Master equation model

Figure A3.13.15. Master equation model for IVR in highly excited The left-hand side shows the quantum levels of the reactive CC oscillator. The right-hand side shows the levels with a high density of states from the remaining 17 vibrational (and torsional) degrees of freedom (from [38]).

For the examples described here, where the temperature dependence of was analyzed both by master equation modeling and by Equation (27), it is interesting to see how well the Tolman theorem approximation compares with the detailed modeling. This is shown in Table 7 for the cases of tetraethylsilane ion and (H20)jCr. It is seen that these two approaches give nearly identical results. 

Fig. 25 Comparison of the predictions of various models for current injection from a metal electrode into a hopping system featuring a Gaussian DOS of variance a = 15 meV as a function of the electric field at different temperatures. The ID continuum and the 3D master equation model have been developed by van der Holst et al. [127]. The calculations based upon the Burin-Ramer and the Arkhipov et al. models are taken from [175] and [170] respectively. Parameters are the sample length L, the intersite separation a and the injection barrierA. From [127] with permission. Copyright (2009) by the American Institute of Physics...

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]. 

For more complicated reaction systems with competing reaction pathways, an additional master equation modeling is necessary to calculate and predict reaction rate coefficients. This treatment [28] includes the collisional energy transfer between rotational and vibrational energy levels of the reactants through activation or collisional deactivation and the different energy amount needed to overcome the transition states. 

Reactions between neutrals include atom/radical + radical and atom/radical + molecule reactions. As discussed above, the intermolecular forces are shorter range than is the case with ion-molecule reactions, so that it is necessary to consider chemical interactions explicitly when modelling a reaction. After a section on experimental methods, the ideas behind transition state (TS) theory and its variational modification are discussed, together with theories of reactions where the TS switches, as the temperature increases, from A-B distances mainly controlled by the potential arising from electrostatic interaction to shorter distances where chemical forces are important. While the pressure in the ISM is too low for pressure dependent reactions, this topic is important in the conditions used to measure rate coefficients and in the chemistry of planetary atmospheres, including those of the exoplanets (see Chap. 5). This topic is discussed in Sect. 3.4.4, which also introduces the ideas that lie behind master equation models, which are widely used for such reactions. These models can also be used for reactions in which the adduct AB from an A + B reaction dissociates into several products, and these ideas are discussed in Sect. 3.4.5. Section 3.4 concludes with discussion of two examples of neutral + neutral reactions. 

K. D. Ball and R. S. Berry, Realistic master equation modeling of relaxation on complete potential energy surfaces Partition function and equilibrium results. J. Chem. Phys. 109(19), 8541-8556 (1998). 

Golden, D.M., Barker, J.R., Lohr, L.L. Master Equation Models for the Pressure- and Temperature-Dependent Reactions HO-1-NO2— HONO2 and HO-1-NO2 — HOONO. J. Phys. Chem. A 107, 11057-11071 (2003)... 

Senosiain, J.P., Musgrave, C.B., Golden, D.M. Temperature and pressure dependence of the reaction of OH and CO Master equation modeling on a high-level potential energy surface. InL J. Chem. Kinet 35, 464-474 (2003)... 

Pilling, M. J. and S. H. Robertson (2003). Master equation models for chemical reactions of importance in combustion. Ann. Rev. Phys. Ghent. 54,245. 

In this chapter we shall examine some of the effects of molecular fluctuations on chemical oscillations, waves and patterns. There are many ways one can attempt to study fluctuation dynamics in reacting systems, the most familiar of which are master equation models [ 1 ]. Here we present results obtained using a specific class of cellular automaton models, termed lattice-gas cellular automata [2-4]. These cellular automaton models provide a mesoscopic description of the spatially-distributed reacting system and are constructed to model the microscopic collision dynamics. The modeling strategy and rule construction are different from those for traditional cellular automata and are based on lattice-gas cellular automaton models for hydrodynamics [5]. However, reactive lattice-gas models differ from the corresponding hydrodynamics models in a number of important respects and are closely related to master equation descriptions of the reactive dynamics. 

Figure 6 Intensity dependence of the infrared laser chemical steady state rate constant over many orders of magnitude in a doubly logarithmic presentation (including prediction of ionization), (a) Semiquantitative prediction. Depending on the molecule, typical ranges of Iq would be 0.1 to 10 W cm with /o = 1 W cm being a good intermediate, (b) Quantitative calculation for a reaction similar to the prototype reaction (equation 1) of CF3I with / c 1 MW cm being a typical condition. The full point.s are calculated from the nonlinear case B/C master equation model of this reference and include all ranges n < 1, n = 1, and n > 1 that have now been found experimentally. Ionization is predicted to occur at intensities where rate.s exceed about 10 -10 s , . see (a) ...

Single-pulse shock tube studies of the thermal dehydrochlorination reactions of chlorocyclopentane and chlorocyclohexane at temperatures of 843-1021 K and pressures of 1.4-2.4 bar have been carried out using the comparative rate technique. Absolute rate constants provided a self-consistent temperature scale of use in comparison with chemical systems studied with different temperature standards. Quantum chemical methods have been used to compute the structure and energies of reactants, products, and transition states. The computations were used, in conjunction with experimentally determined rate constants, to develop Rice-Ramsperger-Kassel-Marcus (RRKM)/ Master Equation models and thereby allow extrapolation of the experimental data over an extended range of temperatures. 

Dames EE. Master equation modeling of the unimolecular decompositions of a-hydroxyethyl (CH3CHOH) and ethoxy (CH3CH2O) Radicals. Int J Chem Kinet. March 2014 46 176-188. 

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