Physical Chemistry Third Edition

12.4 Reaction Mechanisms and Rate Laws 547

Solution

a.The equilibrium approximation gives

K 1 

k 1

k′ 1



[A][A∗]

[A]^2



[A∗]

[A]

(12.4-19)

When Eq. (12.4-19) is substituted into Eq. (12.4-15), we obtain the first-order rate law:

rate

d[B]

dt

k 2 K 1 [A]

k 2 k 1

k′ 1

[A] (12.4-20)

which is the same as Eq. (12.4-14) if thek 2 term in the denominator of the expression in

Eq. (12.4-14) is deleted.

b.The rate law is second order in the case thatk 2 k 1 ′[A]:

rate−

d[A]

dt

k 1 [A]^2 (12.4-21)

Most unimolecular processes are apparently observed in the first-order (high-pressure

or high-concentration) region, and we will continue to assume first-order kinetics

for unimolecular steps in multistep mechanisms. Most proposed mechanisms do not

include unimolecular steps, and there are not very many reactions known to have a

one-step unimolecular mechanism. The first such reaction discovered was the isomer-

ization of cyclopropane to propene. Others are the dissociation of molecular bromine

and the decomposition of sulfuryl chloride.^13

Mechanisms with More than Two Steps

If a proposed mechanism has three steps and if the third step is rate-limiting, the

first two steps are assumed to be at equilibrium. Algebraic equations are written for

these equilibria. If the steady-state approximation is applied, two differential equations

for the rates of change of two reactive intermediates’ concentrations are replaced by

algebraic equations. In either case, there are two algebraic equations and a single

differential equation. Mechanisms with four or more steps are treated analogously.

EXAMPLE12.12

Find the rate law for the liquid-phase mechanisma

(1) H++HNO 2 +NO− 3 
N 2 O 4 +H 2 O (fast)

(2) N 2 O 4 
2NO 2 (fast)

(3) NO 2 +Fe(CN)^36 −−→further intermediates (slow)

(12.4-22)

assuming that step (3) is rate-limiting.

aM. V. Twigg,Mechanisms of Inorganic and Organometallic Reactions, Plenum Press, New York,

1983, p. 39.

(^13) K. J. Laidler,op. cit., pp. 150ff (note 3).