Physical Chemistry Third Edition

544 12 Chemical Reaction Mechanisms I: Rate Laws and Mechanisms

intermediatesare negligibly small. We say that the reactive intermediates are approxi-

mately in a “steady state.” This is generally a good approximation if the concentrations

of the intermediates are small since small variables have small time derivatives if

they do not oscillate rapidly. Numerical simulations of sets of simultaneous rate dif-

ferential equations have shown that the steady-state approximation often gives quite

accurate results.^9 Improvements on the simple steady-state approximation have been

developed.^10

EXAMPLE12.10

Apply the steady-state approximation to the mechanism of Eq. (12.4-4).

Solution

There is one independent equation for each step in any mechanism. One equation must be

written for the rate of change of the concentration of a reactant or product. The other equations

must be for the rate of change of concentrations of reactive intermediates. We write the two

equations

d[O 2 ]

dt

k 1 [O 3 ][M]−k 1 ′[O 2 ][O][M]+ 2 k 2 [O][O 3 ] (12.4-8a)

d[O]

dt

k 1 [O 3 ][M]−k 1 ′[O 2 ][O][M]−k 2 [O][O 3 ] (12.4-8b)

We invoke the steady-state approximation by settingd[O]/dtapproximately equal to zero:

k 1 [O 3 ][M]−k 1 ′[O 2 ][O][M]−k 2 [O][O 3 ]≈ 0 (12.4-8c)

We now have one differential equation and one algebraic equation. We subtract Eq. (12.4-8c)

from Eq. (12.4-8a) to obtain a simpler differential equation:

d[O 2 ]

dt

 3 k 2 [O][O 3 ] (12.4-8d)

We solve Eq. (12.4-8c) for [O]:

[O]

k 1 [O 3 ][M]

k′ 1 [O 2 ][M]+k 2 [O 3 ]

(12.4-9)

We substitute this equation into Eq. (12.4-8d) to obtain the rate law

1

3

d[O 2 ]

dt

k 2 [O 3 ]

k 1 [O 3 ][M]

k′ 1 [O 2 ][M]+k 2 [O 3 ]



k 1 k 2 [O 3 ]^2 [M]

k 1 ′[O 2 ][M]+k 2 [O 3 ]

(12.4-10)

Comparison of Eq. (12.4-10) with Eq. (12.4-7) shows that if

k 1 ′[O 2 ][M]k 2 [O 3 ]

then Eq. (12.4-10) becomes the same as Eq. (12.4-7). This corresponds to the assumption on

which the rate-limiting step approximation was based, that the forward and reverse rates of

the first step are much larger than the rate of the second step.

(^9) L. A. Farrow and D. Edelson,Int. J. Chem. Kinet., 6 , 787 (1974); V. Viossat and R. I. Ben-Aim,J.
Chem. Educ., 70 , 732 (1993); G. I. Gellene,J. Chem. Educ., 72 , 196 (1995); R. A. B. Bond, B. S. Martincigh,
J. R. Mika, and R. H. Simoyi,J. Chem. Educ., 75 , 1158 (1998); V. Viossat and R. I. Ben-Aim,J. Chem.
Educ., 75 , 1165 (1998).
(^10) See for example L. O. Jay, A. Sandu, F. A. Potra, and G. R. Carmichael,SIAM Journal of Scientific
Computing, 18 , 182 (1997).