RRKM rate coefficients

RRKM theory, developed from RRK theory by Marcus and others [20-23], is the most commonly applied theory for microcanonical rate coefficients, and is essentially the formulation of transition state theory for isolated molecules. An isolated molecule has two important conserved quantities, constants of the motion , namely its energy and its angular momentum. The RRKM rate coefficient for a unimolecular reaction may depend on both of these. For the sake... 

The fact that the RRKM rate coefficient is an upper bound on the exact classical solution for the rate coefficient is actually very useful, as it leads to a method for optimizing the position of the transition state. It is not always obvious where on the reaction coordinate the transition state should be located. If the reaction has a well defined potential barrier the natural place to locate the transition state is at the maximum of the potential barrier (the transition structure). However, if there is no barrier, there is no obvious location for the transition state. Since it is guaranteed that the theory wiU overestimate the classical rate coefficient, the position of the transition state along the reaction coordinate can be varied until the calculated rate coefficient passes through a minimum. This minimum value will be the optimal estimate of the rate coefficient and the corresponding location of the transition state will also be optimal. 

Calculation of the RRKM rate coefficient requires knowledge of the density of states of the reactants and the transition state, and a great deal of effort has been expended in working out methods for calculating, simulating or estimating these quantities. 

It is important to note that this assumption yields an RRKM rate coefficient, RRKM, that is an upper bound to the ergodic rate coefficient, ergodic, since every reactive trajectory (with xr J) necessarily has a positive velocity through the dividing surface. Thus, RRKM theory may be implemented in a variational manner, with the best approximation to ergodic obtained from the dividing surface S that provides the smallest rrkm-... 

The RRKM rate coefficient is computed using the expression Equation (6.21). The cumulative state density at the transition state in the semiclassical adiabatic approximation is given by Equation (6.24). The molecular density of states p(E,J) is obtained by the numerical derivative of Equation (6.14). The overall factor of 2/ + 1, not included in these formulae, thus cancels in the final expression. The result for J=0 versus energy above the dissociation limit, 48.4... 

Figure B2.5.7 shows the absorption traces of the methyl radical absorption as a fiinction of tune. At the time resolution considered, the appearance of CFt is practically instantaneous. Subsequently, CFl disappears by recombination (equation B2.5.28). At temperatures below 1500 K, the equilibrium concentration of CFt is negligible compared witli (left-hand trace) the recombination is complete. At temperatures above 1500 K (right-hand trace) the equilibrium concentration of CFt is appreciable, and thus the teclmique allows the detennination of botli the equilibrium constant and the recombination rate [54, M]. This experiment resolved a famous controversy on the temperature dependence of the recombination rate of methyl radicals. Wliile standard RRKM theories [, ] predicted an increase of the high-pressure recombination rate coefficient /r (7) by a factor of 10-30 between 300 K and 1400 K, the statistical-adiabatic-chaunel model predicts a...

Ion-molecule radiative association reactions have been studied in the laboratory using an assortment of trapping and beam techniques.30,31,90 Many more radiative association rate coefficients have been deduced from studies of three-body association reactions plus estimates of the collisional and radiative stabilization rates.91 Radiative association rates have been studied theoretically via an assortment of statistical methods.31,90,96 Some theoretical approaches use the RRKM method to determine complex lifetimes others are based on microscopic reversibility between formation and destruction of the complex. The latter methods can be subdivided according to how rigorously they conserve angular momentum without such conservation the method reduces to a thermal approximation—with rigorous conservation, the term phase space is utilized. 

Transition state theory yields rate coefficients at the high-pressure limit (i.e., statistical equilibrium). For reactions that are pressure-dependent, more sophisticated methods such as RRKM rate calculations coupled with master equation calculations (to estimate collisional energy transfer) allow for estimation of low-pressure rates. Rate coefficients obtained over a range of temperatures can be used to obtain two- and three-parameter Arrhenius expressions ... 

J. Manz Let me add a comment on Professor W. H. Miller s remark that he would never make himself, but I can express this as the chairman of this session. In fact, Professor Miller s extension of the standard RRKM-theory allows to predict not only the statistical mean values of the rate coefficients, but also their fluctuations. This is an important achievement in the theory of chemical reaction theory over the past couple of years and it should be adequate to call it the RRKMM theory (Ramspeiger-Rice-Kassel-Marcus-Miller) [1]. 

An RRKM model was used to calculate a high-pressure limit rate coefficient, and thermochemistry was used to obtain Arrhenius parameters for the reverse reaction, the addition of methyl to isobutane. 

Some of the initial work dealt with the formation of proton-bound dimers in simple amines. Those systems were chosen because the only reaction that occurs is clustering. A simple energy transfer mechanism was proposed by Moet-Ner and Field (1975), and RRKM calculations performed by Olmstead et al. (1977) and Jasinski et al. (1979) seemed to fit the data well. Later, phase space theory was applied (Bass et al. 1979). In applying phase space theory, it is usually assumed that the energy transfer mechanism of reaction (2 ) is valid and that the collisional rate coefficients kx and fc can be calculated from Langevin or ADO theory and equilibrium constants. 

Ethyl bromide, in a static system, was studied at 724.5-755.1 K103. The pressure dependence for the HBr elimination was observed in its fall-off region. Evaluation of the rate coefficients was performed by using the RRKM theory and the values were compared with the experimental observation. The work reported an activation energy of 216.3 kJ moT1 and an Arrhenius A factor of 1012 5. The low-frequency factor was rationalized in terms of the formation of a tight activated complex and a molecular elimination as a prevalent reaction mode. 

A possible alternative decomposition as described in equation 25 was not observed. A four-membered cyclic transition state and an Arrhenius factor similar to that of the HC1 elimination from chlorocyclobutane was assumed for the RRKM calculations. The experimental unimolecular rate coefficients are consistent with the Arrhenius equation log kx... 

More sophisticated treatments of Lindemann s scheme by Lindemann— Hinshelwood, Rice—Ramsperger—Kassel (RRK) and finally Rice— Ramsperger—Kassel—Marcus (RRKM) have essentially been aimed at re-interpreting rate coefficients of the Lindemann scheme. RRK(M) theories are extensively used for interpreting very-low-pressure pyrolysis experiments [62, 63]. 

A more general discussion of the dependence of the decomposition rate on internal energy was developed by Marcus and Rice [4] and further refined and applied by Marcus [5] (RRKM). Their method is to obtain the reaction rate by summing over each of the accessible quantum states of the transition complex. The first-order rate coefficient for decomposition of an energised molecule is shown to be proportional to the ratio of the total internal quantum states of the transition complex divided by the density of states (states per unit energy) of the excited molecule. It is a great advance over previous theory because it can be applied to real molecules, counting the states from the known vibrational frequencies. 

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. 

Fig. 5.—Dependence of the apparent second order rate coefficient at 300 K on total pressure. The open symbols refer to measurements made on gas mixtures containing 0.125 Torr NO2,2.5 Torr H2 and different pressures of He kbi is plotted against the total pressure. The closed symbols were the results of experiments on mixtures which again contained 0.125 Torr NO2 but in which the partial pressure of H2 was varied. To bring these results into agreement with those from the first series of experiments, it was necessary to allow for the greater efficiency of H2 as a third body relative to He and kbi was plotted against pHe+4.0 (pHa—2.5). The curves show the results of RRKM calculations ...

Fig. 6.—Dependence of the apparent second order rate coefficients at 416 K on total pressure. The points are experimental data obtained by measurements on gas mixtures containing 0.19 or 0.38 Torr NO2, 3.8 Torr H2 and different pressures of He. The curves represent the results of RRKM calculations — — model A,----model B (in both cases Psfs = 1 -0) and — model A (Psf = 0.65). The...

According to RRKM theory, the apparent second order rate coefficient is given by... 

The RRKM microcanonical rate coefficient is the rate at which states pass through the transition state (per unit energy) divided by the total density of states of the reactants. 

Chapter 3 deals with an even more fundamental difficulty of RRKM theory. The theory expresses the microcanonical rate coefficient in terms of a sum and... 

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