Steady-state approximations, radical chain

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. 

In this scheme, CHO appears irrelevant we return to it later. The rate law can be derived by making the steady-state approximation for each of the chain-carrying radical intermediates ... 

Derive the rate expression. Make the steady-state approximation for the radical intermediates, and assume that the chains are long. 

Historically, the steady state approximation has played an important role in unraveling mechanisms of apparently simple reactions such as H2 + CI2 = 2HC1, which involve radicals and chain mechanisms. We discuss here the formation of NO from N2 and O2, responsible for NO formation in the engines of cars. In Chapter 10 we will describe how NO is removed catalytically from automotive exhausts. 

In principle there is a competition for the HO2 radical between peroxydisulphate and hydrogen peroxide [reactions (63) and (86)] however, when the stoichiometry is 1 1 reaction (86) can be neglected. Assuming that the chain length is large, with the usual steady-state approximation, we obtain the following rate equation ... 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

Radical chain reactions are complicated because multiple reactions occur, but the overall velocity of the sequence can be given in simplified form by applying steady-state approximations. An important feature of any chain reaction is that the velocities of all propagation steps must be identical because the radicals formed as products in each elementary reaction are the reactants in another elementary... 

The molecular weight distribution and the average molecular weight in a free-radical polymerization can be calculated from kinetics. The kinetic chain length v is defined as the average number of monomers consumed per number of chains initiated during the polymerization. It is the ratio of the propagation rate to the initiation rate (or the termination rate with a steady-state approximation) ... 

In order to determine the factors that control the rate, r, and the chain length, Vp, of the polymerization, it is important to determine the rate of the separate reactions involved, namely initiation, propagation and termination, as well as the rates of any other processes that may compete with them. These rate equations will contain reactive intermediates (i.e. radical concentrations [R-]) that cannot be explicitly determined. The process of solution requires the steady-state approximation, which states that there is no net change in radical population with time during the steady-state polymerization, i.e. d[R ]/df = 0, in order to... 

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. 

The specificity of the reaction mechanism to the chemistry of the initiator, co-initiator and monomer as well as to the termination mechanism means that a totally general kinetic scheme as has been possible for free-radical addition polymerization is inappropriate. However, the general principles of the steady-state approximation to the reactive intermediate may still be applied (with some limitations) to obtain the rate of polymerization and the kinetic chain length for this living polymerization. Using a simplified set of reactions (Allcock and Lampe, 1981) for a system consisting of the initiator, I, and co-initiator, RX, added to the monomer, M, the following elementary reactions and their rates may be... 

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... 

Problem 6.16 Typical values determined from photoinitiated radical chain polymerization with intermittent illumination are in the range 0.1-10 s. Calculate from this the duration of the non-steady-state period and comment on the validity of steady-state approximation made in a typical polymerization study. 

The most important difference between a living ionic polymerization which has no termination or transfer mechanism and free-radical or ionic processes that do have termination or chain transfer steps is that the distributions of the degrees of polymerization are quite different. The distribution function can be derived by a kinetic approach due to Flory [6], which is analogous to that used earlier for free-radical reactions (see Problem 6.44). However, in the present case with no chain termination the simplifying steady-state approximation cannot be used. 

The limit to the chain reaction is determined by the relative values of the rate constants for the propagation step and the branching or transfer reactions involving solvent or inhibitor molecules. As the concentration of the oxidizable molecule falls in the solution, the reaction rate also falls. The reaction is characterized by a steady-state or maximum rate represented by the linear portion of the sigmoidal reaction progress curve. This is achieved when the rate of generation of new initiating radicals is equal to their termination rate. Here, the kinetics is simplified by the steady-state approximation, and the maximum rate is first order with respect to the benzaldehyde concentration. 

Chain polymerization is a complicated radical chain mechanism involving initiation, propagation, and termination steps (see Section 23.4 for the details of this mechanism). The derivation of the overall rate equation utilizes the steady state approximation and leads to the following expression for the average number of monomer units in the polymer chain ... 

A radical chain polymerization is started when the initiator begins to decompose according to Eq. (6.3) and the concentration of radicals in the system, [M ], increases from zero. The rate of termination or disappearance of radicals, being proportional to [M ]- [cf. Eqs. (6.17)-(6.19)], is thus zero in the beginning and increases with time, till at some stage it equals the rate of radical generation. The concentration of radicals in the system then becomes essentially constant (or steady ), as radicals are formed and destroyed at equal rates. This condition, described as steady-state assumption or steady-state approximation , can thus be described by the following two equations ... 

The steady-state approximation applied to chain reactions by Bodenstein cannot describe several time-dependent phenomena, for instance branched chain reactions, leading to explositions. For such cases the semi (or quasi ) steady-state approach, developed by a Nobel prize winner N. Semenov, assumes that concentrations of all radicals except one (the greatest) are considered as steady state. 

The rates of homogeneous reactions (radical or chain) are then computed taking into account the concentration of radicals generated in the initiation step. Usually from the steady-state approximation applied to chain reactions, the initiation rate is equal to the termination rate. An example of a termination reaction given below... 

The steady-state approximation was first enunciated by Bodenstein.> It states that in a reaction in which transient species, such as atoms or radicals, are involved, a steady state sets in, characterized by an equal rate of formation and disappearance of the species. This principle, applied to the case of a polymerization reaction, means that at a certain reaction stage the amount of active centers formed is equal to the amount of growing chains terminated ... 

Franklin [32] and Benson [29] have summarized methods for predicting the rates of chemical reactions involving free radicals and Gavalas [33] has shown how the steady-state approximation and use of the chain propagation reactions alone (long-chain approximation) leads to reasonably simple calculation of the relative concentrations of the nonintermediate species. Also see Benson [34],... 

The previous sections address the kinetics for each of the processes involved in free radical polymerization, as well as the overall polymerization process. A steady-state approximation was used to determine the overall rate of polymerization and the chain length distribution. Practically, there are many exceptions to these approximations, including nonsta-tionary polymerization and dead-end polymerization [50, 51], which are treated in more detail elsewhere. 

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