Multistep reactions, kinetics

The simplest enzymatic system is the conversion of a single substrate to a single product. Even this straightforward case involves a minimum of three steps binding of the substrate by the enzyme, conversion of the substrate to the product, and release of the product by the enzyme (Scheme 4.6). Each step has its own forward and reverse rate constant. Based on the induced fit hypothesis, the binding step alone can involve multiple distinct steps. The substrate-to-product reaction is also typically a multistep reaction. Kinetically, the most important step is the rate-determining step, which limits the rate of conversion. 

Identification of the intermediates in a multistep reaction is a major objective of studies of reaction mechanisms. When the nature of each intermediate is fairly well understood, a great deal is known about the reaction mechanism. The amount of an intermediate present in a reacting system at any instant of time will depend on the rates of the steps by which it is formed and the rate of its subsequent reaction. A qualitative indication of the relationship between intermediate concentration and the kinetics of the reaction can be gained by considering a simple two-step reaction mechanism ... 

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. 

In deriving the kinetics of activation-energy controlled charge transfer it was emphasised that a simple one-step electron-transfer process would be considered to eliminate the complications that arise in multistep reactions. The h.e.r. in acid solutions can be represented by the overall equation ... 

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... 

Rather than always occurring in one step, reactions in the natural world often result from a series of simple processes between atoms and molecules resulting in a set of intermediate steps from reactants to products. The way multistep reactions occur can have a strong effect on the kinetics of the overall reaction. For instance, in... 

The present chapter will cover detailed studies of kinetic parameters of several reversible, quasi-reversible, and irreversible reactions accompanied by either single-electron charge transfer or multiple-electrons charge transfer. To evaluate the kinetic parameters for each step of electron charge transfer in any multistep reaction, the suitably developed and modified theory of faradaic rectification will be discussed. The results reported relate to the reactions at redox couple/metal, metal ion/metal, and metal ion/mercury interfaces in the audio and higher frequency ranges. The zero-point method has also been applied to some multiple-electron charge transfer reactions and, wheresoever possible, these results have been incorporated. Other related methods and applications will also be treated. 

It appears like a miracle how aliphatic chains (mainly olefins and paraffins) are formed from a mixture of CO and H2. But miracle means only high complexity of unknown order (Figure 9.1). Problems in FT synthesis research include the visualization of a multistep reaction scheme where adsorbed intermediates are not easily identified. Kinetic constants of the elemental reactions are not directly accessible. Models and assumptions are needed. The steady state develops slowly. The true catalyst is assembled under reaction conditions. Difficulties with product analysis result from the presence of hundreds of compounds (gases, liquids, solids) and from changes of composition with time. 

Labelling experiments provided the evidence that the Fe1- and Co1-mediated losses of H2 and 2H2 from tetralin are extremely specific. Both reactions follow a clear syn- 1,2-elimination involving C(i)/C(2) and C(3)/C(4), respectively. In the course of the multistep reaction the metal ions do not move from one side of the rr-surface to the other. The kinetic isotope effect associated with the loss of the first H2 molecule, k( 2)/k(Y)2) = 3.4 0.2, is larger than the KIE, WFLj/ATHD) = 1.5 0.2, for the elimination of the second H2 molecule. A mechanism of interaction of the metal ion with the hydrocarbon n-surface, ending with arene-M+ complex 246 formation in the final step of the reaction, outlined in equation 100, has been proposed241 to rationalize the tandem MS studies of the unimolecular single and double dehydrogenation by Fe+ and Co+ complexes of tetraline and its isotopomers 247-251. 

Helfferich FG (2003) Kinetics of Homogeneous Multistep Reactions, 2nd ed. Elsevier, Amsterdam... 

Isotope effects have been used to determine whether the hydride transfer from the enzyme cofactor nicotinamide-adenine dinucleotide (NADH) (reaction (43)) takes place as a hydride ion transfer in a single kinetic step or in a multistep reaction via an uncoupled electron and hydrogen transfer. 

In enzyme-catalyzed kinetics, one must necessarily deal with the behavior of a multistep reaction scheme. For initial rate enzyme processes, one typically deals with collections of rate constants which appear in the form of the maximal velocity Um (shortened to U) or the specificity constant VJK (shortened to VIK). Accordingly, enzyme kineticists will use °V and °V/K as an easy way to indicate the respective isotope effects [(Um)H/(Um)D ] and [(VJK )u/(VJK )b], respectively. 

This nonmatch shows that we must try to develop a multistep reaction model to explain the kinetics. 

One of the important applications of mono- and multimetallic clusters is to be used as catalysts [186]. Their catalytic properties depend on the nature of metal atoms accessible to the reactants at the surface. The possible control through the radiolytic synthesis of the alloying of various metals, all present at the surface, is therefore particularly important for the catalysis of multistep reactions. The role of the size is twofold. It governs the kinetics by the number of active sites, which increase with the specific area. However, the most crucial role is played by the cluster potential, which depends on the nuclearity and controls the thermodynamics, possibly with a threshold. For example, in the catalysis of electron transfer (Fig. 14), the cluster is able to efficiently relay electrons from a donor to an acceptor, provided the potential value is intermediate between those of the reactants [49]. Below or above these two thresholds, the transfer to or from the cluster, respectively, is thermodynamically inhibited and the cluster is unable to act as a relay. The optimum range is adjustable by the size [63]. 

For a recent comprehensive treatment see Helfferich, F.F. (2004) Kinetics of multistep reactions. In N.J.B. Green (Ed.) Comprehensive Chemical Kinetics (vol 40). Eslevier, Amsterdam. 

Mechanistic Multiphase Model for Reactions and Transport of Phosphorus Applied to Soils. Mansell et al. (1977a) presented a mechanistic model for describing transformations and transport of applied phosphorus during water flow through soils. Phosphorus transformations were governed by reaction kinetics, whereas the convective-dispersive theory for mass transport was used to describe P transport in soil. Six of the kinetic reactions—adsorption, desorption, mobilization, immobilization, precipitation, and dissolution—were considered to control phosphorus transformations between solution, adsorbed, immobilized (chemisorbed), and precipitated phases. This mechanistic multistep model is shown in Fig. 9.2. 

Chemical kinetics is a powerful tool that provides unique mechanistic information and deep insight into the activation process that is at the heart of every chemical transformation. This chapter is structured around some of the most important types of information obtained from kinetic studies. The rate law provides the composition of the transition state (TS), kinetic isotope effects (KIEs) can establish whether a specific bond is involved in the activation process, and activation parameters provide information about the energy and entropy requirements. Independent generation, characterization, and reactivity studies of potential intermediates allow one to search for and identify such intermediates in multistep reactions by spectroscopic means or by use of chemical traps. 

Finally, because the addition of Br2 to cyclohexene is 27 kcal/mol - 11 kcaFmol = 16 kcal/ mol more exothermic than the substitution of Br2 on cyclohexene, can we conclude that the first reaction also takes place more rapidly Not necessarily The (fictitious) substitution reaction of Br2 on cyclohexene should be a multistep reaction and proceed via a bromonium ion formed in the first and also rate-determining reaction step. This bromo-niurn ion has been demonstrated to be the intermediate in the known addition reaction of Br2 to cyclohexene (Section 3.5.1). Thus, one would expect that the outcome of the competition of substitution vs. addition depends on whether the bromonium ion is converted— in each case in an elementary reaction—to the substitution or to the addition product. The Hammond postulate suggests that the bromonium ion undergoes the more exothermic (exergonic) reaction more rapidly. In other words, the addition reaction is expected to win not only thermodynamically but also kinetically. 

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... 

A reaction is described as proceeding by a kinetic template effect if it provides a route to a product that would not be formed in the absence of the metal ion and where the metal ion acts by coordinating the reactants. An alternative description for this process is the coordination template effect that more aptly describes how the stereochemistry imposed by the metal ion, through coordination, promotes a series of controlled steps in a multistep reaction, e.g., Scheme 1 17, 136, 137). 

While concentrating on methods, the book uses a number of reactions of industrial importance for illustration. However, no comprehensive review of multistep homogeneous reactions is attempted, simply because there are far too many reactions and reaction mechanisms to present them all. Instead, the book aims at providing the tools with which the practical engineer or chemist can handle his specific reaction-kinetic problems in an efficient manner, and examples of how problems unique to a specific reaction at hand can be overcome. Some examples drawn from my own laboratory experience have been construed or details have been left out, in order to protect former employers or clients proprietary interests. In particular, the omission of information on exact structure and composition of catalysts is intentional. 

As seen in Table 2.1, the overall order of an elementary step and the order or orders with respect to its reactant or reactants are given by the molecularity and stoichiometry and are always integers and constant. For a multistep reaction, in contrast, the reaction order as the exponent of a concentration, or the sum of the exponents of all concentrations, in an empirical power-law rate equation may well be fractional and vary with composition. Such apparent reaction orders are useful for characterization of reactions and as a first step in the search for a mechanism (see Chapter 7). However, no mechanism produces as its rate equation a power law with fractional exponents (except orders of one half or integer multiples of one half in some specific instances, see Sections 5.6, 9.3, 10.3, and 10.4). Within a limited range of conditions in which it was fitted to available experimental results, an empirical rate equation with fractional exponents may provide a good approximation to actual kinetics, but it cannot be relied upon for any extrapolation or in scale-up. In essence, fractional reaction orders are an admission of ignorance. 

The kinetics of a multistep reaction is described by a set of simultaneous rate equations, one for each participant. The equations are independent of one another in their algebraic forms and values of their coefficients. Each equation is the summation of the contributions from all steps in which the respective species participates. Each such contribution is the product of the stoichiometric coefficient of the species, the rate coefficient of the step, and the concentration (or concentrations) of the reactant (or reactants) of the step, raised to the power corresponding to the molecularity. 

This section has concentrated on relatively simple cases. More detail can be found in texts on kinetics and reaction engineering (see general references). Establishment of empirical rate equations and coefficients for multistep reactions will be discussed in Chapter 7. 

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