Rate equation, complicated

In Section 1.2 we distinguished between elementary and complex reactions. We now make a distinction between simple and complicated rate equations. A simple rate equation has the form of Eq. (1-11). A complicated rate equation has a form different from Eq. (1-11) it may be a sum of terms like that in (1-11), or it may have quantities in the denominator. We have seen that there is no necessary relationship between the complexity of the reaction and the form of the experimental rate equation. Simple rate equations are treated in Chapter 2 and complicated rate equations in Chapter 3. 

This ability to reduce the reaction order by maintaining one or more concentrations constant is a veiy valuable experimental tool, for it often permits the simplification of the reaction kinetics. It may even allow a complicated rate equation to be transformed into a simple rate equation. 

In Chapter 1 we distinguished between elementary (one-step) and complex (multistep reactions). The set of elementary reactions constituting a proposed mechanism is called a kinetic scheme. Chapter 2 treated differential rate equations of the form V = IccaCb -., which we called simple rate equations. Chapter 3 deals with many examples of complicated rate equations, namely, those that are not simple. Note that this distinction is being made on the basis of the form of the differential rate equation. 

There is no general explicit mathematical treatment of complicated rate equations. In Section 3.1 we describe kinetic schemes that lead to closed-form integrated rate equations of practical utility. Section 3.2 treats many further approaches, both experimental and mathematical, to these complicated systems. The chapter concludes with comments on the development of a kinetic scheme for a complex reaction. 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

DERIVATION OF MORE COMPLICATED RATE EQUATIONS. So far, the rate equations that describe one-substrate enzyme systems have been fairly simple, and the usual algebraic manipulations of substitution and/or addition of simultaneous equations have permitted us to obtain the pertinent rate law. When the number of steps increases and especially when there are branched pathways involved, these manual methods become cumbersome, and more systematic procedures are required. The next two sections should allow the reader to develop a working knowledge of effective methods for obtaining multisubstrate enzyme rate expressions. 

Only a few kinetic studies on dehydrochlorination and dehydrobrom-ination have been published. They are summarised in Table 7 and the general impression is that the more complicated rate equations have resulted... 

If the reactor is to be operated isothermally, the rate of reaction diA can be expressed as a function of concentrations only, and the integration in equation 1.24 or 1.25 carried out. The integrated forms of equation 1.25 for a variety of the simple rate equations are shown in Table 1.1 and Fig. 1.8. We now consider an example with a rather more complicated rate equation involving a reversible reaction, and show also how the volume of the batch reactor required to meet a particular production requirement is calculated. 

In fact the reaction scheme is considerably more complicated than suggested by equation (3) [9] and consequently more complicated rate equations are proposed in literature [3,9]. For the purpose of this work, however, equation (5) was found to be sufficiently accurate. Other CO2 converting reactions, as well as the hydrolysis of the carbamate ion, are slow compared to reaction (3) and hence are not incorporated in the model. 

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