The Kinetics of Chain Reactions

In this chapter, we will expand on the modeling of chain reactions, which are of considerable practical importance in the fields of polymer chemistty and combustion, of molecnles snch as hydrocaibons for example, and explosions. Our study will examine the chain reactions that can lead to psendo-steady state modes. The extension to non-psendo-steady state modes will be discussed in Chapter 15. 

A chain reaction is a reaction in which a smaii nnmber of eiementaiy steps are repeated numerous times with the heip of a smaii number of intermediate species that are constantly formed and destroyed. The repetitive sequence is called a Unk in the chain. The best known example is the reaction of hydrogen bromide synthesis, according to reaction [12.R1], whose mechanism may be represented by steps [12.Rla] to [12.Rld]  

By extension, a chain reaction is also called a linear reaction, which consists of a large number of veiy similar steps. Thus, the polymerization and polycondensation reactions are also considered chain reactions, each step being veiy close to the previous since it corresponds to the addition of a small entity to the molecular chain. Another example of such a chain reaction consists of all the steps of soUd nnclens formation from a liquid solution or a solid (see Chapter 14). 

In homogeneous kinetics, chain reactions can be arranged into three large groups  

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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 ... 

Exercise 2.6 Steady State Assumption in the Kinetics of Chain Reactions... 

The kinetics of chain-reaction polymerization is illustrated in Fig. 3.28 for a free radical process. Analogous equations, except for termination, can be written for ionic polymerizations. Coordination reactions are more difficult to describe since they may involve solid surfaces, adsorption, and desorption. Even the crystallization of the macromolecule after polymerization may be able to influence the reaction kinetics. The rate expressions, as given in Appendix 7, Fig. A7.1, are easily written under the assumption that the chemical equations represent the actual reaction path. Most important is to derive an equation for the kinetic chain length, v, which is equal to the ratio of propagation to termination-reaction rates. This equation permits computation of the molar mass distribution (see also Sect. 1.3). The concentration of the active species is very small and usually not known. First one must, thus, ehminate [M ] from the rate expression, as shown in the figure. The boxed equation is the important equation for v. 

The kinetics of chain reactions of small molecules is much harder to follow (and prove) than chain-reaction polymerization. Once the reaction is over, the sttucture of the produced macromolecule can be studied as permanent documentation of the reaction (see Chaps. 1 and 3). 

A number of original monographs and textbooks [3-31] are dedicated to chemical chain reactions. Here we provide a minimum of information on the kinetics of chain reactions, which is primarily intended to create the necessary basis for stating in subsequent Chapters the eoneept of the value description for the kinetics of chain reactions. 

As follows from a brief review of the fundamentals on the kinetics of chain reactions, an elegant one-centered model of the process, which was offered at the begiiming of the development of the chain reaction theory, provides a phenomenological description. Initiation and inhibition of reactions by small additions of compormds, critical phenomena, etc. may serve as examples. However, it should be noted that modeling a chain reaction is usually complicated, if the one-centered approximation is not justified and in the cases when consumption of initial compormds should be taken into accormt, or when the intermediates are participating in the chain process, in other words if one has to deal with chain processes of a more complicated nature. So the efforts aimed at developing the special theoretical approaches that are thought to help for a better orientation in a complex chain chemical process are justified, in particular, rmder the conditions of multi-centered, and consequently multi-routed occurrance. 

Condensation reactions follow kinetic schemes similar to those of small molecule reactions. They are simple first-order, second-order, etc. reactions. In contrast, the kinetics of chain reactions, such as free-radical polymerization or ionic polymerization, are much more complicated. 

Chain reactions provide energetically favorable reaction routes for reactions between stable molecules. The kinetics of chain reactions can be solved with the steady-state approximation. 

Methods for studying chain reactions appeared and were improved in parallel with their discovery and study. Experimental evidence for the chain mechanism of the process and participation in it of free atoms and radicals gained significance at the first stage of development of the kinetics of chain reactions (1920-1940). It seems reasonable to consider briefly these proofs. 

The study of the kinetics of chain reactions in the stationary regime allows one to determine a combination of the rate constants for bimolecular chain termi-... 

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