Chemical kinetics steady-state approximation

In this section, the different behavior of processes with coupled noncatalytic homogeneous reactions (CE and EC mechanisms) is discussed in comparison with a catalytic process. We will consider that the chemical kinetics is fast enough and in the case of CE and EC mechanisms K (- c /cf) fulfills K 1 so that the kinetic steady-state and even diffusive-kinetic steady-state approximation can be applied. 

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

One useful trick in solving complex kinetic models is called the steady-state approximation. The differential equations for the chemical reaction networks have to be solved in time to understand the variation of the concentrations of the species with time, which is particularly important if the molecular cloud that you are investigating is beginning to collapse. Multiple, coupled differentials can be solved numerically in a fairly straightforward way limited really only by computer power. However, it is useful to consider a time after the reactions have started at which the concentrations of all of the species have settled down and are no longer changing rapidly. This happy equilibrium state of affairs may never happen during the collapse of the cloud but it is a simple approximation to implement and a place to start the analysis. 

Kinetics is one of the key issues of catalysis together with selectivity and catalyst stability. Chemical kinetics has been discussed in several dedicated works [1] and the readers will be aware of its basics [2], In the following sections several commonly used concepts are mentioned such as steady state approximation, rate-determining step, determination of selectivity, and a few points of particular interest to catalysis will be high-lighted such as incubation. 

As for the quasi (pseudo)-steady-state case, the basic assumption in deriving kinetic equations is the well-known Bodenshtein hypothesis according to which the rates of formation and consumption of intermediates are equal. In fact. Chapman was first who proposed this hypothesis (see in more detail in the book by Yablonskii et al., 1991). The approach based on this idea, the Quasi-Steady-State Approximation (QSSA), is a common method for eliminating intermediates from the kinetic models of complex catalytic reactions and corresponding transformation of these models. As well known, in the literature on chemical problems, another name of this approach, the Pseudo-Steady-State Approximation (PSSA) is used. However, the term "Quasi-Steady-State Approximation" is more popular. According to the Internet, the number of references on the QSSA is more than 70,000 in comparison with about 22,000, number of references on PSSA. 

The quasi steady state approximation is a powerful method of transforming systems of very stiff differential equations into non-stiff problems. It is the most important, although somewhat contradictive technique in chemical kinetics. Before a general discussion we present an example where the approximation certainly applies. 

This section presents the solutions for CE and EC mechanism in DDPV technique at planar electrodes under the approximation of kinetic steady state, which are applicable to fast chemical reactions [72], To obtain these solutions, a mathematical procedure similar to that presented in Sects. 3.4.2 and 3.4.3 has been followed for which it has been assumed that the perturbation of the chemical equilibrium is independent of time (i.e., d[Pg.305]

The series model can be extended to longer series and to the inclusion of reversibility to illustrate a variety of fundamental kinetic phenomena in an especially simple and straightforward manner. Depending on the relative rates employed, one can demonstrate the classic kinetic phenomena of a rate-limiting step and preequilibrium,72 and one can examine the conditions needed for the validity of the steady-state approximation commonly used in chemical kinetics.70... 

Bowen, J. R.. A. Acrivos and A. K. Oppenheim, Singular perturbation refinement to quasi-steady state approximation in chemical kinetics. Chem. Eng. Sci., 18, 177-188 (1963). 

The three-state (three-potential-energy surface) problem is of interest for redox chains, chemical triad model systems, DNA electron transfer, and the primary charge separation in photosynthesis. As there are two energy-gap fluctuations in these reactions, and the fluctuations are not simply related to each other (in contrast to the case of two-electron transfer in two-center systems, vide infra), the problem is intrinsically two-dimensional. Marchi et al. [54], Zusman and Beratan [55], and Okada and Bandyopadhyay [56] have analyzed the nature of these potential energy surfaces and the electron-transfer kinetics. In the steady-state approximation for species 2,... 

The steady-state approximation provides a simple framework for analyzing the kinetics of many chemical reactions. 

There are many ways one can try to reduce the computational burden. Ideally, one would find numerical methods which are guaranteed to retain accuracy while speeding the calculations, and it would be best if the procedure were completely automatic i.e. it did not rely on the user to provide any special information to the numerical routine. Unfortunately, often one is driven to make physical approximations in order to make it feasible to reach a solution. Common approximations of this type are the quasi-steady-state approximation (QSSA), the use of reduced chemical kinetic models, and interpolation between tabulated solutions of the differential equations (Chen, 1988 Peters and Rogg, 1993 Pope, 1997 Tonse et al., 1999). All of these methods were used effectively in the 20th century for particular cases, but all of these approximated-chemistry methods share a serious problem it is hard to know how much error is... 

Compilation of a kinetic scheme containing a limited number of chemical equations and corresponding differential equations is another possible form of simple models. The chemical part of it cannot be considered as a detailed mechanism, but serves only as a phenomenological description of the process. As to the system of differential equations, on certain assumptions it can be solved and reduced to an algebraic form (for instance, in steady-state approximation). Models of this type are widely presented in the scientific literature. 

One approach commonly used in the integration of such chemical kinetics problems is the pseudo-steady-state approximation (PSSA) (see Chapter 3). For example, instead of solving a differential equation for short-lived species like O, OH, and N03, one calculates and solves the corresponding PSSA algebraic equations. For example, McRae et al. (1982a) estimated that nine species (O, RO, OH, R02, N03, RCO, H02, HN04, and N2Os) with characteristic lifetimes less than 0.1 min in the environment of interest could be... 

This is essentially the classic case of the steady-state approximation and pre-equilibrium conditions for a two-step reaction with a reversible step. See Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed. McGraw-Hill New York, 1995 pp 77-90. 

The term steady state is also used to describe a situation where some, but not all, of the state variables of a system are constant. For such a steady state to develop, the system does not have to be a flow system. Therefore, such a steady state can develop in a closed system where a series of chemical reactions take place. Literature on chemical kinetics usually refers to this case, calling it steady-state approximation. Steady-state approximation, occasionally called stationary-state approximation, involves setting the rate of change of a reaction intermediate in a... 

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. 

Unfortunately, there exists no general theory that does for a generad sequence of elementary steps what has been done here for the simple sequence of first-order reactions. Yet the general ideas are clear. While exceptions to the validity of the steady-state approximation are known, they are rare and the steady-state approximation can be considered as the most important general technique of applied chemical kinetics. The treatment of long sequences becomes a simple problem as will now be shown. 

Noyes, R. (1978). Generalized kinetics of chemical change some conditions for validity of the steady state approximation. Supplement of the Progr. Theor. Phys., 64, 295-306. 

For (1.4.1-l)-(1.4.1-3) to be a useful approximation of the complete scheme, the induction time should be very short, meaning that the concentration of the intermediate must be very small. From (1.3.2-6) it is seen how the maximum in the curve Cq versus t moves towards t = 0 as k2 k. Quite frequently the existence of an intermediate is chemically logical, but it is difficult or impossible to measure its concentration. The pseudo steady state approximation is then a very useful tool. Examples will be encountered in Section 1.5 on bio-kinetics and Section 1.6 on complex reactions. 

It has been stated by Boudart that the steady-state approximation (SSA) can be considered as the most important general technique of applied chemical kinetics [9]. A formal proof of this hypothesis that is applicable to all reaction mechanisms is not available because the rate equations for complex systems are often impossible to solve analytically. However, the derivation for a simple reaction system of two first-order reactions in series demonstrates the principle very nicely and leads to the important general conclusion that, to a good approximation, the rate of change in the concentration of a reactive intermediate, X, is zero whenever such an intermediate is slowly formed and rapidly disappears. 

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