Unimolecular master equation

Formal Solution of the Unimolecular Master Equation Eigenvalue Separation... 

An important example for the application of general first-order kinetics in gas-phase reactions is the master equation treatment of the fall-off range of themial unimolecular reactions to describe non-equilibrium effects in the weak collision limit when activation and deactivation cross sections (equation (A3.4.125)) are to be retained in detail [ ]. 

Troe J 1977 Theory of thermal unimolecular reactions at low pressures. I. Solutions of the master equation J. Chem. Phys. 66 4745-57... 

Venkatesh P K, Dean A M, Cohen M H and Carr R W 1999 Master equation analysis of intermolecular energy transfer in multiple-well, multiple-channel unimolecular reactions. II. Numerical methods and application to the mechanism of the C. + O2 reaction J. Chem. Phys. Ill 8313... 

C3.3.5.1 MASTER EQUATION ANALYSIS OF UNIMOLECULAR REACTION DYNAMICS... 

At these low pressures the reaction is in the second-order region of the unimolecular falloff, and low-pressure-limit rate coefficients, k0, are obtained. A master equation calculation was used to obtain the critical energy, E0, and average energy transferred per collision, from which an expression for the high-pressure rate coefficient was obtained. 

Knyazev et al. have reported the decomposition of the 1-chloroethyl radical by photoionization mass spectrometry [127]. Rate coefficients, determined as a function of temperature (848-980 K) and bath gas density (3-22 x 1016 molecules cm"3) in He, Ar, and N2, were in the unimolecular falloff. The falloff behavior was modeled by a master equation analysis. 

Unimolecular reactions such as this one involve conformational changes of a single molecule without explicit interaction with other molecules. Implicit interactions with the solvent, of course, are assumed. As for the generic master equation introduced above, rather than asking the question What is the concentration of species A at time / ." we ask What is the probability that a single molecule is in state A at time tl ... 

These estimates bracket the NASA-JPL and lUPAC recommendations of 6.5x10 and 7.7 x 10 cm molecule s [9,60]. It is therefore possible fo reconcile fhe thermochemistry proposed here with the observed lO + NO2 recombination kinetics while employing reasonable input parameters for the unimolecular model. Nevertheless it must be stressed, as emphasized earlier [16], that there is considerable uncertainty in some of the input parameters to an RRKM analysis, especially the Frot term. It is of interest to compare the present kinetic calculations with the Multiwell [61] Master Equation calculations on this system by Golden [16]. He used a Morse potential to locate the centrifugal maximum, and from the bond extension Frot 2.1 is derived, about 1/7 of fhaf used here. On the other hand, the higher Eo value yields a density of sfafes larger by a facfor of 6, and fhese two factors largely cancel. 

UNIMOL Calculation of Rate Coefficients for Unimolecular and Recombination Reactions. R. G. Gilbert, T. Jordan, and S. C. Smith, Department of Theoretical Chemistry, Sydney, NSW 2006, Australia, 1990. FORTRAN computer code for calculating the pressure and temperature dependence of unimolecular and recombination (association) rate coefficients. Theory based on RRKM and numerical solution of the master equation. See Theory of Unimolecular and Recombination Reactions, by R. G. Gilbert and S. C. Smith, Blackwell Scientific Publications, Oxford, 1990. 

The parametric representation of unimolecular rate coefficients, both at the low pressure limit and in the fall-off region has been the result of a monumental work by Troe and his coworkers [67-71]. This work is tremendously useful for applications of unimolecular reactions to complex systems, for example to combustion or to atmospheric chemistry, as it permits accurate representation of the fall-off without having to to a time-consuming and difficult Master Equation calculation. 

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]. 

Solutions of the Master Equation.—In the low-pressure limit of a thermal unimolecular... 

This treatment of the interplay of relaxation and randomisation in thermal unimolecular reactions has assumed that the two processes are completely separable, on the rather plausible grounds that the randomisation rates are very much faster than the relaxation rate. The conditions for such separability are, in fact, known. We can write the full master equation for reaction as... 

No other analytic solution to the master equation for a weak collision system over the whole range of pressures has yet been found. A solution is known, at the low pressure limit only for a rather limited exponential probability model of a unimolecular reaction [77.T2 80.F1], and Troe has developed empirical schemes for determining the pressure range over which the fall-off exhibits curvature and for joining smoothly the high and low pressure limiting solutions [77.Q 79.T2]. 

By about 1972 also, we had unearthed most of the salient features of the diatomic dissociation problem [73.A 73.K], and I now had some appreciation of the properties of the master-equation approach to the chemical reaction problem. Consequently, I spent a sabbatical term at the Physical Chemistry Laboratory in Oxford in the autumn of 1972 planning the beginnings of the state-to-state treatment of unimolecular reactions described in the following pages. The unimolecular dissociation... 

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. 

In this section we summarize methods for solution of the master equation, which couples the collisional relaxation of the highly excited unimolecular species with the microcanonical dissociation rates to determine, for a given temperature and pressure, the non-equilibrium probability distribution for the molecular population over energies and angular momenta, and thence the thermal rate coefficient k(T, P). The separability of molecular interactions in the gas phase into unimolecular events and bimolecular events enables the overall thermal dissociation process to be modeled by the two-dimensional master equation, expressed in continuum notation as... 

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