Long-chain approximation

Application of a steady state approximation (that / t = / j, eq. 2) and a long chain approximation (negligible monomer consumption in the initiation or reinitiation steps) provides a number of useful relationships. ... 

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

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... 

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. 

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 rate law for each, with the steady-state and long-chain approximations, is... 

Linear free energy relations (see LFER) Lineweaver-Burk treatment, 91 Long-chain approximation, 183... 

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. 

Most of the linoleic acid is used in the propagation step, Equation 7. (This is the so-called long-chain approximation.) Thus, the rate of autoxidation, RQxi, is given by Equation 10 (fii,fi >). 

If the polymers have long chains, then the consumption of the monomer in polymerization, other than propagation, is negligible (long chain approximation) ... 

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... 

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