Branching chain reaction

Branching steps are of major importance in flames and explosions, in both gas and liquid phases. The best-established examples are for gas-phase reactions. One reaction in which they play a pivotal role is a seemingly simple one, 

The reaction is actually quite complex, even now not completely understood. The following are thought to be important chain-carrying steps  

Although the first reaction is of an ordinary sort, the next two are unusual in that one propagating intermediate is converted into two. These are branching reactions. As each occurs, the total rate speeds up. When that happens, even more branching occurs, and so on. If unchecked, the exponential growth of chain carriers leads to explosion, just as in nuclear chain reactions. 

One should note that this happens without the occurrence of reactions between pairs of intermediates that have inherently lower rates under steady-state conditions. 

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Chain reactions begin with the initiation of a reactive intermediate that propagates the chain and concludes with termination when radicals combine. Branching chain reactions can be explosively fast. 

If the three product neutrons strike three other fissile nuclei, then after the next round of fission there will be nine neutrons, which can induce fission in nine more nuclei. In the language of Section 13.9, neutrons are carriers in a branched chain reaction (Fig. 17.24). 

Both of these reactions involve the production of two active centers where there was only one before. When reactions of this type occur to a significant extent, the total number of active centers present in the system can increase very rapidly, since a multiplication effect sets in as the chains propagate. The growth of chain carriers in a branched chain reaction is pictured below. 

Chain reactions can lead to thermal explosions when the energy liberated by the reaction cannot be transferred to the surroundings at a sufficiently fast rate. An explosion may also occur when chain branching processes cause a rapid increase in the number of chains being propagated. This section treats the branched chain reactions that can lead to nonthermal explosions and the physical phenomena that are responsible for both branched chain and thermal explosions. 

The generalized mechanism by which branched chain reactions proceed provides a basis for a semiquantitative understanding of explosions resulting from chain branching. ... 

As already noted (see Chapter 4), autoxidation is a degenerate branching chain reaction with a positive feedback via hydroperoxide the oxidation of RH produces ROOH that acts as an initiator of oxidation. The characteristic features of inhibited autoxidation, which are primarily due to this feedback, are the following [18,21,23,26,31-33] ... 

The hydrogen-chlorine chain reaction has proved to be one of the most controversial systems yet studied. After thirty years of investigation Bodenstein43 was able to say in 1931 that every worker on the photochemical synthesis of HC1 had produced his own mechanism even as late as 1940 little positive information had been obtained. However, the accumulated techniques and experience had firmly established the importance of atom chain reactions. The mechanism of photo-initiation and propagation is the same as for the hydrogen bromide photosynthesis, a non-branching chain reaction... 

The cyclodimerization depicted in Scheme 7.19 is one of the many examples concerning cation-radicals in the synthesis and reactions of cyclobutanes. An authoritative review by Bauld (2005) considers the problem in detail. Dimerization is attained through the addition of an olefin cation-radical to an olefin in its neutral form one chain ends by a one-electron reduction of the cyclic dimer cation-radical. Unreacted phenylvinyl ether acts as a one-electron donor and the transformation continues. Up to 500 units fall per one cation-radical. The reaction has an order of 0.5 and 1.5 with respect to the initiator and monomer, respectively (Bauld et al. 1987). Such orders are usual for branched-chain reactions. In this case, cyclodimerization involves the following steps ... 

Explosion may occur as the result of a chain reaction, when the reaction of a "chain carrier , such as a free atom or radical, with a molecule frees another such particle to continue the chain, for example, -OH+H2 H20+H - Particularly effective is a branching chain reaction, such as... 

Such reactions have been used to explain the three limits found in some oxidation reactions, such as those of hydrogen or of carbon monoxide with oxygen, with an "explosion peninsula between the lower and the second limit. However, the phenomenon of the explosion limit itself is not a criterion for a choice between the critical reaction rate of the thermal theory and the critical chain-branching coefficient of the isothermal-chain-reaction theory (See Ref). For exothermic reactions, the temperature rise of the reacting system due to the heat evolved accelerates the reaction rate. In view of the subsequent modification of the Arrhenius factor during the development of the reaction, the evolution of the system is quite similar to that of the branched-chain reactions, even if the system obeys a simple kinetic law. It is necessary in each individual case to determine the reaction mechanism from the whole... 

In recent years Emanuel, Neiman, and their respective schools have greatly contributed to the theory of antioxidant action by studying the phenomenon of the critical antioxidant concentration in terms of a degenerate branched chain reaction. The critical antioxidant concentration, a well-established feature of phenolic antioxidants, is one below which autoxidation is autocatalytic and above which it proceeds at a slow and steady rate. Since the theory allowed not only a satisfactory explanation of the critical antioxidant concentration itself but elucidation of many refinements, such as the greater than expected activity of multifunctional phenolic antioxidants (21), we wondered whether catalyst-inhibitor conversion could be fitted into its framework. If degenerate chain branching is assumed to be the result of... 

This phenomenon of multiple pressure limits of explosion can be explained on the basis of branching chain reactions. For example, a series of reactions comprising a branching chain may be illustrated by resort to the well-known hydrogen-oxygen reaction (13, 31, 36, 52). 

This is of the same form as Equation 30, but involves the mixed diffusion coefficient, Jci9, instead of the thermal conductivity of the mixture. However, as seen from the kinetic theory of gases, the thermal conductivity is proportional to the diffusion coefficient. Therefore agreement of experimental results with either Equation 30 or 53a is not an adequate criterion for distinguishing between first explosion limits in branching chain reactions and purely thermal limits. It has been reported (52), that, empirically,... 

Question. In a branched chain reaction with a = 2, there is a consequent build-up of radicals with each cycle of branched chain propagation ... 

A highly schematic and simplified mechanism for a branched chain reaction... 

This mechanism includes all the features necessary to describe a branched chain reaction. For simplicity, the initiation and second propagation steps are taken to be in... 

A typical branched chain reaction showing explosion limits... 

The reaction between hydrogen and oxygen is a branched chain reaction which shows explosive regions. It has been very extensively studied, and appears to have a very complex mechanism. 

PHENOMENOLOGICAL MODEL OF BRANCHED-CHAIN REACTIONS ON A CATALYST SURFACE... 

In the 1950s, Semenov and Voevodskii [148] made an attempt to apply the concepts of the branching-chain reaction theory to the kinetics of heterogeneous catalysts. They applied the concept of free valencies migrating over the catalyst surface and of "semi-chemisorbed radicals. But their attempt was criticized (see, for example, ref. 149 where Temkin, using hydrogenation of ethylene on palladium as an example, proved experimentally the inapplicability of the chain theory concepts). 

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