Single rate-determining step

It is not always possible to determine intrinsic isotope effects. However, other useful information about the reaction can still be obtained. Above we assumed a single rate determining step sensitive to each isotope substitution. More frequently, however, the isotope sensitivity is found in different steps. Studies with multiple isotope effects can be used to determine the sequence of steps. To illustrate, a more complicated reaction scheme is needed ... 

Such a single rate-determining step scarcely occurs in ordinary reactions usually, the overall reaction affinity is distributed in multiple rate-determining steps rather than localized at a single step as is described in Sec. 7.4. 

When the overall rate of a multistep reaction is determined solely by a single elementary step whose rate is extremely small compared with the rates of the other elementary steps, the multistep reaction is called the reaction of a single rate-determining step. In such a multistep reaction, as shown in Fig. 7-11 (a), all the elementary steps except for the rate-determining step are cmisidered to be in quasi-equilibrium. Note that the multistep reaction of a sin le rate-determining step is rather uncommon in practice. 

Fig. 7-11. Potential energy curves for a mialtistep reaction of (a) single rate-determining step and (b) multiple rate-determining steps v = stoichiometric number of a single rate-determining step v = mean stoichiometric number of multiple rate-determining steps.

With a single rate-determining step, the affinity of elementary steps other than the rate-determining step is negligible, and the overall reaction affinity — AG approximately equals the affinity - 4gr multiplied by the stoichiometric number of the rate-determining step in Eqn. 7-44 as has been shown in Eqn. 

The concept of categorizing carcinogens into threshold carcinogens and non-threshold carcinogens is a pragmatic approach that simplifies the reality of dose-response relationships. The observed dose-response curve for tumor formation in some cases represents a single rate-determining step however, in many cases it may be more complex and represent a superposition of a number of dose-response curves for the various steps involved in the mmor formation. It is therefore more realistic to assume that there is a continuum of shapes of dose-response relationships which cannot be easily differentiated by data and information usually available. 

While the pre-equilibrium scheme may be an important source of an intermediate r value, any given reaction with an intermediate r value cannot necessarily be said to proceed by the single rate-determining step mechanism or the pre-equilibrium, two-step mechanism. 

All these methods stem from the fact that for a given reaction sequence, such as that in Eqs. (209) and (210), which involve a single rate-determining step, all the kinetics and thus the shape and location of the voltammogram depends only on the dimensionless rate constant parameter X in Eq. (212). As a result, any modification of the experimental conditions that keep X constant does not modify the dimensionless voltammogram. Thus quantitative information on the chemical mechanism [Eq. (210) may be a succession of chemical steps] is obtained without mathematical derivation, but only from dimensionless analysis (compare Chapter 2). 

The steady-state approximation is a more general approach than those considered earlier and can be used when no single rate-determining step exists. 

Deduce the rate law from a mechanism characterized by a single rate-determining step (Section 18.4, Problems 25-30). 

The quasi-equilibrium approximation relies on the assumption that there is a single rate-determining step, the forward and reverse rate constants of which are at least 100 times smaller than those of all other reaction steps in the kinetic scheme. It is then assumed that all steps other than the rds are always at equilibrium and hence the forward and reverse reaction rates of each non-rds step may be equated. This gives simple potential relations describing the varying activity of reaction intermediates in terms of the stable solution species (of known and potential-independent activity) that are the initial reactants or final products of the reaction. The variation of the activities of reaction intermediates is, however, restricted by either the hypothetical solubility limit of these species or, in the case of surface-confined reactions and adsorbed intermediates, the availability of surface sites. In both these cases, saturation or complete coverage conditions would result in a loss of the expected... 

Finally we note that not all reactions have a single rate-determining step. A reaction may have two or more comparably slow steps. The kinetic analysis of such reactions is generally more involved. 

Most oxygen transfer reactions, such as CO -f HjO = CO2 + H2 or 2CO -h O2 = 2CO2 involve several consecutive steps. Thus, it is especially relevant to investigate reactions involving a single rate-determining step, e.g., the isotope exchange reactions... 

In the foregoing examples, rate equations were developed on the basis of a single rate-determining step. It is possible that many steps of a cycle are simultaneously controlling, as in the Wacker process. The rate equation for such a reaction tends to be more complicated but can be developed by the methods discussed in Chapter 7. Thus for the oxidation of triphenylphosphine with a Pt complex, a rate equation can be developed based on the catalytic cycle shown in Figure 8.9 (Halpern and Pickard, 1970 Birk et al., 1968a,b) ... 

Thus in the limiting cases where the rate constants of two (or more) irreversible steps differ greatly, it is desirable to distinguish one of these steps as the rate-determining step. If, on the other hand, both reaction steps have comparable rate constants, the concept of a single rate-determining step, of course, makes no more sense, so that kinetics of the process would be substantially dependent on the parameters of both steps [see Eqs. (6) and (9)]. 

Figure 8.8 displays the three reactions in the WGS model with the largest normalized sensitivity coefficients. These were computed using Equation (8.35), except that f. is the preexponential factor of reaction j, This is similar to the degree of rate control defined in Equation (8.34) except that the overall conversion is used rather than the reaction rate. Important insights from this plot are that the sensitivity (kinetic relevance) of a reaction depends on reaction conditions, for example, temperature, and that there is not always a single rate-determining step rather, multiple reactions can be simultaneously kinetically important. 

So far in this chapter, we have considered only traditional cyclic voltammetric experiments where the I-E response is recorded. Moreover, we have taken a rather simplistic view of the mechanisms of homogeneous chemical reactions, assuming that they always follow simple, limiting pathways with a single rate determining step. Studies based on such principles remain important to electrochemistry and are widespread in the literature, but they ignore many mechanistic nuances and lead to imprecise data. This was also the state of the art maybe ten years ago. 

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