Physical Chemistry , 1st ed.

The rate of change in concentrations of A and C should appear reasonable

from inspection of the sequential reaction. The rate of change of concen-

tration of B is a combination of two effects: an increase in [B] due to for-

mation fromA, and a decrease in [B] due to the formation ofC. The posi-

tivity and negativity of each term reflects the corresponding increase or

decrease.

The three equations 20.46 can be integrated to yield

[A]t[A] 0 ek^1 t

[B]t

k

k

1

2





[A

k

]

1

^0 (ek^1 tek^2 t) (20.47)

[C]t[A] (^0)  1 
k 1 

1

k 2

(k 2 ek^1 tk 1 ek^2 t)

where again,trepresents the elapsed time and [A] 0 is the original amount of

reactant A in the system. (Remember, we are assuming that the original amounts

of B and C are zero.) The expression for [A]tshould look familiar; it is exactly

the same as equation 20.15.

However, the expressions for [B]tand [C]tare more complicated. Both of

them, as a matter of fact, depend on both rate constants for both reactions.

Furthermore, they also depend on the differences in the rate constants (indi-

cated by the k 2 k 1 and k 1 k 2 terms in the denominators of the expressions).

If, for example, the second reaction is much slower than the first reaction (that

is,k 1



 k 2 ), there is an initial buildup of product B as shown by the left part

of Figure 20.12. Only over long periods of time will product B eventually re-

act to form the final product C, which is illustrated on the right part of Figure

20.12. (Although there may be some tendency to refer to B as the kinetically

favored product and C as the thermodynamically favored product, this is not

an accurate parallel.)

When k 1 k 2 , the second reaction is much faster than the first one and

the penultimate product B reacts to the ultimate product C almost as fast as B

itself is formed. The amounts of products over time are illustrated by Figure

20.13. Notice that very little of B is present at any time. The reaction system

will remain this way unless a large amount of B is favored by the equilibria of

the two reactions.

700 CHAPTER 20 Kinetics

1.0

0

0

Time (s)

[A]

[C]

[B]

2500

Fraction present

0.8

0.6

0.4

0.2

500 1000 1500 2000

Figure 20.12 For consecutive reactions in which k 1 is much greater than k 2 , there is a short-

term buildup of the intermediate product, B. But over longer periods of time, the final product

C is formed. In this plot,k 1 0.01 s^1 and k 2 0.0005 s^1.