Physical Chemistry Third Edition

12.4 Reaction Mechanisms and Rate Laws 549

The proposed mechanism is

(1) N 2 O 5 
NO 2 +NO 3

(2) NO 2 +NO 3 −→NO+O 2 +NO 2

(3) NO+NO 3 −→2NO 2

(12.4-30)

Find the rate law using the steady-state approximation.

Solution

We write three differential equations and impose the steady-state approximation on the reactive interme-

diates NO 3 and NO:

−

d[N 2 O 5 ]

dt

k 1 [N 2 O 5 ]−k′ 1 [NO 2 ][NO 3 ] (12.4-31)

d[NO 3 ]

dt k^1 [N^2 O^5 ]−k

′

1 [NO^2 ][NO^3 ]−k^2 [NO^2 ][NO^3 ]−k^3 [NO][NO^3 ]≈^0 (12.4-32a)

d[NO]

dt

k 2 [NO 2 ][NO 3 ]−k 3 [NO][NO 3 ]≈ 0 (12.4-32b)

The second equalities in Eqs. (12.4-32a) and (12.4-32b) give two simultaneous algebraic equations,

which can be solved for [NO 3 ] and [NO]. The results are

[NO]k^2 [NO^2 ]

k 3

(12.4-33a)

and

[NO 3 ]( k^1 [N^2 O^5 ]

k 1 ′+ 2 k 2

)

[NO 2 ]

(12.4-33b)

When these two equations are substituted into Eq. (12.4-31), we obtain

rate

2 k 1 k 2

k 1 ′+ 2 k 2 [N^2 O^5 ]kapp[N^2 O^5 ] (12.4-34)

so that the reaction is first order. Although the steady-state approximation usually leads to a rate law with

a two-term denominator, this rate law corresponds to a definite order.

Exercise 12.13

a.Verify Eq. (12.4-34).

b.Assume that the second step of the mechanism of Example 12.13 is rate-limiting. Find the

rate law for the reaction.

c.What assumptions will cause the steady-state result to become the same as the result of

part b?

Deducing a Mechanism from a Rate Law

It is not a routine matter to decide what mechanism to propose once a rate law has been

determined. However, you can sometimes use the following procedure:^14

(^14) J. O. Edwards, E. F. Greene, and J. Ross,J. Chem. Educ., 45 , 381 (1968).