Physical Chemistry Third Edition

12.4 Reaction Mechanisms and Rate Laws 553

Exercise 12.16

a.Show that the proposed mechanism in the previous example leads to the correct rate law.

b.If H 2 O and NO 3 are the products of step 2, and if no further HNO 3 enters in later steps,

propose steps 3 and 4 to complete the mechanism and give the correct stoichiometry.

Fractional orders can occur if one of the substances in the rate-limiting step is

produced in a previous step with a stoichiometric coefficient greater than unity, as in

Eq. (12.4-28).

The Temperature Dependence of Rates of Nonelementary

Reactions

The rate law of a reaction corresponding to an assumed mechanism contains the rate

constants of some or all of the steps of the mechanism. If the temperature dependence

of these rate constants is known, the temperature dependence of the overall rate can

be deduced. The rate law for the ozone decomposition of Eq. (12.4-4) is given by

Eq. (12.4-7):

rate

1

3

(

d[O 2 ]

dt

)

kapp

[O 3 ]^2

[O 2 ]

k 2

k 1 [O 3 ]^2

k′ 1 [O 2 ]

(12.4-45)

If each of the elementary rate constants is governed by the Arrhenius formula,

Eq. (12.3-2), then the temperature dependence of the overall rate constant,kapp,is

given by

kapp

A 2 A 1

A 1 ′

e−Ea2/RTe−Ea1/RT

e−Ea1′/RT

Aappexp

(

−

Ea2+Ea1−Ea1′

RT

)

Aappexp

(

−

Ea,app

RT

)

(12.4-46)

where the apparent activation energyEa,appis

Ea,appEa 2 +Ea 1 −Ea 1 ′

Since the ratiok 1 /k′ 1 is equal to an equilibrium constant for the first step, we can rewrite

Eq. (12.4-46) as

kappA 2 exp

(

−

Ea2+∆G◦ 1

RT

)

(12.4-47)

It is possible that the apparent activation energy of a reaction with a multistep mech-

anism is negative. This occurs in the recombination of iodine atoms to form I 2 ,as

described in Problem 12.41.

Exercise 12.17

The temperature dependence of a rate law corresponding to a steady-state approximation is more

complicated than that of the previous example. Write the temperature dependence of the apparent

first-order rate constant for the decomposition of N 2 O 5 given in Eq. (12.4-34).