Physical Chemistry , 1st ed.

Equations 20.41 and 20.42 may look a bit complicated, but the ratio of [B]t

and [C]thas a very simple expression:



[

[

C

B]

]

t

t



k

k

1

2

 (20.43)

This expression is valid for any time t, as long as the reverse reactions are

negligible.

When the reactions approach equilibrium, our analysis of the situation

must change. Near equilibrium, product converts back to reactant, which can

then convert over to product again. But there is no requirement that the reac-

tant molecule react to make the same product! (That is, molecules do not have

memory.) We need to consider the following scheme:

k 1

A B

k 1

k 2 k 2

C

Not only have we defined forward rate constants k 1 and k 2 , but we also define

reverse rate constants k 1 and k 2. According to the previous section, we can

also define equilibrium constants for each reaction,K 1 and K 2 ,as

K 1 

k

k



1

1



K 2 

k

k



2

2



Without presenting the derivation, we simply state that at equilibrium (that is,

as t→), the concentrations of products and reactant are

[A]eq

K 1 

[A

K

]

2

0

 1



[B]eq

K 1

K



1 

K

[

2

A



] 0

1

 (20.44)

[C]eq

K 1

K



2 

K

[

2

A



] 0

1



Additionally,



[

[

C

B]

]

e

e

q

q

K

K

1

2

 (20.45)

Equation 20.45 can be considered a special case of equation 20.43. At the

limit of small times, the effects of the reverse reactions are negligible and equa-

tion 20.43 is applicable. Only in the case of long times, where reverse reactions

have an appreciable effect on concentrations and equilibria are being estab-

lished, does equation 20.45 become applicable. However, there is no reason to

presume that the ratios [B]t/[C]t, the product ratio at some intermediate time,

and [B]eq/[C]eq, the product ratio at “infinite time,” are the same.

When two parallel reactions begin initially, the reaction that goes faster is

the one with the larger rate constant (assuming that both reactions are simple

first-order reactions). Therefore, the specific ratio of products initially,given

by [B]t/[C]t, will be given by the ratio of the forward rate constants k 1 /k 2.

Because the initial product ratio is determined by the kinetics of the forward

reactions (that is, equation 20.43), we say that the ratio of products is kineti-

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698 CHAPTER 20 Kinetics