Chemical kinetics, simplification approximation

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

There are a number of possible approaches to the calculation of influences of finite-rate chemistry on diffusion flames. Known rates of elementary reaction steps may be employed in the full set of conservation equations, with solutions sought by numerical integration (for example, [171]). Complexities of diffusion-flame problems cause this approach to be difficult to pursue and motivate searches for simplifications of the chemical kinetics [172]. Numerical integrations that have been performed mainly employ one-step (first in [107]) or two-step [173] approximations to the kinetics. Appropriate one-step approximations are realistic for limited purposes over restricted ranges of conditions. However, there are important aspects of flame structure (for example, soot-concentration profiles) that cannot be described by one-step, overall, kinetic schemes, and one of the major currently outstanding diffusion-flame problems is to develop better simplified kinetic models for hydrocarbon diffusion flames that are capable of predicting results such as observed correlations [172] for concentration profiles of nonequilibrium species. 

Here we have used the approximation that can be replaced by Dj y and that variations of D y can be ignored within the averaging volume. The fact that only a single tortuosity needs to be determined by equations 1.152 and 1.153 represents the key contribution of this study. It is important to remember that this development is constrained by the linear chemical kinetic constitutive equation given by equation 1.113. The process of diffusion in porous catalysts is normally associated with slow reactions and equation 1.93 is satisfactory however, the first-order, irreversible reaction represented by equation 1.113 is the exception rather than the rule, and this aspect of the analysis requires further investigation. The influence of a non-zero mass average velocity needs to be considered in future studies so that the constraint given by equation 1.97 can be removed. An analysis of that case is reserved for a future study which will also include a careful examination of the simplification indicated by equation 1.117. 

Thus, both the small parameters and the multidimensional nature complicate the investigation of mathematical models of chemical kinetics. On the other hand, the small parameters often lead to a simplification of similar problems of classical mechanics by means of asymptotic approximations. From this point of view, the role of the small parameters is positive. 

A general thrust of the chapter will be SIMPLICITY. Differences in terminology have been eliminated wherever possible. In this analysis Availability, Available Energy, Exergy, and Work will be used as equivalent. This means that kinetic and potential energy effects and the potential work to be derived from the diffusion of chemical species into equilibrium with the environment have been ignored. This simplification may introduce significant inaccuracies in some studies, but is not important here. The intent is to demonstrate that simplified - perhaps even approximate - analysis can have valuable practical applications. 

Finally, neither the effect of external noise, which affects nonequilibrium transitions in chemical and biological systems (Lefever, 1981 Horsthemke Lefever, 1984 Lefever Turner, 1986), nor the stochastic aspects of these transitions (Nicolis, Baras Malek-Mansour, 1984) are considered - with the exception of the glycolytic system (chapter 2). Such a simplification, justified in the first approximation by the absence of systematic noise in the biological systems considered, permits us to avoid complicating from the outset the analysis of systems whose kinetics is already complex. 

Computed kinetic parameters are incorporated into simulations to model the chemical mechanism or part of it. In some instances, simplifications can be introduced, such as the steady-state approximation, allowing the formulation of an analytical kinetic model, involving only few kinetic parameters (Section 7.3.2). In the general case, the rate constants of all participating reactions need to be included. 

In this chapter we first consider a mathematically tractable model mechanism and demonstrate that, depending upon the relative magnitudes of the rate constants, there are two chemical approximations that may be appropriate for simplifying analysis the preequilibrium and the steady-state assumptions. We then demonstrate how hypotheses based upon these simplifications are used to interpret rate law data and to develop chemically reasonable mechanistic descriptions for gas- and solution-phase reactions. Finally we consider the problem of catalysis, i.c., how addition of trace amounts of an intermediate permits a sluggish or kinetically forbidden reaction to become rapid if a new mechanistic pathway can be created. 

Next, similar data for individual components of the system are to be found. One approach that has been often used in electrochemical investigations assumes that concentrations of components are close to the equilibrium ones when the system is sufficiently labile. Then expressions for the respective stability constants together with material balance equations are sufficient to obtain the required data. Though this approach sets unacceptable constraints on the EAC composition, it can be used as a satisfactory approximation when the rate constants of chemical steps exceed some critical values. Otherwise, the mass transfer problem should be solved without any simplifications accounting for the kinetics of chemical steps. 

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