Rate equations of multistep reactions

The temperature dependence of rate coefficients of elementary steps generally follows the Arrhenius equation in good approximation. That of apparent rate coefficients of empirical rate equations of multistep reactions, however, may deviate in several ways The activation energy may be negative (slower rate at higher temperature) or have different values in different temperature regions. 

Rate equations of multistep reactions often are not power laws. Reaction orders therefore may vary with concentrations, and attempts at accurate determination would be futile. Unless reaction orders are integers, or integer multiples of one half in special cases, only their ranges (such as between zero and plus one) are of interest. 

Rate equations of multistep reactions with reverse steps contain additive terms in the denominator, but not the numerator. Examples from previous chapters include nitration of aromatics and olefin hydroformylation (see Examples 4.4 in Section 4.4 and 6.2 in Section 6.3, respectively). In all such cases, the number of phenomenological coefficients can be reduced by one if numerator and denominator are divided by one of the terms of the latter. The result is a "one-plus" rate equation, with a "1" as the leading term in the denominator. (This procedure is superfluous if all terms in the denominator consist only of coefficients, or of coefficients multiplied with the same concentration or concentrations, so that they can be combined to produce a true power-law rate equation.)... 

As this chapter has shown, rate equations of multistep homogeneous catalysis are still relatively simple if the catalyst-containing intermediates are at trace level, but the free catalyst is not. In heterogeneous catalysis this corresponds to an almost entirely unoccupied catalyst surface. Since adsorption is prerequisite for reaction, low surface coverage results in low rates and therefore is of practical interest only in exceptional situations. Heterogeneous catalysis cannot avoid dealing with substantially covered... 

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. 

Rate constant The proportionality constant in the rate equation for a reaction, 288 Rate-determining step The slowest step in a multistep mechanism, 308 Rate expression A mathematical relationship describing the dependence of reaction rate upon the concentra-tion(s) of reactant(s), 288,308-309 Rayleigh, Lord, 190... 

Another rapidly progressing field is that of multistep reactions which occur in ordered sequences chemo-, regio- and stereo-selectively on a transition metal species. To this end, it is necessary to delay release of the desired product until the whole series of steps has been completed competitive terminations (such as hydride elimination) must be prevented or must only occur at low rates compared to the main sequence. An example, reported by Chiusoli in the late 50s, is offered by the nickel-catalyzed synthesis of methyl 2,5-heptadienoate from 2-butenyl chloride, acetylene, CO and methanol. The reaction is chemo-, regio- and stereo-selective the four molecules react in the order shown in Equation A3.4 (chemoselectivity) the butenyl group attacks the terminal allylic carbon rather than the internal one (regioselectivity) and acetylene insertion leads to a Z double bond (stereoselectivity). 

In principle, a multistep reaction with any network is fully described by the complete set of rate equations of all participants, compiled as shown in Section 2.4. However, if the reaction is complex, the experimental work required to verify the assumed mechanism and to determine all its coefficients and their activation energies can get out of hand. A reduction of complexity then is imperative, and is also desirable for both a better understanding of reaction behavior and more efficient numerical modeling. The present chapter describes ways of achieving this. 

Rigorous rate equations for multistep catalytic reactions in terms of total amount of catalyst material are enormously cumbersome. Just the reduction to the level complexity of the Christiansen formula calls for the Bodenstein approximation of quasi-stationary behavior of the intermediates, requiring these to remain at trace concentrations, and that formula still entails a lot more algebra than does the general rate equation for noncatalytic simple pathways For a reaction with three intermediates, the Christiansen denominator contains sixteen terms instead of four for a reaction with six intermediates, forty-nine instead of seven Although the mathematics is simple and easy to program for modeling purposes and, usually, some... 

The postulate of quasi-equilibrium of all steps except a single one that controls the rate is very powerful. It reduces the mathematical complexity of kinetics even of large networks to quite simple rate equations and has become a favorite tool, employed today in a great majority of publications on kinetics of multistep reactions, sometimes uncritically. In many cases, a sharp distinction between fast and slow steps cannot be justified. A more general approach that avoids the postulate of a single rate-controlling step and contains the results obtained with it as special cases will be described in Sections 4.4 and 6.3 and widely used in later chapters. 

Many reactions occur by a multistep mechanism, involving two or more elementary reactions, or steps. An intermediate is produced in one elementary step, is consumed in a later elementary step, and therefore does not appear in the overall equation for the reaction. When a mechanism has several elementary steps, the overall rate is hmited by the slowest elementary step, called the rate-determining step. A fast elementary step that follows the rate-determining step will have no effect on the rate law of the reaction. A fast step that precedes the rate-determining step often creates an equilibrium that involves an intermediate. For a mechanism to be valid, the rate law predicted by the mechanism must be the same as that observed experimentally. 

Let us address now the issue of identifying the kinetic significance of individual steps. It is easy to see that according to equation (2.7) the rate of each reverse reaction may be estimated from experimentally measured data on the rates of reaction paths. Hence, the role of (y -s) steps is revealed. It should be mentioned that in the framework of pathway theory, M.I. Temkin [18] offers a method to derive the kinetic equations. This is when the rate on reaction path, and further according to (2.6) the rate of multistep reaction, is expressed through the rate constants of individual steps and characteristics, and more often concentrations of the initial substances, determined in kinetic experiments. In the final kinetic model the dependency on the rate constants of steps specifies their kinetic participation in the total chemical process [17]. 

Not all reactions can be fitted by the Hammett equations or the multiparameter variants. There can be several reasons for this. The most common is that the mechanism of the reaction depends on the nature of the substituent. In a multistep reaction, for example, one step may be rate-determining in the case of electron-withdrawing substituents, but a different step may become rate-limiting when the substituent is electron-releasing. The rate of semicarbazone formation of benzaldehydes, for example, shows a nonlinear Hammett... 

From the intercept at AG° = 0 we find AGo = 31.9 kcal mol , and the slope is 0.77. As we have seen, if Eq. (5-69) is applicable, the slope should be 0.5 when AG = 0. In this example either the data cover too small a range to allow a valid estimate of the slope to be made or the equation does not apply to this system. Such a simple equation is not expected to be universally applicable. Recall that it was derived for an elementary reaction, so multistep reactions, even if showing simple rate-equilibrium behavior, introduce complications in the interpretation. The simple interpretation of Eq. (5-69) also requires that AGo be constant within the reaction series, but this condition may not be met. Later pages describe another possible reason for the failure of Eq. (5-69). 

Multistep reactions Previous considerations have been based on a simple one-step reaction involving one electron, but if the reaction occurs by a series of steps of which one is significantly slower than all the others, which may be regarded as at equilibrium, and is thus rate determining, equation 20.61 is not valid and becomes... 

This chapter takes up three aspects of kinetics relating to reaction schemes with intermediates. In the first, several schemes for reactions that proceed by two or more steps are presented, with the initial emphasis being on those whose differential rate equations can be solved exactly. This solution gives mathematically rigorous expressions for the concentration-time dependences. The second situation consists of the group referred to before, in which an approximate solution—the steady-state or some other—is valid within acceptable limits. The third and most general situation is the one in which the family of simultaneous differential rate equations for a complex, multistep reaction... 

For most real systems, particularly those in solution, we must settle for less. The kinetic analysis will reveal the number of transition states. That is, from the rate equation one can count the number of elementary reactions participating in the reaction, discounting any very fast ones that may be needed for mass balance but not for the kinetic data. Each step in the reaction has its own transition state. The kinetic scheme will show whether these transition states occur in succession or in parallel and whether kinetically significant reaction intermediates arise at any stage. For a multistep process one sometimes refers to the transition state. Here the allusion is to the transition state for the rate-controlling step. 

From Equation 2.9 we can also derive the following the SN1 product is produced with the rate constant of the first reaction step. Thus the rate of product formation does not depend on the rate constant kattlck of the second reaction step. In a multistep reaction, a particular step... 

According to the Equation (30) the experimental isotope effect depends not only on the intrinsic isotope effects a,-, but also on the rate constants k2 and k. The intrinsic isotope effects describe the structure of the transition states and the commitment reflects the relative heights of energetic barriers of competitive reactions. If k2i. k -[li(x l), the formation of intermediate B is the rate-limiting step and experimental isotope effect is equal to ai(aexp = i). When intermediate B returns to substrate much faster than forms the product k2, k it (x 1), the experimental isotope effect is aexp = (a1a2)/a 1. For more complex multistep reactions the analysis of isotope effects is analogous, however the commitment factor become a complex collection of kinetic terms.54... 

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