Third-order kinetic model

Since cubic autocatalytic reaction requires reaction between three molecules, modelling autocatalytic reactions by cubic form often is criticized. Aris et al. have considered two different models of autocatalytic reactions [26]. The first model considers only two variables, but requires a third-order kinetic model ... 

Numerical solutions to equation 11.2.9 have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for first-order kinetics, it is most appropriate for use when the dimensionless dispersion group is small. Fan and Bailie (19) have solved the equations for quarter-order, half-order, second-order, and third-order kinetics. Others have used perturbation methods to arrive at analogous results for the dispersion model (e.g. 20,21). 

There remains one objection—of a less precise kind but felt by many chemists. It is that third-order kinetics as embodied in the representation of step (1) are intrinsically objectionable. If the equations had to be interpreted as representing elementary steps, this would be a weightier consideration, but it has also been asserted that the oscillatory properties of certain other model schemes collapse completely (King, 1983 Gray and Morley-Buchanan, 1985) if the third-order steps therein are replaced. Accordingly it is most desirable to establish whether oscillations and other exotic behaviour arising from a cubic rate-law of the form k ab2 can also arise from a series of successive second-order or bimolecular steps. Similar interests have been expressed previously by Tyson (1973) and Tyson and Light (1973). 

The kinetics of nucleophilic substitution at the silicon atom assisted by uncharged nucleophiles have been studied by Corriu et at. (248-251). Hydrolysis of triorganochlorosilanes induced with HMPA, DMSO, and DMF was used as the model. The reaction proceeded according to the third-order kinetic law, first order with respect to the nucleophile, the silane, and the silylation substrate. Very low values of activation enthalpy and high negative entropy of activation were observed (Table VI). These results were taken as evidence for the intermediacy of silicon hypervalent species (249,251) however, they are also perfectly consistent with... 

Flory [3, 48] found that the esterification reactions between model compounds on the one hand and polyfunctional reactants on the other are substantially identical. Thus the reaction of two monofunctional compounds (lauric acid, lauryl alcohol), of a bifunctional compound with a monofunctional one (adipic acid, lauryl alcohol) and two bifunctional compounds (adipic acid, decamethylene glycol) followed essentially third-order kinetics. In the absence of added strong-acid catalyst a second molecule of the carboxylic acid functions as catalyst. Thus when the concentrations (C) of the reacting groups are identical, the rate is given by ... 

We and others have demonstrated that association of short strands containing a single guanine-repeat seems to obey a fourth-order kinetics model. Third or fourth-order reactions are not common in biochemistry, and the practical consequences of this reaction order are important. A fourth-order reaction does not imply that an elementary kinetic step involves a four-body collision. Such mechanism is extremely unlikely and other processes could lead to this fourth order. The structure of these elusive intermediates remains unknown Stefl et have recently demonstrated that a Hoogsteen G-G duplex is an improbable intermediate. Its identification will be experimentally difficult, as numerical simulations indicate that it may not be present at detectable levels. 

Reference was made earlier to some specific numerical studies reported for the axial dispersion model employing rate expressions other than first order. Some results given by Fan and Balie for half-, second-, and third-order kinetics are illustrated in Figure 5.21. In these results the parameter R is defined as... 

MOREAU - A third order kinetics is indeed necessary for the existence of two locally stable stationary states in the homogeneous case. Although I am not convinced that three-bodies interactions can never occur/ it is sure that the kinetics which is used here (and in many other works) is only the overall result of several bimolecular elementary reactions (this is clear in the detained models proposed to represent/ for instance/ the Belousov-Zhabotinski reaction). 

With these kinetic data and a knowledge of the reactor configuration, the development of a computer simulation model of the esterification reaction is iavaluable for optimising esterification reaction operation (25—28). However, all esterification reactions do not necessarily permit straightforward mathematical treatment. In a study of the esterification of 2,3-butanediol and acetic acid usiag sulfuric acid catalyst, it was found that the reaction occurs through two pairs of consecutive reversible reactions of approximately equal speeds. These reactions do not conform to any simple first-, second-, or third-order equation, even ia the early stages (29). 

The best fit, as measured by statistics, was achieved by one participant in the International Workshop on Kinetic Model Development (1989), who completely ignored all kinetic formalities and fitted the data by a third order spline function. While the data fit well, his model didn t predict temperature runaway at all. Many other formal models made qualitatively correct runaway predictions, some even very close when compared to the simulation using the true kinetics. 

A new assumption to be discussed in this section is that the fourth-order kinetics in SatAr by amines in aprotic solvents is due to the formation of the substrate-catalyst molecular complex. Since 1982, Forlani and coworkers149 have advocated a model in which the third order in amine is an effect of the substrate-nucleophile interaction on a rapidly established equilibrium preceding the substitution process, as is shown in Scheme 15 for the reaction of 4-fluoro-2,4-dinitrobenzene (FDNB) with aniline (An), where K measures the equilibrium constant for ... 

Kinetic studies of the reaction of Z-phenyl cyclopropanecarboxylates (1) with X-benzylamines (2) in acetonitrile at 55 °C have been carried out. The reaction proceeds by a stepwise mechanism in which the rate-determining step is the breakdown of the zwitterionic tetrahedral intermediate, T, with a hydrogen-bonded four-centre type transition state (3). The results of studies of the aminolysis reactions of ethyl Z-phenyl carbonates (4) with benzylamines (2) in acetonitrile at 25 °C were consistent with a four- (5) and a six-centred transition state (6) for the uncatalysed and catalysed path, respectively. The neutral hydrolysis of p-nitrophenyl trifluoroacetate in acetonitrile solvent has been studied by varying the molarities of water from 1.0 to 5.0 at 25 °C. The reaction was found to be third order in water. The kinetic solvent isotope effect was (A h2o/ D2o) = 2.90 0.12. Proton inventories at each molarity of water studied were consistent with an eight-membered cyclic transition state (7) model. 

This concludes a discussion of exactly solvable second-order processes. As one can see, only a very few second-order cases can be solved exactly for their time dependence. The more complicated reversible reactions such as 2Apt C seem to lead to very complicated generating functions in terms of Lame functions and the like. This shows that even for reasonably simple second- and third-order reactions, approximate techniques are needed. This is not only true in chemical kinetic applications, but in others as well, such as population and genetic models. The actual models in these fields are beyond the scope of this review, but the mathematical problems are very similar. Reference 62 contains a discussion of many of these models. A few of the approximations that have been tried are discussed in Ref. 67. It should also be pointed out at this point that the application of these intuitive methods to chemical kinetics have never been justified at a fundamental level and so the results, although intuitively plausible, can be reasonably subject to doubt. 

A kinetic model based on homogeneous polymerization was developed to describe the polymerization in CO2 [51, 54]. A model based on the reaction scheme in Fig. 3 adequately described the polymerization rates and the poly-dispersity of the polymer. Monomer inhibition was incorporated into the model to account for the observed deviation from first-order kinetics. However, imperfect mixing of the higher viscosity medium is an alternative explanation. It was concluded that termination was by combination, for three reasons. First, there was no existing literature to support termination by disproportionation for PVDF. Second, the polydispersity was approximately 1.5 at low monomer concentrations. Third, NMR studies showed no evidence of unsaturation. 

The kinetic model of styrene auto-initiation proposed by Hui and Hameilec [27] was used as a starting point for this work. The Mayo initiation mechanism was assumed (Figure 7.2) but the acid reaction was of course omitted. After invoking the quasi-steady-state assumption (QSSA) to approximate the reactive dimer concentration, Hui and Hameilec used different simplifying assumptions to derive initiation rate equations that are second and third order in monomer concentration. 

Figure 1. Comparison of the extent of delignificatlon predicted by the homogeneous (first-order, second-order and third-order) models and by the reaction-diffusion kinetic model. The open circles are the measured Klason lignin contents In the residues obtained from methylamine extraction of red spruce at 276 bar, 185 C and 1 g/mln solvent flow rate.

Because / is a multivariate Gaussian distribution its higher-order moments can easily be computed (e.g. using the moment-generating function). However, the reader should keep in mind that the kinetic model ensures only that the moments up to second order are the same as with Eq. (6.109). Third- and higher-order moments may therefore be poorly approximated when the true velocity-distribution function is far from equilibrium. 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

This rate-limiting step is third order in the forward direction (i.e., adsorption) and second order in the reverse direction (i.e., desorption). Since two active sites are required for dual-site adsorption, one predicts that the kinetic model is proportional to the square of the vacant-site fraction. Substitution for a from (14-144) leads to... 

Using a master equation to model the VER process as a multistep reaction, the excess energy flow kinetics in FeP was examined, where the third order Fermi resonance parameters served as approximate reaction rate constants [88]. It was found that the subsequent relaxation is slow relative to relaxation of the initially excited system mode, providing an explanation for the observed difference in relaxation timescales. 

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