Physical Chemistry Third Edition

500 11 The Rates of Chemical Reactions

whereaandbrepresent stoichiometric coefficients and where A and B represent

chemical formulas. The rate law is

r−

1

a

d[A]

dt

kf[A][B] (11.3-2)

We first consider the case that the reactants’ concentrations are initially in the stoichio-

metric ratio:

[A] 0

[B] 0



a

b

(11.3-3)

The concentrations will remain in this ratio during the reaction.

Exercise 11.10

Show that if Eq. (11.3-3) holds then

[A]t

[B]t



a

b

(11.3-4)

for all values oftgreater than 0.

We define a single dependent variablex(t):

x(t)

[A] 0 −[A]t

a



[B] 0 −[B]t

b

(11.3-5)

Exercise 11.11

Show that both versions of Eq. (11.3-5) correspond to the same value ofx.

We can now write Eq. (11.3-2) in the form

dx

dt

kf([A] 0 −ax)([B] 0 −bx) (11.3-6a)

kfab

(

[A] 0

a

−x

)(

[B] 0

b

−x

)

kfab

(

[A] 0

a

−x

) 2

(11.3-6b)

where we have used the fact that[B] 0 /b[A] 0 /a.

The variables can be separated in Eq. (11.3-6b) by dividing by[[A] 0 /a−x]^2 and

multiplying bydt:

dx

(

[A] 0

a

−x

) 2 kfabdt (11.3-7)

We integrate both sides of Eq. (11.3-7) fromt0tott′:

1

[A] 0

a

−x(t′)

−

1

[A] 0

a

kfabt′