Kinetic equations singlet

Oxygen radical anion forms excited-singlet oxygen in different pathways, e.g. by a reaction with copper-cysteine-oxygen complex to yield the excimer (02)2- The computerized kinetic equations derived from this scheme allowed predictions in respect of the chemiluminescence intensity as a function of the oxygen and cysteine concentrations and as a function of time these were satisfactorily confirmed by the ex-... 

Photosensitization of diaryliodonium salts by anthracene occurs by a photoredox reaction in which an electron is transferred from an excited singlet or triplet state of the anthracene to the diaryliodonium initiator.13"15,17 The lifetimes of the anthracene singlet and triplet states are on the order of nanoseconds and microseconds respectively, and the bimolecular electron transfer reactions between the anthracene and the initiator are limited by the rate of diffusion of reactants, which in turn depends upon the system viscosity. In this contribution, we have studied the effects of viscosity on the rate of the photosensitization reaction of diaryliodonium salts by anthracene. Using steady-state fluorescence spectroscopy, we have characterized the photosensitization rate in propanol/glycerol solutions of varying viscosities. The results were analyzed using numerical solutions of the photophysical kinetic equations in conjunction with the mathematical relationships provided by the Smoluchowski16 theory for the rate constants of the diffusion-controlled bimolecular reactions. 

In conclusion, it appears that the application of recent theories of nonequilibrium statistical mechanics to transport in dense media confirms Eyring s theory and provides in addition a convenient framework for possible extensions and refinements for instance, Allen et al. have recently combined the original PNM model with an approximate kinetic equation for the singlet distribution function and obtained a stilt better agreement with experiment. 

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... 

Since the derivation of the pair equation exactly parallels that for the singlet kinetic equation, the details are sketched in Appendix D and not given here. It is quite easy to derive a kinetic equation for the general reversible reaction case the calculations need only be carried out in matrix form. To avoid this more complex notation and to present the results in simple form, however, we again give only the results for the irreversible decay of the AB pair field. 

Consider first the kinetic equation for the singlet field in a nonreactive system. It has the general form given in (7.2a), which now reduces (suppressing species labels) to... 

An equation with the form of the macroscopic law in (2.16) can be obtained from the singlet field kinetic equation by projecting out the velocity dependence of the phase-space correlation functions. A comparison of the resulting equation with this macroscopic law can then yield a microscopic correlation function expression for the rate kernel. 

Using these results, the kinetic equation for the A singlet phase-space correlation function takes the form... 

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... 

Equation (35) can be regarded as a generalized kinetic equation. The relationship of this equation to the Boltzmann equation is well under-stood. Using linear response theory, one can show that the linear deviation of the singlet distribution function from equilibrium, for a system initially held in constrained equilibrium, is proportional to... 

Here ho is the kinetic energy and nuclear attraction operator while and 1C are the coulomb and exchange operators, respectively. The coefficients X and Y are solutions of the RPA equations, which for the / singlet transition with excitation energy can be written as... 

Therefore, this method allows for the determination of relative rate constants for the excitation step in a complex reaction system, where this step cannot be observed directly by kinetic measurements. The singlet quantum yield at infinite activator concentrations (high-energy intermediates formed interact with the activator, is also obtained from this relationship (equation 5). 

An indirect method has been used to determine relative rate constants for the excitation step in peroxyoxalate CL from the imidazole (IM-H)-catalyzed reaction of bis(2,4,6-trichlorophenyl) oxalate (TCPO) with hydrogen peroxide in the presence of various ACTs18. In this case, the HEI is formed in slow reaction steps and its interaction with the ACT is not observed kinetically. However, application of the steady-state approximation to the reduced kinetic scheme for this transformation (Scheme 6) leads to a linear relationship of 1/S vs. 1/[ACT] (equation 5) and to the ratio of the chemiluminescence parameters /ic vrAi), which is a direct measure of the rate constant of the excitation step. Therefore, this method allows for the determination of relative rate constants for the excitation step in a complex reaction system, where this step cannot be observed directly by kinetic measurements18. The singlet quantum yield at infinite activator concentrations ( °), where all high-energy intermediates formed interact with the activator, is also obtained from this relationship (equation 5). 

Shin and Kapral have applied the kinetic theory of reactions in solution to the case of two radicals (e.g. iodine atoms) recombining with one another [286]. As it is the behaviour of both radicals which is of interest, Shin and Kapral seek to evaluate the doublet density of A and B, t), rather than the singlet density as used in the case of homogenous reactions of the type [eqn. (306)] where one species is not transformed. The doublet density changes as a result of collision with the solvent and so the triplet density, / BS(123, f), is of concern and the equation for the doublet density is like that of eqn. (295) with a = A, j3 = B and p = S. The triplet density, /f 8, itself depends on a quartet distribution, that of the radical reactants A and B and any two solvent molecules. The second solvent molecule can collide with A, B or the first solvent molecule and thereby change f BS. Following the usual procedure, the triplet density... 

The UT equation (3.289a) continues to describe the kinetics of bimolecular charge separation, but the total survival probability of charged products is the sum of fractions originating from the triplet and singlet RIPs ... 

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