Physical Chemistry Third Edition

12.4 Reaction Mechanisms and Rate Laws 543

Equation (12.4-6) is now solved for [O]:

[O]K 1

[O 3 ]

[O 2 ]

This formula is substituted into Eq. (12.4-5) to obtain

d[O 2 ]

dt

∣

∣

∣

∣

step 2

 2 k 2 [O 3 ][O] 2 k 2

k 1

k′ 1

[O 3 ]^2

[O 2 ]

 2 k 2 K 1

[O 3 ]^2

[O 2 ]

We now write the rate law for the reaction by noting that step 1 produces one molecule

of O 2 for every two molecules produced by step 2:

rate

1

3

d[O 2 ]

dt



1

3

(

d[O 2 ]

dt

∣

∣

∣

∣

step 2

+

d[O 2 ]

dt

∣

∣

∣

∣

step 1

)

(

1

3

)(

3

2

)

d[O 2 ]

dt

∣

∣

∣

∣

step 2



(

1

3

)(

3

2

)

2 k 2

k 1

k′ 1

[O 3 ]^2

[O 2 ]

k 2 K 1

[O 3 ]^2

[O 2 ]

(12.4-7)

The forward rate of the first step exceeds the reverse rate by an amount equal to half

of the rate of step 2. Since both the forward and the reverse rate of the first step must

be large compared to the rate of the second step, the forward and reverse rates are still

nearly equal to each other. The concentration of O 2 (a product) occurs in this forward

rate law, which holds only if some O 2 is already present. In general a product will occur

in a forward rate law only if it is produced in a step prior to the rate-limiting step.

Exercise 12.9

For the gas-phase reaction

2NO+2H 2 −→N 2 +2H 2 O

the mechanism has been proposed

(1) 2NO+H 2 
N 2 +H 2 O 2

(2) H 2 O 2 +H 2 −→2H 2 O

a.Find the rate law if the second step is rate-limiting.

b.Find the rate law if the reverse reaction of step 1 is omitted and this step is rate-limiting.

We can summarize the rate-limiting step approximation: If the first step is rate-

limiting, the rate law is the rate law of the first step. If a step after the first step is

rate-limiting, the rate law of the rate-limiting step is written. The concentrations of

any reactive intermediates in the rate law of the rate-limiting step are replaced by

expressions obtained by assuming that the steps prior to the rate-limiting step are at

equilibrium. The result is the final approximate rate law.

The Steady-State Approximation

This approximation, which is also called thequasi-steady-state approximation, con-

sists of the assumption that the rates of change of the concentrations of allreactive