Rate expressions termination

To deal with the case of termination by combination, it is convenient to write some reactions by which an n-mer might be formed. Table 6.5 lists several specific chemical reactions and the corresponding rate expressions as well as the general form for the combination of an (n - m)-mer and an m-mer. On the assumption that all kj values are the same, we can write the total rate of change of [M -] ... 

LDPE polymerization reaction consists of various elementary reactions such as initiation, propagation, termination, chain transfer to polymer and monomer, p-scission and so forth [1-3], By using the rate expression of each elementary reaction in our previous work [4], we can construct the equations for the rate of formation of each component. 

The termination constants kt found previously (see Table XVII, p. 158) are of the order of 3 X10 1. mole sec. Conversion to the specific reaction rate constant expressed in units of cc. molecule" sec. yields A f=5X10". At the radical concentration calculated above, 10 per cc., the rate of termination should therefore be only 10 radicals cc. sec., which is many orders of magnitude less than the rate of generation of radicals. Hence termination in the aqueous phase is utterly negligible, and it may be assumed with confidence that virtually every primary radical enters a polymer particle (or micelle). Moreover the average lifetime of a chain radical in the aqueous phase (i.e., 10 sec.) is too short for an appreciable expectation of addition of a dissolved monomer molecule by the primary radical prior to its entrance into a polymer particle. 

Because the steady-state assumption leads to the equilibrium relation for the bromine atom concentration (4.2.20), it does not matter what mechanism is assumed to be responsible for establishing this equilibrium. Alternative elementary reactions for the initiation and termination processes, which give rise to the same equilibrium relationship, would also be consistent with the observed rate expression for HBr formation. For example, the following reactions give rise to the same equilibrium ... 

If equations 4.2.25 and 4.2.26 are substituted for equations 4.2.11 and 4.2.15, respectively, in the mechanism described above, the effect is to replace kx by k [M] and k5 by /c 5[M] everywhere that they appear. Since these quantities appear as a ratio in the final rate expression, the third body concentration will drop out and kjks) becomes identical with k /k 5 The necessity for the use of the third body concentration thus is not obvious in kinetic studies of the thermal reaction. However, from studies of photochemical reaction between hydrogen and bromine, there is strong evidence that the termination reaction is termolecular. This fact and... 

A first-order rate expression results when the termination process involves the dissimilar radicals, Rl and R2. That is, reaction (4b) is the chain breaking step. 

A reaction rate expression that is proportional to the square root of the reactant concentration results when the dominant termination step is reaction (4c), that is, the termination reaction occurs between two of the radicals that are involved in the unimolecular propagation step. The generalized Rice-Herzfeld mechanism contained in equations 4.2.41 to 4.2.46 may be employed to derive an overall rate expression for this case. 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

Approaches to the determination of the concentration-dependent terms in expressions for reversible reactions are often based on a simplification of the expression to limiting cases. By starting with a mixture containing reactants alone and terminating the study while the reaction system is still very far from equilibrium, one may use an initial rate study to determine the concentration dependence of the forward reaction. In similar fashion one may start with mixtures containing only the reaction products and use the initial rates of the reverse reaction to determine the concentration dependence of this part of the rate expression. Additional simplifications in these initial rate studies may arise from the use of stoichiometric ratios of reactants and/or products. At other times the use of a vast... 

The choice of reaction (72) as the termination reaction is dictated by the overall order of unity, if reaction (69) is in its first-order pressure region. This is probably true above 200 torr. With the usual approximations the mechanism leads to the rate expression... 

Considering that for a steady system, the termination and initiation steps must be in balance, the definition of chain length could also be defined as the rate of product formation divided by the rate of termination. Such a chain length expression would not necessarily hold for the arbitrary system of reactions (3.1)—(3.6), but would hold for such systems as that written for the H2—Br2 reaction. When chains are long, the types of products formed are determined by the propagating reactions alone, and one can ignore the initiation and termination steps. 

A 2 is inserted in Equation 6.11 because, in this presentation, for each initiator molecule that decomposes, two radicals are formed. The 2 is omitted from Equation 6.6 because this rate expression describes the rate of decomposition of the initiator, but not the rate of formation of free radicals R . Similarly, in Equations 6.18 and 6.20, each termination results in the loss of two growing chains, thus a 2 appears in the descriptions. 

A chain reaction, 4 —> 6 + C obeys the rate expression r = What initiation and termination steps will predict these kinetics Repeat for r = kC. ... 

The first termination step occurs homogeneously, while the second occurs by adsorption of R on the walls of the reactor. Formulate an expression for the rate of this reaction expected in a tube of diameter D with laminar flow (Shp = 8/3) and a diffusion coefficient Dr. What effective rate expressions are obtained in the limits of when... 

In this paper the rate expressions have all been corrected for nitrogen evolution from the azo initiator, oxygen absorption by initiator radicals, and oxygen evolution in termination. It is assumed that the initiator which decomposes without starting oxidation chains does not react with oxygen (21). This correction involves the addition of (l-e)Ri/2e to the measured rate, where e is the efficiency of chain initiation, found to be 0.5 at 30 °C. and 0.6 at 56 °C. The rate constant for Reaction 7 has been written as 4ktCT in order that the three termination constants may be comparable (26, footnote 27).]... 

Each of these termination reactions has a similar rate expression that is second order in radical concentration, so the overall termination rate can be expressed as the rate of disappearance of the radicals due to both types of termination ... 

It has been known for some time that UV photopolymerization of multifunctional monomers does not obey the classical rate expression, Rp proportional to I0 5, but follows an approximately first-order relationship [196,197]. These results have been explained by postulating that, in these viscous monomers, radical occlusion competes with bimolecular termination. 

At the time this mechanism was proposed it was considered that boron fluoride catalysis showed rather similar kinetics and activation energy, differing from the stannic chloride only in that chain transfer could occur but termination apparently was very slow or absent (13). Subsequent work with boron fluoride suggests however (14) that the apparent fractional order dependence on monomer may be spurious and thus tends to discredit the data for stannic chloride in spite of the fact that the rate expressions give an excellent account of the rate of disappearance of epoxide both within and between the individual experiments. The difference in behaviour between stannic chloride und boron fluoride remains one of the unsolved problems in this field. 

Current views on polymerization of acrylonitrile in homogeneous solution are illustrated by a description of the reaction in N,N-dimethyl-formamide (DMF) as initiated by azobisisobutyronitrile (AIBN) at about SO to 60°. Primary radicals from the decomposition of AIBN react with monomer to start a growing chain. About one-half of the primary radicals are effective, the others being lost in side reactions not leading to polymer. Bevington and Eaves (32) estimated initiator efficiency by use of AIBN labelled with C-14, whereas Bamford, Jenkins and Johnson (13) used the FeCls termination technique. Both of these methods require that the rate of AIBN decomposition be known, and the numerical value of this rate has undergone a number of revisions that require recalculation of efficiency results. From recently proposed rate expressions for AIBN decomposition at 60° (22, 136) one calculates an efficiency of about 40% by the tracer technique and 60—65% by the FeCl3 method. 

Once a mechanism has been proposed, a kinetic analysis is required to see whether the kinetics predicted by the mechanism fits the experimentally determined kinetic rate expression. The predicted rate expression is found generally from a steady state analysis. Most mechanisms have at least three termination steps, and for a first analysis only one step is included at a time. 

All three expressions, Equations (6.144), (6.150) and (6.152), are the same, as they should be. If the long chains approximation had not been used, the terms involving the rates of termination and initiation would not have dropped out, and a more complex expression would have resulted. 

At partial pressures of oxygen greater than approximately 100 Torr, chain termination occurs exclusively via the mutual destruction of two alkylperoxy radicals [reaction (6)]. The cross-termination reaction (5) may be neglected. The predicted rate expression, under steady-state conditions, is then given by... 

Termination. Two radicals react to form one or two molecules having no unpaired spins. If the radicals are assumed randomly distributed in space, the termination rate is second order in radical concentration. Under conditions where diffusion is restricted, however, the twin chains initiated by radicals from the same initiator molecule may remain close together for a time, so that termination has a higher probability than in the case of random distribution of radicals. This case has been discussed by Allen and Patrick (I, 3). It results in a rate expression formally equivalent to a reaction first order in radical concentration, with half-life t. Termination by initiator radicals is also taken into account here. 

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