QRRK

CHEMACT A Computer Code to Estimate Rate Constants for Chemically-Activated Reactions, Dean, A. M Bozzelli, J. W. and Ritter, E. R. Combust. Sci. Tech. 80, 63-85 (1991). A computer code based on the QRRK treatment of chemical activation reactions to estimate apparent rate constants for the various channels that can result in addition, recombination, and insertion reactions. 

To illustrate the application of the QRRK method, consider the unimolecular decomposition of SiH we considered previously. The parameters... 

The QRRK approach illustrated above also constitutes the basis to analyze the behavior of the reverse, i.e., association, reactions that proceed through chemically activated transition states. Recently Dean (1985) reformulated the unimolecular quantum-RRK method of Kassel and devised a practical method for the proper description of the fall-off behavior of bimolecular reactions, including reactions when multiple product channels are present. The method developed was shown to describe the behavior of a large variety of bimolecular reactions with considerable success (Dean and Westmoreland, 1987 Westmoreland et ai, 1986). 

Fig. 21. Unimolecular QRRK predictions for the rate of decomposition of SiH4...

To discuss the major features of the bimolecular QRRK, consider the following reaction mechanism ... 

To illustrate the utility of the bimolecular QRRK theory, consider the recombination of CHjCl and CHjCl radicals at temperatures in the range 800-l,5(X) C. This recombination process is important in the chlorine-catalyzed oxidative pyrolytic (CCOP) conversion of methane into more valuable C2 products, and it has been studied recently by Karra and Senkan (1988a). The following composite reaction mechanism represents the complex process ... 

Other reactions, such as C-H bond scissions and Hj eliminations, although possible, have been shown to be unimportant under the conditions studied. In Table XIV, the parameters needed for the QRRK analysis of the recombination of CH2CI radicals are presented. The methods and sources used to obtain these data are the same as those noted in the discussion of the unimolecular QRRK method. In Fig. 22, the apparent rate coefficients for... 

Fig. 22. Bimolecular QRRK predictions for the rates of recombination of CH2CI—CH2CI.

Dean, A. M, and Westmoreland, P, R., Bimolecular QRRK analysis of methyl radical reactions, Int. J. Chem. Kinetics 19, 207 (1987). 

Karra, S. B., and Senkan, S. M., Analysis of the chemically activated CH2CI/CH2CI and CH3/CH2CI recombination reactions at elevated temperatures using the QRRK method, Ind. Eng. Chem. Research 27, 447 (1988b). 

Rice, Ramsperger, and Kassel [206,333,334] developed further refinements in the theory of unimolecular reactions in what is known as RRK theory. Kassel extended the model to account for quantum effects [207] this treatment is known as QRRK theory. 

Fig. 10.5 Reaction pathways in the QRRK analysis of unimolecular reactions.

The reaction scheme in the QRRK theory for unimolecular decomposition can be written... 

A schematic of the reactions and energy levels involved in this scheme is shown in Fig. 10.5. The QRRK reaction scheme differs in several respects from Lindemann s treatment, reactions 10.99 and 10.100. The rate constant for the excitation step 10.136 is written to explicitly include the dependence on the amount of energy e — nhv transferred to C (n). The vibrational energy obtained in reaction 10.136 is assumed to be randomly or statistically distributed over s identical vibrational modes of the molecule. The rate constant kact (n,m) in reaction 10.137 is for formation of the activated complex, in which at least m quanta of vibrational energy have accumulated in a critical bond (out of the total of n). This rate constant depends on both n and m, and is derived below. 

The QRRK model postulates that vibrational energy can freely flow (internally) from one vibrational mode in the molecule to another. This is a very significant assumption. For a collection of harmonic oscillators, energy in a particular vibrational mode will stay in that mode it cannot flow into other vibrational modes of the system. That is, a system of harmonic oscillators is uncoupled. 

The QRRK reaction scheme 10.136 through 10.138 can be rewritten equivalently as... 

Therefore the QRRK expression for the unimolecular rate constant is... 

But the left-hand side equals K(n, m), the fraction of excited C at energy level n in the Boltzmann distribution. By Eqs. 10.141 and 10.163 we obtain the desired QRRK expression for the excitation rate constant ke(n) ... 

Fig. 10.7 QRRK analysis [207] of azomethane, CH3N2CH3, unimolecular decomposition at 603 K (solid curve), and comparison with experimental data (points) from Ramsperger [326].

As an example calculation using QRRK theory, we consider the unimolecular decomposition of azomethane, CH3N2CH3, from Kassel s original paper [207], Kassel tested... 

The QRRK rate constant in Fig. 10.7 certainly fits the experimental data well. However, this is to be expected given the origin of the parameters in the model. Specifically, the high-pressure Arrhenius parameters were obtained from fits to the experimental data. The number of oscillators was taken as an adjustable parameter, as was the collision cross section used in ks. Thus the QRRK curve in Fig. 10.7 should match the experiment in the high-pressure limit, and two parameters were varied to enable a fit to the pressure fall-off behavior. 

Summary of QRRK Unimolecular Rate Constant In summary, the QRRK result for the observed unimolecular reaction rate constant fcun was given by Eq. 10.154 as... 

The modem theory theory of unimolecular reactions was established by Marcus, who built upon QRRK theory [260,261,431]. This work is known as the RRKM theory. We will... 

Because QRRK theory was developed long before computing became readily available, it had to employ significant physical approximations to obtain a tractable result. The most significant assumption was that the molecule is composed of s vibrational modes with identical frequency i and that other molecular degrees of freedom are completely ignored. RRKM theory relies on neither approximation and thus has a much sounder physical basis. In the limit of infinite pressure, RRKM theory matches the transition state theory discussed in Section 10.3. 

In RRKM theory, the activation rate constant kact of Eq. 10.146 in QRRK theory is replaced by the more rigorous... 

This section treats the theory of chemical activation reactions more rigorously, at the same level of approximation as in the discussion of unimolecular reactions in Section 10.4.4. That is, the QRRK theory of chemical activation reactions is developed here. This theory for bimolecular reactions was set out by Dean and coworkers [93,428],... 

The QRRK treatment of bimolecular chemical activation reactions considers in more detail the energy-dependence of the rate coefficients. Begin by modifying the chemical activation reaction scheme of Eqs. 9.132 to 9.134 to account for the specific energy levels of the rate constants and activated species. 

For this QRRK analysis we will define the zero of energy as the ground-state energy of the stabilized C molecule. As in QRRK the analysis of unimolecular reactions, assume that the excited C molecule consists of s identical oscillators, each with vibrational frequency v. When we write C ( ), this indicates that the excited intermediate species has been formed with n quanta of vibrational energy thus, its total energy is E = nhv above the ground-state energy of C (which we have arbitrarily set to zero). 

Therefore the derived expression for the QRRK chemical activation bimolecular rate constant for formation of products D + E is... 

It is interesting to examine these two QRRK rate constants in the limits of very high and very low pressures. First, look at the high-pressure limit of Eq. 10.198 ... 

Show that in the classical limit, the QRRK expression for kd(n, m), Eq. 10.171, gives the result derived by Kassel s earlier paper [206]... 

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