Steady state chain reactions

Rate of polymerization In chain polymerization, the rate at which the monomer (M), of concentration [M], is converted to polymer, —d[M]/df. Since the rate of propagation occurs hundreds of times more frequently than initiation, rates of polymerization and propagation are the same and the symbol J p is used for both. For a steady state chain polymerization reaction, J p is defined by the following equation... 

Burton has also discussed the steady state in reaction chains (8). Where it exists in a cell, and where source and sink are exteraal to the cell... 

Transition to steady-state etching. The surface becomes sufficiently disordered to dismpt tlie quasi-equilibrium, and tlie reaction layer becomes a tree stmcture of fluorosilyl chain stmctures tenninated by SiF., groups. 

New radicals are introduced by thermolysis of the hydroperoxide by chain-branching decomposition (eq. 4). Radicals are removed from the system by chain-termination reaction(s) (eq. 5). Under steady-state conditions, the production of new radicals is in balance with the rate of radical removal by termination reactions and equation 8 appHes for the scheme of equations 1—5 where r. = rate of new radical introduction (eq. 4). 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

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. 

The kinetics and mechanism of living radical polymerization have been reviewed by Fischer,21 Fukuda et at.,22 and Goto and Fuktida.23 In conventional radical polymerization, new chains are continually formed through initiation w hile existing chains are destroyed by radical-radical termination. The steady state concentration of propagating radicals is 10"7 M and an individual chain will have a lifetime of only 1-10 s before termination within a total reaction lime that is... 

The new pathway, too, is a chain reaction Note that the first term of Eq. (8-31) does not give a meaningful transition state composition. Since the scheme in Eqs. (8-20M8-23) seems valid for the Cu2+-free reaction, we can seek to modify it to accommodate the new result. This approach is surely more logical than inventing an entirely new sequence. To arrive at the needed modification, we simply replace Eq. (8-23) by a new termination step, Eq. (8-30). With that, and the steady-state approximation, the rate law is... 

Also, the rates of the propagation steps are equal to one another (see Problem 8-4). This observation is no surprise The rates of all the steps are the same in any ordinary reaction sequence to which the steady-state approximation applies, since each is governed by the same rate-controlling step. The form of the rate law for chain reactions is greatly influenced by the initiation and termination reactions. But the chemistry that converts reactant to product, and is presumably the matter of greatest importance, resides in the propagation reactions. Sensitivity to trace impurities, deliberate or adventitious, is one signal that a chain mechanism is operative. 

Derive the expression for -d[ ] W)]/dt at constant [H+] by making steady-state and long-chain approximations. The fourth reaction is very fast that is, every H202 produced converts U(IV) to U(VI) directly without forming U(V). Termination proceeds by... 

Derive the expression for v (= —d[0 j/df), making steady-state and long-chain approximations. How is Ea for the overall reaction related to the activation energies of the elementary reaction steps ... 

The inlet monomer concentration was varied sinusoidally to determine the effect of these changes on Dp, the time-averaged polydispersity, when compared with the steady-state case. For the unsteady state CSTR, the pseudo steady-state assumption for active centres was used to simplify computations. In both of the mechanisms considered, D increases with respect to the steady-state value (for constant conversion and number average chain length y ) as the frequency of the oscillation in the monomer feed concentration is decreased. The maximum deviation in D thus occurs as lo 0. However, it was predicted that the value of D could only be increased by 10-325S with respect to the steady state depending on reaction mechanism and the amplitude of the oscillating feed. Laurence and Vasudevan (12) considered a reaction with combination termination and no chain transfer. 

Once the copolymerisation reaction has become properly established, the radical chains A- and B- each achieve a steady-state concentration. The absolute rate of conversion of A- to B- and of B- to A- becomes the same ... 

In the above reactions, I signifies an initiator molecule, Rq the chain-initiating species, M a monomer molecule, R, a radical of chain length n, Pn a polymer molecule of chain length n, and f the initiator efficiency. The usual approximations for long chains and radical quasi-steady state (rate of initiation equals rate of termination) (2-6) are applied. Also applied is the assumption that the initiation step is much faster than initiator decomposition. ,1) With these assumptions, the monomer mass balance for a batch reactor is given by the following differential equation. 

Catalytic reactions (as well as the related class of chain reactions described below) are coupled reactions, and their kinetic description requires methods to solve the associated set of differential equations that describe the constituent steps. This stimulated Chapman in 1913 to formulate the steady state approximation which, as we will see, plays a central role in solving kinetic schemes. 

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. 

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

Steady-state treatment for the transients (H02- and UO2 ) leads to the observed rate law. The chain reaction is indicated by (/) strong catalysis by Cu " ions and ill) partial and complete inhibition respectively by added Cl" and Ag" ions. The inhibition by Ag" is not indefinite, however, and takes the form of an induction period, during which time metallic silver is deposited. 

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 ... 

Radicals of type Mi- are formed by primary initiation and by reaction (2,1) above. They are destroyed by the reaction (1,2) and in termination reactions. At the steady state, the rates of generation and of disappearance of these radicals are practically equal. If the chains are long, initiation and termination are of exceedingly rare occurrence compared with the reactions (1), and it suffices therefore to consider the latter only for the present where we are concerned merely with the relative concentrations of the two types of chain radicals. The steady-state condition reduces in this approximation to... 

In this reaction scheme, the steady-state concentration of peroxyl radicals will be a direa function of the concentration of the transition metal and lipid peroxide content of the LDL particle, and will increase as the reaction proceeds. Scheme 2.2 is a diagrammatic representation of the redox interactions between copper, lipid hydroperoxides and lipid in the presence of a chain-breaking antioxidant. For the sake of clarity, the reaction involving the regeneration of the oxidized form of copper (Reaction 2.9) has been omitted. The first step is the independent decomposition of the Upid hydroperoxide to form the peroxyl radical. This may be terminated by reaction with an antioxidant, AH, but the lipid peroxide formed will contribute to the peroxide pool. It is evident from this scheme that the efficacy of a chain-breaking antioxidant in this scheme will be highly dependent on the initial size of the peroxide pool. In the section describing the copper-dependent oxidation of LDL (Section 2.6.1), the implications of this idea will be pursued further. 

In 1919 Christiansen (25), Herzfeld (26), and Polanyi (27) all suggested the same mechanism for this reaction. The key factor leading to their success was recognition that hydrogen atoms and bromine atoms could alternately serve as chain carriers and thus propagate the reaction. By using a steady-state approximation for the concentrations of these species, these individuals were able to derive rate expressions that were consistent with that observed experimentally. 

ILLUSTRATION 4.3 USE OF THE BODENSTEIN STEADY-STATE APPROXIMATION TO DERIVE A RATE EXPRESSION FROM A CHAIN REACTION MECHANISM... 

Because there are two positive terms in the denominator of equation 4.2.85 (either of which may be associated with the dominant termination process), this equation leads to two explosion limits. At very low pressures the mean free path of the molecules in the reactor is quite long, and the radical termination processes occur primarily on the surfaces of the reaction vessel. Under these conditions gas phase collisions leading to chain breaking are relatively infrequent events, and fst fgt. Steady-state reaction conditions can prevail under these conditions if fst > fb(a — 1). 

Kinetic vs. material chain. Kinetically, a chain reaction exists throughout the "life" of the radical, that is, from the initiation of a radical up to its termination by recombination or by disproportionation. The lifetime of a radical determines the so-called kinetic chain length Lp defined as the number of monomers consumed per initiating radical. Lp, by definition, can be calculated from the ratio between the propagation rate Rp to the initiation rate R, or, using steady-state hypothesis (Equation (1)), from the ratio between propagation rate to the termination rate Rt (Equation (3)). 

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