QRRK theory

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 ... 

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. 

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

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

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],... 

In summary, the QRRK theory result for the observed bimolecular reaction rate constant fcbimol was given by Eq. 10.198 as... 

Compute the fall-off curve using QRRK theory. For this calculation, assume a collision diameter of 4.86 A. Assume that the average energy transfer per N2-C-C5H5 collision is -0.69 kcal/mol (needed to calculate the parameter /5 used in the model). Take the number of oscillators to be, v = actual, with the frequency calculated above. Assume the reaction barrier to be E0, given above. 

Use QRRK theory to calculate kan as a function of pressure for the decomposition of azomethane at T = 563 and 603 K. Parameters needed for the calculation are given in Section 10.4.4, in the discussion of Fig. 10.7. Plot calculated rate constant in units of 1/s versus pressure (in atm) include in the same plot a comparison with experimental data of Ramsperger [326], which can be found in the data file azomethanedata.csv. 

Use QRRK theory to calculate kstab and kprod at 1000 K as a function of bath-gas concentration [M] over a range 10-13 to 1012 mol/m3. 

Calculate the stabilization rate constant kstab and bimolecular product formation rate constant ftprod using QRRK theory, as specified below. 

Using QRRK theory, calculate kstab and fcprod over the temperature range 270 to 2500 K for a total pressure of 1 atm. Display the calculated rate constants in an Arrhenius-type plot. 

These problems arise because of the use of the classical density of states rather than the proper vibrational energy levels, and, of course an alternative would be to use the more difficult QRRK theory. If this is done, acceptable fits may be obtained to experimental data using the full number of oscillators. However, QRRK theory is not easily applicable with a realistic spectrum of vibrational frequencies, and it is preferable to use an alternative theory such as the RRKM theory instead. 

The Bozzelli and Dean [113] mechanism used QRRK theory and placed the transition state for isomerization at even lower energies. An alternative, high energy four membered ring transition state was also proposed which leads to CH3CHO -l- OH. 

Multifrequency Quantum Rice-Ramsperger-Kassel (QRRK) is a method used to predict temperature and pressure-dependent rate coefficients for complex bimolecular chemical activation and unimolecular dissociation reactions. Both the forward and reverse paths are included for adducts, but product formation is not reversible in the analysis. A three-frequency version of QRRK theory is developed coupled with a Master Equation model to account for collisional deactivation (fall-off). The QRRK/Master Equation analysis is described thoroughly by Chang et al. [62, 63]. 

The single frequency versions of RRK and QRRK" theories predict experimental fall-off curves in most cases reasonably well if s is identified with the number of effective oscillators, which is often about one-half of the number of actual oscillators. Several ways to calculate the number of effective oscillators s are suggested in the literature. For example, Troe and Wagner [3] use... 

CARRA CARRA, for chemically activated reaction rate analysis, calculates apparent rate constants for multi-well, multi-channel systems based on QRRK theory. It uses either the MSC (CAR-RA MSC) or the steady-state ME (CARRA ME) approach. The original concept was based on a single frequency representation of the active modes of each isomer [35,36]. Later, the code was updated to handle three representative frequencies. Descriptions of these earlier versions as well as applications can be found in Refs. [7,37]. CARRA is a modihed version of these older codes, which is currently still under development [38]. 

For reactions that are unimolecular in one or both directions, the reaction rate is expected to be pressure dependent, as discussed in detail in an earlier chapter of this text. In the high-pressure limit, conventional transition state theory as described in the previous section can be applied to estimate the rate constant. The only change in equation (20) is that only a single reactant partition function appears in the denominator. The pressure dependence can then be described at various levels of sophistication, from QRRK theory to RRKM theory, to full master equation treatments using microcanonical rate constants from RRKM theory, as described in the chapter by Carstensen and Dean. Because these approaches have been described in detail there, they are not treated in the present chapter. 

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