Physical Chemistry Third Edition

586 13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics

corresponds to the mechanism

(1) A+F
C

(2) C→2F

(13.3-2)

where C is a reactive intermediate. Assuming that the second step is rate-limiting, the

rate law is

d[F]

dt

 2 k 2 [C] 2 k 2 K 1 [A][F]kapp[A][F] (13.3-3)

where the initial concentration of F must be nonzero for the reaction to proceed. This

differential equation can be integrated. We let

[A][A] 0 −x

and

[F][F] 0 +x

where the initial concentrations are labeled with a subscript 0. The rate law now

becomes

dx

dt

kapp([A] 0 −x)([F] 0 +x) (13.3-4)

or

dx

([A] 0 −x)([F] 0 +x)

kappdt (13.3-5)

We apply the method ofpartial fractions, setting

1

([A] 0 −x)([F] 0 +x)



G

[A] 0 −x

+

H

[F] 0 +x

where a theorem of algebra guarantees thatGandHare constants. We obtain two

simultaneous equations forGandHby settingx0 to obtain one equation,

1

[A] 0 [F] 0



G

[A] 0

+

H

[F] 0

(13.3-6a)

and by settingx[F] 0 to obtain the other equation,

1

([A] 0 −[F] 0 )2[F] 0



G

[A] 0 −[F] 0

+

H

2[F] 0

(13.3-6b)

The solution to these simultaneous equations is

GH

1

[A] 0 +[F] 0

(13.3-7)

so that

dx

([A] 0 +[F] 0 )([A] 0 −x)

+

dx

([A] 0 +[F] 0 )([F] 0 +x)

kappdt (13.3-8)