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Chain Reactions

Chain mechanisms involve reactive intermediates that are called chain carriers. A chain mechanism usually contains an initiation step, in which chain carriers are formed; one or more chain propagation steps in which products are formed and in which chain carriers are produced as well as being consumed; and a chain termination step in which chain carriers are consumed without being replaced. Since chain carriers are produced as well as being consumed, the reaction can continue without further initiation steps.
    The following gas-phase reaction has been identified as a chain reaction:
Equation 1
The empirical rate law for the forward reaction in the presence of some HBr is
Equation 2
where ka and kb are temperature-dependent parameters. The accepted mechanism for the forward reaction is
Equation 3
Step 1 is the initiation step. The forward reactions of steps 2 and 3 are chain propagation steps, producing the two chain carriers, Br and H. The reverse reaction of step 1 is the termination step. The reverse reactions of steps 2 and 3 regenerate chain carriers but consume the product. They are called inhibition processes.
    Once bromine atoms are formed in the initiation step, the reaction can proceed without further initiation. The chain length γ is defined as the average number of times the cycle of the two propagation steps is repeated for each initiation step. It is possible to have a chain length as large as 106. In this reaction, the initiation step gives two Br atoms, and each of these gives two molecules of HBr per cycle, so that the number of molecules of product for each initiation step is equal to 4 times the chain length.
    To obtain the rate law we apply the steady-state approximation. For a three-step mechanism we must write three differential equations. We choose the time derivatives of [H2] and the concentrations of the two chain carriers [H] and [Br]. We choose [H2] instead of [Br2] or [HBr] because H2 occurs in only one step of the mechanism and will give a simpler differential equation. The simultaneous differential equations are
Equation 4
We have applied the steady-state approximation and set the time derivatives of the concentration of the chain carriers H and Br equal to zero. To solve the algebraic versions of Eqs. (4b) and (4c), we add Eqs. (4b) and (4c) to give
Equation 5a
which is the same as
Equation 5b
Equation (5b) is the same equation that would result from assuming that step 1 is at equilibrium. The relation of Eq. (5b) is substituted into Eq. (4b) or Eq. (4c) to give (after several steps of algebra)
Equation 6
We now simplify Eq. (4a) by noticing that the first two terms in Eq. (4c) are the same as the two terms on the right-hand side of Eq. (4a), so that
When Eq. (6) is substituted into this equation, we have
Equation 7
which reproduces the empirical rate law with the following expressions for the empirical parameters:
Equation 8

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