Physical Chemistry Third Edition

12.4 Reaction Mechanisms and Rate Laws 541

be incorrect. If they match, the mechanism might be correct, although other mechanisms

might give the same rate law. There are sometimes other types of experiments that can be

done, including direct detection of reactive intermediates, molecular beam experiments,

and radioactive tracer experiments. However, we must regard even a well-accepted

mechanism as tentative.

Since a proposed mechanism consists of elementary steps, we can write a rate

law for each step from the fact that for an elementary process the order equals the

molecularity. Every multistep mechanism leads to a set of simultaneous differential

equations analogous to those for the simple mechanism of Eq. (11.5-1). There is one

independent differential equation for each elementary step of the mechanism.

Consider the reaction of Eq. (12.1-1):

2NO 2 +O 3 −→N 2 O 5 +O 2

If the reaction were elementary, the rate law for the forward reaction would be:

rate

d[N 2 O 5 ]

dt

−

d[O 3 ]

dt

k[NO 2 ]^2 [O 3 ]

(rate law for the

one-step mechanism)

(12.4-1)

This rate law does not agree with the experimentally determined rate law, which is

second order overall and first order with respect to each reactant.

The accepted mechanism of the forward reaction is

(1) NO 2 +O 3 −→NO 3 +O 2

(2) NO 3 +NO 2 −→N 2 O 5

(12.4-2)

where the reverse reactions of both steps are assumed to be negligible. For this mech-

anism we can write two simultaneous differential equations. One possible choice is

d[O 3 ]

dt

−k 1 [NO 2 ][O 3 ] (12.4-3a)

d[NO 5 ]

dt

k 2 [NO 3 ][NO 2 ] (12.4-3b)

We have some choice as to which concentration time derivatives are used in writing the

equations, but for a two-step mechanism there are only two independent differential

equations.

Exercise 12.8

Write the differential equation ford[NO 3 ]/dt(it will have two terms on the right-hand side) and

show that the right-hand side of this equation is equal to a linear combination (weighted sum

or difference) of the right-hand sides of Eqs. (12.4-3a) and (12.4-3b), and that this equation is

therefore not independent of the other two equations.

We do not attempt an exact solution of the set of equations in Eq. (12.4-3), but

introduce two approximation schemes.

The Rate-Limiting Step Approximation

Let us assume that the second step of the mechanism in Eq. (12.4-3) is much more

rapid than the first step. We mean by this assumption that the NO 3 produced by the