Kinetic Characteristics of Chain Reactions

Certain kinetic aspects of free-radical reactions are unique in eomparison with the kinetie eharaeteristics of other reaction types that have been eonsidered to this point. The underlying difference is that many free-radieal reactions are ehain reaetions that is, the reaetion meehanism consists of a cycle of repetitive steps whieh form many produet molecules for eaeh initiation event. The hypothetieal mechanism below illustrates a ehain reaetion. 

SECTION 12.2. CHARACTERISTICS OF REACTION MECHANISMS INVOLVING RADICAL INTERMEDIATES 

The step in which the reactive intermediate, in this ease A-, is generated is ealled the initiation step. In the next foin equations in the example above, a sequenee of two reactions is repeated this is the propagation phase. Chain reaetions are eharaeterized by a chain length, which is the number of propagation steps that take plaee per initiation step. Finally, there are termination steps, which include any reactions that destroy one of the reactive intermediates necessary for the propagation of the chain. Clearly, the greater the frequency of termination steps, the lower the chain length will be. 

The overall rate of a chain process is determined by the rates of initiation, propagation, and termination reactions. Analysis of the kinetics of chain reactions normally depends on application of the steady-state approximation (see Section 4.2) to the radical intermediates. Such intermediates are highly reactive, and their concentrations are low and nearly constant throughout the course of the reaction  

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

The application of the steady-state approximation and the resulting equality of the propagation and termination rates permits formulation of a rate law for the reaction mechanism above. When the chain length is long, the stoichiometry is 

Setting the rate of initiation equal to the rate of termination, assuming that ka. is the dominant rate constant for termination  

Termination reactions involving coupling or disproportionation of two radicals ordinarily occurs at diffusion-controlled rates. Since the concentration of the reactive 

After the steady state approximation, both propagation steps must proceed at the same rate or the concentration of A- or C- would build up. By substimting for the concentration of the intermediate C-, we obtain 

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On the basis of the kinetic characteristics of chain polymerisation reactions, it is possible to predict the final microstructures available by a so-called random process from a simple mixture of two comonomers. Indeed, the global mechanism of copolymerisation can be illustrated as presented in Figure 30. 

The analysis of the thermodynamic and kinetic characteristics of these reactions will allow a better understanding of the biological importance of chain breaking in the antioxidant actions of polyphenols. 

Lastly, let us point out that in 1953 the photochemical oxidations of mixtures of benzaldehyde and of n-decanal were studied by Ingles and Melville. The kinetic characteristics of the reactions indicate that in mixtures these aldehydes do not undergo oxidation independently of one another the two molecules are involved in a single kinetic chain, exactly as in a copolymerization reaction. 

Different methods for determination of the number of active centers during catalytic olefin polymerization are proposed. There are two basic groups of method applied to determine the kinetic characteristics of propagation reactions and transfer reactions of polymer chains (values of Cp, kp, and K ) for catalytic olefin polymerization. 

The pyrolysis of CH2CI-CH2CI is of great industrial importance, as it is a method of manufacturing vinyl chloride monomer CH2=CHC1. The experimental kinetic characteristics of this reaction (sensitivity to the nature of the walls, existence of an induction period...) permit this reaction to be classed as a straight chain radical reaction. 

In examining the kinetics of this type of chain reaction (18), it was found that the major characteristics of all the curves of Figure 2 could be reproduced by assuming ... 

The thermodynamic and kinetic characteristics of an intramolecular transimination reaction observed in solutions containing pyridoxal-5-phosphate and ethylenediamine have been investigated (75JA6530). The open-chain structure Schiff bases and the cyclic tautomers such as 54 are in equilibrium in aqueous solution over the pH range 7.5-14, but these equilibria are rather complex owing to the different states of the ionization (protonation) in both tautomers. The ring-chain equilibrium constant (the sum of all cyclic tautomers versus all open-chain tautomers) varies by less than a factor of 4 over the pH range 7-14. At pH 14, KT = 1.2 at pH 10,... 

A reaction mechanism is a sequence of elementary processes proposed to account for experimental kinetic results. Pure chemical kinetics proposes a classification of various types of mechanism (non-chain mechanisms, straight-chain and branched-chain mechanisms, etc.), establishes relationships between the properties of a global reaction and those of the elementary processes involved in the corresponding mechanism, and provides rules for writing a priori a reaction mechanism from a knowledge of the thermochemical and kinetic characteristics of the... 

The kinetic characteristics of the solvolysis of long-chain phenyl esters catalyzed by polyvinylimidazole are totally different from those of simpler substrates. Over-beiger etal found a dramatic rate enhancement for the solvolysis of NDBA13 catalyzed by poly-4-vinylimidazole 1 in ethanol-water mixture (5S). The reaction obeys the Michaelis-Menten kinetics when the ethanol content is below ca. 40%, while solvolyses in 60,70, and 80% ethanol are satisfactorily described by a simple second-order rate equation. In term of the half-life, t fi, the solvolysis in 20% ethanol... 

C. S. Marvel, J. Dec, and H. G. Cooke, Jr. [/ Am. Chem. Soc., 62, 3499 (1940)] employed optical rotation measurements to study the kinetics of polymerization of vinyl-1-phenylbutyrate. In dioxane solution the specific rotation angle represents a linear combination of contributions from the monomer proper and those of the polymerized monomer units. The contribution of the polymerized units can be viewed as independent of chain length. The reaction takes place in a constant-volume system and may be viewed as irreversible. The stoichiometry of the reaction may be viewed as A -I-Pjj Pn+i where A represents the monomer and P the polymer. The following data are characteristic of this reaction. 

The closed sequence radical reaction introduces a multiplicative kinetic factor with respect to the initiation so that there is a certain analogy with homogeneous and heterogeneous catalysis, which is also characterized by the existence of closed sequences. However, a big difference exists between chain reactions and so-called catalytic reactions. In fact, in catalytic reactions, the total number of active centres is fixed, whereas that of chain reactions is determined by the competition between the initiation and termination reactions. The rate of a chain reaction is therefore a function of both the kinetic characteristics of the initiation, propagation and termination reactions. 

The fcatalytic role of interface layer in 3 -D polymerization is certainly connected with the decreasing chain termination rate. Both rate of radical trapping and also the kinetic order of the reaction are decreased from the second to the first. This can be explained by several possible causes for this effect. In accordance with Ref. [7] chain termination represents the diffusion-control reaction and its rate depends on the mobility of macroradicals. If this mobility is decreased, so that condition tytp 1 takes place (fj and fp are characteristic moving times of the active center of macroradicals as a result of... 

As efficiency of inhibitor it is implied its capability of providing the most prolonged and deep deceleration of a chain reaction. In the hteratuie proper attention is paid to this issue [1-16], and different quantitative characteristics are provided, which delrne the efficiency of chain reaction inhibitors. It seems to us as positive in these approaches that a solution of the inhibitor efficiency problem is turned into the kinetic analysis of the model for inhibited reaction. Nevertheless, in most cases, the sorting out of the quantitative characteristics for inhibitor efficiency is based on occurring a definite set of reactions in the inhibited process and carmot be considered as a versatile one. In other words, their apphcation is restricted by the degree of complexity of the used kinetic model. 

As an alternative, this chapter describes methods for predicting small-molecule diffusivity that are based on transition-state theory (TST) [82-84] and kinetic Monte Carlo (KMQ [85,86]. These methods capitalize on the proposed penetrant jump mechanism. TST was described in Chapter 1 and is typically used to estimate the rates of chemical reactions from first principles here we use TST to calculate the rate of characteristic jumps for each penetrant in a host polymer matrix. The collection of jump rates can be combined with the penetrant jump topology and KMC to obtain the penetrant diffusion coefficient. Other results obtainable from these simulations are physical aspects related to the jump mechanism the sizes and shapes of voids accessible to penetrant molecules [87], enthalpic and entropic contributions to the penetrant jump rate [88,89], the extent and characteristics of chain motions that accompany each jump [90], and the shape and structure of the jump network itself [91]. 

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