500
11 The Rates of Chemical Reactions
where a and b represent stoichiometric coefficients and where A and B represent
chemical formulas. The rate law is
r −
1
a
d[A]
dt
k f [A][B]
(11.3-2)
We first consider the case that the reactants’ concentrations are initially in the stoichiometric 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 of t greater than 0.
We define a single dependent variable x(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 of x.
We can now write Eq. (11.3-2) in the form
dx
dt
k f ([A] 0 − ax)([B] 0 − bx)
(11.3-6a)
k f ab
[A] 0
a
− x
[B] 0
b
− x
k f ab
[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 by dt:
dx
[A] 0
a
− x
2 k f abdt
(11.3-7)
We integrate both sides of Eq. (11.3-7) from t 0 to t t :
1
[A] 0
a
− x(t )
−
1
[A] 0
a
k f abt