First-Order Chemical Kinetic Mechanisms

So far we have considered the so-called E mechanism where a simple one-electron transfer is the only process that takes place within the time scale of the experiments. However, in many situations the electrochemical system is more complex and includes more than one electron transfer and/or coupled chemical reactions, as has been discussed in Chapter 1 (1-3). These processes affect the concentration of the electroactive species and thus the electrochemical response, which is the basis of the use of such techniques for the characterisation of reaction mechanisms and kinetics. 

From a theoretical point of view, the existence of coupled processes complicates the numerical resolution of the problem. Depending on the reaction mechanism, these complications may affect simply the form of the coefficients of the Thomas algorithm (as in the first-order ECirre mechanism, see Section 5.1) or, more profoundly, the resolution method due to coupled (e.g., first-order catalytic and ECrev mechanisms, see Sections 5.2 and 5.4) and non-linear (see Chapter 6) equation systems. 

In this chapter, the case of first-order mechanisms will be considered, analysing the changes in the corresponding differential equations, boundary conditions and problem-solving methodology. Strategies for the resolution of the linear equation systems resulting from discretisation will be detailed for some representative mechanisms. The case of multiple-electron transfers will also be considered. This will enable us to simulate other common situations where the electroactive molecule transfers more than one electron. 

---

The mechanisms considered above are all composed of steps in which chemical transformation occurs. In many important industrial reactions, chemical rate processes and physical rate processes occur simultaneously. The most important physical rate processes are concerned with heat and mass transfer. The effects of these processes are discussed in detail elsewhere within this book. However, the occurrence of a diffusion process in a reaction mechanism will be mentioned briefly because it can lead to kinetic complexities, particularly when a two-phase system is involved. Consider a reaction scheme in which a reactant A migrates through a non-reacting fluid to reach the interface between two phases. At the interface, where the concentration of A is Caj, species A is consumed in a first-order chemical rate process. In effect, consecutive rate processes are occurring. If a steady state is achieved, then... 

In direct analogy to the Michaelis-Menten mechanism for reaction of enzyme with a substrate, the inactivator, I, binds to the enzyme to produce an E l complex with a dissociation constant K. A first-order chemical reaction then produces the chemically reactive intermediate with a rate constant k. The activated species may either dissociate from the active site with a rate constant to yield product, P, or covalently modify the enzyme ( 4). The inactivation reaction should therefore be a time-dependent, pseudo-first-order process which displays saturation kinetics. This is verified by measuring the apparent rate constant for the loss of activity at several fixed concentrations of inactivator (Fig. lA). The rate constant for inactivation at infinite [I], itj act (a function of k2, k, and k4), and the Ki can be extracted from a double reciprocal plot of 1/Jfcobs versus 1/ 1 (Fig. IB) (Kitz and Wilson, 1962 Jung and Metcalf, 1975). A positive vertical... 

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

The reactant R2 can also be considered to be a solvent molecule. The global kinetics become pseudo first order in Rl. For a SNl mechanism, the bond breaking in R1 can be solvent assisted in the sense that the ionic fluctuation state is stabilized by solvent polarization effects and the probability of having an interconversion via heterolytic decomposition is facilitated by the solvent. This is actually found when external and/or reaction field effects are introduced in the quantum chemical calculation of the energy of such species [2]. The kinetics, however, may depend on the process moving the system from the contact ionic-pair to a solvent-separated ionic pair, but the interconversion step takes place inside the contact ion-pair following the quantum mechanical mechanism described in section 4.1. Solvation then should ensure quantum resonance conditions. 

In chemical equilibria, the energy relations between the reactants and the products are governed by thermodynamics without concerning the intermediate states or time. In chemical kinetics, the time variable is introduced and rate of change of concentration of reactants or products with respect to time is followed. The chemical kinetics is thus, concerned with the quantitative determination of rate of chemical reactions and of the factors upon which the rates depend. With the knowledge of effect of various factors, such as concentration, pressure, temperature, medium, effect of catalyst etc., on reaction rate, one can consider an interpretation of the empirical laws in terms of reaction mechanism. Let us first define the terms such as rate, rate constant, order, molecularity etc. before going into detail. 

Increased conversion and product purity are not the only benefits of simultaneous separation during the reaction. The chromatographic reactor was also found to be a very suitable tool for studying kinetics and mechanisms of chemical and biochemical reactions. Some recent publications describe the results on investigation of autocatalytic reactions [135], first-order reversible reactions [136], and estimation of enantioselectivity [137,138]. It is beyond the scope of this chapter to discuss the details, but the interested reader is referred to an overview published by Jeng and Langer [139]. 

A distinction between "molecularity" and "kinetic order" was deliberately made, "Mechanism" of reaction was said to be a matter at the molecular level. In contrast, kinetic order is calculated from macroscopic quantities "which depend in part on mechanism and in part on circumstances other than mechanism."81 The kinetic rate of a first-order reaction is proportional to the concentration of just one reactant the rate of a second-order reaction is proportional to the product of two concentrations. In a substitution of RY by X, if the reagent X is in constant excess, the reaction is (pseudo) unimolecular with respect to its kinetic order but bimolecular with respect to mechanism, since two distinct chemical entities form new bonds or break old bonds during the rate-determining step. 

The investigation of the kinetics of a chemical reaction serves two purposes. A first goal is the determination of the mechanism of a reaction. Is it a first order reaction, A—or a second order reaction, 2A— Is there an intermediate A—>/— and so on. The other goal of a kinetic investigation is the determination of the rate constant(s) of a reaction. 

This set includes all reaction mechanisms that contain only first order reactions, as well as very few mechanisms with second order reactions. Any textbook on chemical kinetics or physical chemistry supplies a list. A few examples for such mechanisms are given below ... 

The reaction is acid-catalyzed and yields isocyclosporin A (iso-CsA, 6.58, Fig. 6.23) as the major product. At 50°, the kinetics of the first-order reaction were k=l x 10 6 s 1 (tV2 ca. 1.1 d) at pH 1, and k= 1. 7 x 10 8 s 1 (tv2 ca. 1.2 y) at pH 4. Iso-CsA (i.e., the O-peptide) had a much greater chemical stability than CsA (i.e., the A-peptide) under acidic conditions, in contrast to other findings where the opposite was true. Interestingly, O-acclyl-CsA did not yield iso-CsA and exhibited a much greater stability than CsA, consistent with the nucleophilic mechanism mentioned above. 

In an aprotic solution, the mechanism of oxidation of diaryl disulfides was shown to be more complex than a direct cleavage of the S—S linkage [116,117,123], The occurrence of two consecutive reactions being of second kinetic order if potential-determined and of first order if current-determined, was established for the two-electron transfer steps. Dimerization of the cation radicals occurs on the ArS fragment, whose contribution to the HOMO is more important, and produces an intermediate disulfonium dication. The subsequent cleavage of the latter results in two ArS+ cations and a molecule of a disulfide (the same as a starting disulfide in the case of symmetrical compounds). This mechanism, EC2C1E (E = electrochemical, C = Chemical), has... 

Chemical clastogenesis and mutagenesis both involve a complex series of processes, including pharmacokinetic mechanisms (uptake, transport, diffusion, excretion), metabolic activation and inactivation, production of DNA lesions and their incomplete repair or misrepair, and steps leading to the subsequent expression of mutations in surviving cells or individuals (Thble 7.1). Each of the steps in these processes might conceivably involve first order kinetics at low doses (e.g., diffusion, MichaeUs-Menten enzyme kinetics) and hence be linear. In principle, therefore, the overall process edso might be linear and without threshold. 

Arguments for the presentation of kinetic theory and chemical kinetics as the first topics taught in the initial physical chemistry course are presented. This presentation allows the first topic in physical chemistry to be mathematically more accessible, to be highly relevant to modem physical chemistry practice, and to provide an opportunity to make valuable conceptual connections to topics in quantum mechanics and thermodynamics. Preliminary results from a recent survey of physical chemistry teaching practice are presented and related to the primary discussion. It was found that few departments of chemistry have adopted this order of topical presentation. 

---