Physical Chemistry Third Edition

542 12 Chemical Reaction Mechanisms I: Rate Laws and Mechanisms

first step is immediately consumed by the second step, which has a large rate constant.

The rate of the second step is controlled by the first step since the second step cannot

consume the NO 3 more rapidly than it is formed by the first step. We call the slow first

step therate-limiting steporrate-determining step. The second step does not affect

the rate law of the forward reaction, which is just the differential equation of the slow

first step:

−

d[O 3 ]

dt

k 1 [NO 2 ][O 3 ]

This rate law agrees with experiment. The mechanism of Eq. (12.1-2) is possibly correct,

but other mechanisms might be found that produce the same rate law.

If a step other than the first step is rate-limiting, the steps prior to the rate-determining

step will play a role in determining the rate law, but any steps after the rate-limiting step

will play no role in determining the forward rate. We illustrate this with the gas-phase

reaction

2O 3 −→3O 2

This reaction is thought to proceed by the mechanism

(1) O 3 +M
O 2 +O+M (fast)

(2) O+O 3 −→2O 2 (slow)

(12.4-4)

where M stands for any molecule, such as an O 2 molecule or a molecule of another

substance that is present. An inelastic collision with the molecule M is needed to provide

the energy for breaking the bond in the O 3 molecule. We assume that the second step is

rate-determining. We do not include the reverse reaction for the rate-determining step

and will obtain only the forward rate law. The rate differential equation for the second

step is

d[O 2 ]

dt

∣

∣

∣

∣

step 2

 2 k 2 [O 3 ][O] (12.4-5)

where the subscript means that only the contribution of step 2 to the production of O 2 is

included. The factor 2 is included because two molecules of O 2 occur in the chemical

equation for the second step.

We now replace the differential equation for the first step by an algebraic equation.

Both the forward and the reverse reaction of the first step are assumed to be rapid

compared with the second step. We invoke theequilibrium approximationorquasi-

equilibrium approximation, which is that all steps prior to the rate-limiting step are

assumed to be at equilibrium. With this approximation the forward and reverse rates

of the first step are set equal to each other:

k 1 [O 3 ][M]k′ 1 [O 2 ][O][M]

where the rate constant for a reverse reaction is labeled with a prime (′). This is

equivalent to

K 1 

k 1

k 1 ′



[O 2 ][O][M]

[O 3 ][M]



[O 2 ][O]

[O 3 ]

(12.4-6)

whereK 1 is the equilibrium constant for the step (ignoring activity coefficients). This

corresponds to the assumption that the slow second step removes a product of step 1 so

slowly that step 1 can nearly relax to equilibrium as each product molecule is removed.