinetic equations for the electron tunneling reactions is reduced, as a matter of fact, to the calculation of the value... 

When the concentrations of reagents have comparable values, it is necessary to pay attention to the correlation effect in the decay of different donors, i.e. to consider the fact that the spatial distribution of acceptors near the chosen donor can change as a result of the decay of the acceptors in the reactions with other donors neighbouring the chosen one. The rigorous derivation of kinetic equations with the consideration of such a correlation is, as far as we know, unavailable. The approximate description of the kinetics of a biomolecular electron tunneling reaction at n(t) = N t) can be given in terms of the pair density method with the help of eqn. (19) in which, however, N is not a constant quantity but depends on time in the same way as n(t), i.e. 

Kinetic equations for the recombination luminescence intensity in the presence of a permanent electric field for the arbitrary non-pair initial spatial distribution of the reagents in the case when the concentration of one of them considerably exceeds that of the other, have been obtained in ref. 21. These equations have the form... 

According to (4.38), the correlation function (5.4) determines the collision integral of the relevant kinetic equation for the distribution of pairs. 

The experimental curves in Fig. 2.9 are compared with the curves calculated according to the second-order kinetic equation for two different initial temperatures. The divergence between the pairs of curves at the final stages of the process reaches 15 %, while the experimental is less than 3 %. This proves that the equation is inadequate. On the other hand, a kinetic equation that takes into account the effect of self-deceleration fits the experimental data along the whole curve. Therefore, we can predict that the reaction in the system under discussion will be incomplete. 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

To develop the kinetic equations in condensed phases the master equation must be formulated. In Section 3 the master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The latter set of equations permits consideration of history of formation of the local solid structure as well as its influence on the subsequent elementary stages. The many-body problem and closing procedure for kinetic equations are discussed. The influence of fast and slow stages on a closed system of equations is demonstrated. The multistage character of the kinetic processes in condensed phase creates a problem of self-consistency in describing the dynamics of elementary stages and the equilibrium state of the condensed system. This problem is solved within the framework of a lattice-gas model description of the condensed phases. 

Eq. (28) are used initially for obtaining the kinetic equations for condensed phase processes. Let us discuss a fairly general case for the theory of surface processes and assume the lattice to be inhomogeneous and the radius of the adspecies interaction pair potential to be equal to R (R> 1). 

A presence of the kinetic equations for pair correlators in closed system of equation allows to describe properties about a prehistory of the experimental system (such as the shape and size of spatio-temporal distribution of the reagents along the surface or interface area) and the time of evolution during the considered processes. 

In the second case, if the species mobilities differ greatly, the dimensionality of the system of kinetic equations decreases [103], Let all the components be divided into two groups of species a slow (5) and a rapid (r) one. This yields three types of pair functions. For the rapid species the condition of the equilibrium distribution can be considered as satisfied. Then, for the pair functions of types sr and rr instead of the kinetic Eqs. (32) algebraic relations in Appendix A apply, whose dimensionality can be lowered using the method of substitution variables according to Appendix B. In this case the kinetic Eqs (31) for the local concentrations and Eq. (32) for the pair functions type ss do not change. A similar situation remains in passing to the one-dimensional discrete and point-like models. 

---