Page 1009 - Advanced Organic Chemistry Part A - Structure and Mechanisms, 5th ed (2007) - Carey _ Sundberg

k i . 993
A A 2A initiation
. k P1 . SECTION 11.2
A + B C A B + C
Characteristics of
Reactions Involving
repeated C . + A A k P2 C A + A . propagation Radical Intermediates
many times . k .
A + B C P1 A B + C
C . + A A k P2 C A + A .
k T1
2A . A A
k
. T2
2C C C termination
A . + C . k T3 A C
overall A A + B C A B + A C
reaction
.
The step in which the radical intermediate, in this case A , is generated is called the
initiation step. In the next four equations of the example, a sequence of two reactions
is repeated; this is the propagation phase. Chain reactions are characterized by a
chain length, which is the number of propagation steps that take place per initiation
step. Finally, there are termination steps, which include all reactions that destroy one
of the reactive intermediates necessary for the propagation of the chain. Clearly, the
greater the frequency of termination steps, the smaller the chain length will be. The
stoichiometry of a free radical chain reaction is independent of the initiating and
termination steps because the reactants are consumed and products are formed almost
entirely in the propagation steps.

A 2 + B C A B + A C

The 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 3.2.3) to the radical
intermediates. Such intermediates are highly reactive, and their concentrations are low
and nearly constant through the course of the reaction. A 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 reaction rate is given by

d A −B d A −C −d A −d B−C
2
Rate = = = =
dt dt dt dt
Setting the rate of initiation equal to the rate of termination and assuming that k is
t2
the dominant termination process gives
k A = 2k C· 2
t2
i
2
1/2 1/2

k i
C· = A
2
2k t2
Termination reactions involving coupling or disproportionation of two radicals
ordinarily occurs at diffusion-controlled rates. Since the concentration of the reactive