The Long-Chain Approximation

To apply the long-chain approximation to the preceding ethylene hydrogenation mechanism, we write a rate expression that includes only the rate of ethylene consumption in the propagation steps, i.e., in the steps that carry the chain. 

To eliminate the concentrations of the active centers, we first apply the steady-state approximation to the total concentration of all active centers. Since there is no net creation or destruction of active centers in either propagation step, the resulting equation is 

The factors of 2 in front of both terms in this equation reflect the fact that two active centers are created or destroyed each time one of these reactions proceeds. Stated differently, in formulating this equation, it was assumed that the rate constant ki was based on C2H4 or H2 and that the rate constant 4 was based on C2H5 orH. This expression for tac is valid whether or not the long-chain approximation is applied. Except in certain pathological cases, the rate of creation of active centers is balanced by the rate of destruction of active centers. In fact, this balance determines the total concentration of active centers, in the same way that the SSA on a specific active center determines the concentration of that center. 

we recognize that the rates of the two propagation steps must be the same if the amount of ethylene consumed is to be the same as the amount of hydrogen consumed and ethane produced. 

This equation is a direct csonsequence of the long-chain approximation. 

---

With the long-chain approximation (see Problem 8-2), an expression for the rate is obtained that agrees with the experimental findings,... 

Chain termination. The chlorination of alkanes by rm-butyl hypochlorite is believed to follow a chain mechanism, but there is a dispute about the termination step.10 Derive the steady-state rate equation for each, making the long-chain approximation. 

Frequently function R can be written as a single term having the simple form of equation 1. For Instance, with the aid of the long chain approximation (LCA) and the quasi-steady state approximation ((JSSA), the rate of monomer conversion, I.e., the rate of polymerization, for many chain-addition polymerizations can be written as... 

Note well here it would not be correct to use the long chains approximation here, even though the reaction has long chains. This is because there is the inhibition step, which removes and creates each of the chain carriers, and so the rates of the two propagation steps are not equal, as is demonstrated by Equation (6.90). It will be shown later that if initial rates were used, where there is very little inhibition, then the long chains approximation becomes valid (Section 6.10.3). 

This can be simplified. The first term can be dropped, since rate of initiation -C rate of propagation—the long chains approximation—leaving the rate as... 

This analysis has used the long chains approximation ... 

A few chain reactions occur where the number of cycles in propagation, the chain length, is small and the long chains approximation will no longer be applicable. Such reactions will not be considered here. 

The long chains approximation must never be used when one or other or both of the chain carriers are removed or formed in steps other than propagation, initiation and termination. For example, when inhibition is present and a product removes or forms one or other of the chain carriers, the long chains approximation is invalid see the H2/Br2 reaction, Section 6.10. 

The term in k relates to a rate of initiation. The long chains approximation assumes that rate of initiation [Pg.232]

The first two terms reduce to 2k [I], which refers to a rate of initiation which by the long chains approximation can now drop out, leaving... 

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. 

This is a consequence of the long chains approximation, which states that initiation occurs only very infrequently compared to propagation. 

C2H ] appears in step 3 and this term will drop out from the overall rate because it is not a propagation step, and although it produces C3H this step is not essential for the continuance of the chain. Rate of propagation rate of step 3. This will leave a two-term equation, which can be approximated by the long chains approximation, since rate of initiation [Pg.404]

However, equation (VI) differs from equation (II). Here the rate term for the inhibition step is included, so that the rates of the two propagation steps are no longer equal. The long chains approximation in this form cannot be used for the inhibition mechanism, though it is valid for the reaction without inhibition. 

The mathematical techniques most commonly used in chemical kinetics since their formulation by Bodenstein in the 1920s have been the quasi-stationary state approximation (QSSA) and related approximations, such as the long chain approximation. Formally, the QSSA consists of considering that the algebraic rate of formation of any very reactive intermediate, such as a free radical, is equal to zero. For example, the characteristic equations of an isothermal, constant volume, batch reactor are written (see Sect. 3.2) as... 

Long-chain approximation. In most chain reactions, a short sequence of steps, once initiated, repeats itself many times until it is terminated. The initiation and termination reactions then contribute very much less to product formation than do the self-repeating steps. The long-chain approximation neglects these minor contributions. It will be taken up in the context of chain reactions (see Section 9.3) and also used in chain-growth polymerization (Sections 10.3 and 10.4). 

Approximations based on the concept of relative abundance of catalyst-containing species in trace-level catalysis and of propagation centers in ionic polymerization will be discussed in Chapters 8 and 10, respectively the long-chain approximation in chain reactions will be an important topic in Chapter 9. 

Equation 9.9, derived with the long-chain approximation, shows the rates of the two propagation steps under quasi-stationary conditions to be equal Each of the two steps consumes as many chain carriers as the other produces. Since the total number of chain carriers is not altered by the propagation steps, the overall rates of chain carrier production and consumption (of X plus Y) must be equal in magnitude ... 

We now use the long-chain approximation (LCA). The LCA is that the rate of propagation is much greater than the rate of Initiation ... 

The long-chain approximation holds when is large. 

If the long-chain approximation is valid, this becomes... 

Substituting this result into the rate equation for monomer consumption and making the long-chain approximation (nr, is small) yields... 

The pseudo steady state hypothesis and the long chain approximation are encountered constantly in publications concerning chain reactions. Find in the recent literature a paper using this analysis. Describe the problem addressed by the authors, the reaction and chain mechanism proposed, and the analysis employed. Then see whether you agree with the result and the interpretation proposed in the paper. 

---