Linear chain reactions

In the case of the hydrogen-bromine reaction, each of the elementary propagation reactions led to the formation of a single chain carrier. This type of reaction is said to be a straight or linear chain reaction. Some mechanisms involve elementary propagation reactions in which more than a single chain carrier is formed by the reaction. This type of reaction is known as a branching reaction. Examples of such reactions are... 

Most combustion reactions involve chain branching reaction steps. Under conditions where these steps are less significant than linear chain reactions, the reaction appears to be stable, but when the chain branching steps dominate, the overall reaction rate can accelerate uncontrollably. 

We can constmct a simple model of a chain branching reaction that is a simple variation of the linear chain reaction we used previously. Consider the reaction system A B + C, which proceeds by the steps... 

For the linear chain reaction case, a free radical kinetic mechanism (e.g., polymeriza-... 

Experiments with a flow-system confirmed that in the early stages the only process in operation is a linear chain reaction giving acetone and hydrogen peroxide [22—24]. 

The features of a linear chain reaction may be recognized in the above table. The initiation step creates free radicals and the termination step destroys them. The rate-determining step will be the propagation reaction, which is the product-forming step and has the characteristic feature of a linear chain reaction, viz. one radical is formed for every radical that reacts. Noting these features, we have... 

This relation is of fundamental importance in free-radical polymerization since the kinetic chain length decreases with an increase in the rate of initiation. Thus an attempt to accelerate polymerization by adding more initiator will produce a faster reaction but the polymer will have shorter chains. This can also be seen as a consequence of the steady-state approximation in a linear chain reaction since the rate of termination is equal to the rate of initiation and, if the rate of termination increases to match the rate of initiation, the chains must necessarily be shorter. 

One of the obvious features of the oxidation of polypropylene is the formation of hydroperoxides (reaction (3) in Scheme 1.55) as a product. The initiation of the oxidation sequence is usually considered to be thermolysis of hydroperoxides formed during synthesis and processing (shown as the bimolecular reaction (1 ) in Scheme 1.55). The kinetics of oxidation in the melt then become those of a branched chain reaction as the number of free radicals in the system continually increases with time (ie the product of the oxidation is also an initiator). Because of the different stabilities of the hydroperoxides (e.g. p-, s- and t- isolated or associated) under the conditions of the oxidation, only a fraction of those formed will be measured in any hydroperoxide analysis of the oxidizing polymer. The kinetic character of the oxidation will change from a linear chain reaction, in which the steady-state approximation applies, to a branched-chain reaction, for which the approximation might not be valid since the rate of formation of free radicals is not... 

For the exploitation of chain reactions, we follow the discussion of Basov et al. l>. With reference to the discussion of population inversions in Section 4 (14), we consider first the relationship between chemical pumping P(t) and relaxation L(t) which determines whether the reaction proceeds with or without an inversion. The temporal dependence of the level populations is given by the set of balance equations (16). Various types of temporal dependence of the pumping function P (<) now have to be investigated. If the initial external energy input produces a certain concentration of active centers n, the pumping rate in the case of a linear chain reaction is given as... 

Figure 12.2. Schematic representation of a linear chain reaction with only one active center...

We will consider a linear chain reaction with a single active center, presenting an even initiation rate v in the volume of the reactor, first order breaking in the volume in relation to the active centers, and breaking on the external walls. 

Branching chain reactions are distinct from linear chain reactions because of the presence of another type of reaction step the branching step. Hence, for an active center X there are three reaction possibilities, which are illustrated in Figure 12.8 with their corresponding a, P and 5 probabilities for the propagation, breaking, and branching steps respectively ... 

If we represent a link in the chain by a dash, a linear chain reaction would be... 

The chemistry of furfuryl alcohol polymerization has received much attention over the years. Several recent reviews have been written (5,6,54). Based on the accumulated data, furfuryl alcohol has to be considered a bifimctional monomer in the initial stage and its "normal" reactions give linear chains or oligomers containing essentially two repeating units (15,16) with (16) predominating. 

Esters made from the reaction of acids (or anhydrides) with alcohols. 1-Alkanols produce linear chains, hence the L designation in plasticizer names. 

As pointed out by Flory [16], the principle of equal reactivity, according to which the opportunity for reaction (fusion or scission) is independent of the size of the participating polymers, implies an exponential decay of the number of polymers of size / as a function of /. Indeed, at the level of mean-field approximation in the absence of closed rings, one can write the free energy for a system of linear chains [11] as... 

Ketten-glied. n. link (or member) of a chain, -isomerie, /. chain isomerism, -kokken, m.pl. streptococci, -linie,/. catenary (curve), -moldktil, n. chain molecule, linear molecule, -polymer, n. chain polymer, linear polymer, -rad, n. sprocket wheel, -reaktion, /. chain reaction, -trieb, m. chain drive. 

A polymer molecule may have just a linear chain or one or more hranches protruding from the polymer hackhone. Branching results mainly from chain transfer reactions (see Chain Transfer Reactions later in this chapter) and affects the polymer s physical and mechanical properties. Branched polyethylene usually has a few long hranches and many more short hranches... 

Elementary considerations indicate that with appropriate substitutions some of the reactions mentioned above can be eliminated. Indeed, when 5-methyl-2-vinyl-furan was used, no alkylation was observed, the positions C-3 and C-4 being rather unreactive16, and the polymer was a mixture of linear chains with polyunsaturations and linear saturated chains, i.e. only structures like 21, 23 and 26 were present, with a 5-methyl ring instead of the 5-unsubstituted one. When 2-isopropenylfuran was used, no hydride transfer took place since this requires a hydrogen atom in the a-position to the ring, which this monomer does not have the polymers were white and gave electronic spectra transparent down to 280 nm. Alkylation at C-5, how-... 

The Hammett equation is the best-known example of a linear free-energy relationship (LFER), that is, an equation which implies a linear relationship between free energies of reaction or activation for two related processes48. It describes the influence of polar meta-or para-substituents on reactivity for side-chain reactions of benzene derivatives. 

The Tafel slopes obtained under concentrations of the chemical components that we suspect act on the initiation reaction (monomer, electrolyte, water contaminant, temperature, etc.) and that correspond to the direct discharge of the monomer on the clean electrode, allow us to obtain knowledge of the empirical kinetics of initiation and nucleation.22-36 These empirical kinetics of initiation were usually interpreted as polymerization kinetics. Monomeric oxidation generates radical cations, which by a polycondensation mechanism give the ideal linear chains ... 

This material, previously thought to be Hg CAsP Ij, contains infinite linear chains of Hg atoms in two mutually perpendicular directions. Each Hg atom bears a charge of 0.35. On further reaction of AsFj with Hg in liq SO2, a deep red solution is obtained as well as the golden crystals. From this red solution two further polynuclear Hg derivatives, HgjfAsF Ij and Hg4(AsF5)2 crystallize ... 

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