Physical Chemistry , 1st ed.

order with respect to its reactant, and that the initial amount of reactant A is

given by [A] 0 while [B] 0 0.

The rate of change of concentration of A is given by

d[

d

A

t

]k

f[A] kr[B] (20.30)

The first term on the right is negative because it relates the disappearanceof A,

and the second term is positive because it contributes to an increasein A. At

any time t, we can modify the concentrations in equation 20.30 to show that

the overall rate depends on the concentrations of A and B at any time:



d[

d

A

t

]t

kf[A]tkr[B]t (20.31)

Now, [A]tand [B]timply that the rate is dependent on the instantaneous con-

centrations of A and B at any time. Notice that the rate is also written in terms

of the instantaneous concentration [A]t. Because of the 11 stoichiometry in

the balanced chemical reaction and the law of conservation of mass, there is a

relationship between [A] and [B] at any time:

[B]t[A]t[A] 0 or [B]t[A] 0 [A]t

Substituting for [B]tand rearranging:



d[

d

A

t

]

kf[A]tkr([A] 0 [A]t) kr[A] 0 (kr kf)[A]t (20.32)

Although this equation may look complicated, it is simply a first-order differ-

ential equation in one variable, [A]t. It has a solution that involves exponential

functions, and the equations that satisfy this differential equation (that is, the

“solutions” to the differential equation) are

[A]t

(kf

[A



] 0

kr)

(krkfe(kfkr)t) (20.33)

It is actually fairly easy to show that at the beginning of the process, when the

reverse reaction’s effect is negligible, the above equation reduces to

[A]t[A] 0 ekft

This is equivalent to equation 20.15.

Another integrated form of the rate law in equation 20.32 has the form

ln 

[

[

A

A

]

]

0

t



[

[

A

A

]

]

e

e

q

q

(kfkr) t (20.34)

which has the form of a straight line—plotting the logarithm term in equation

20.34 versus time—with a slope of (kfkr) and a y-intercept of zero. Again,

at the limit of initial reaction conditions this plot will turn into the expected

plot for a first-order reaction, but now we are extending the plot to consider

longer time periods in which the reverse reaction has an effect on [A]t.

Figure 20.9 shows a plot of [A]tthat illustrates an expected behavior: an ex-

ponential decrease toward some asymptotic minimum. At the same time, the

amount of the product [B]tis shown as increasingtoward some asymptotic

maximum. The actual asymptotic values of [A]tand [B]tare dependent on the

initial value [A] 0 as well as the values ofkfand kr.

But the asymptotic values of [A]tand [B]tover time are the expected equi-

librium concentrations of A and B, labeled [A]eqand [B]eq. There must be

20.4 Equilibrium for a Simple Reaction 695

Figure 20.9 As a reaction approaches equilib-
rium, the amounts of reactant, [A]t, and product,
[B]t, approach their equilibrium values (which
depend on the reaction). Compare this plot of
[A]twith Figure 20.1: at long values for time, the
plots differ. This plot of [A]tfollows equation
20.33, not equation 20.15.

[A] 0

[A]eq

[B]eq

[B] 0

Time

Concentration