Physical Chemistry Third Edition

13.3 Autocatalysis and Oscillatory Chemical Reactions 587

Exercise 13.13

Solve the simultaneous equations in Eq. (13.3-6) to obtain Eq. (13.3-7).

Integration of Eq. (13.3-8) fromx0toxx(t′)x′gives

1

[A] 0 +[F] 0

ln

(

[A] 0

(

[F] 0 +x′

)

[F] 0 ([A] 0 −x′)

)

kappt′ (13.3-9)

wherex′x(t′). This expression can be solved forx′:

x[F] 0

eβt− 1

1 +[F][A]^00 eβt

(13.3-10)

whereβ([A] 0 +[F] 0 )kappand where we drop the prime symbol (′). The rate of

the reaction begins at a small value and increases as the concentration of the product

becomes larger. It then declines as the concentration of the reactant decreases. Since

the mechanism does not include reverse reactions, all of the reactant will eventually

be consumed and the rate will vanish.

Oscillatory Chemical Reactions

It was once thought that in any chemical reaction the concentrations of reactants would

decay smoothly, that concentrations of products would rise smoothly, and that con-

centrations of reactive intermediates would rise and fall only once, as in the two-step

mechanism of Chapter 11. The first known oscillatory chemical reaction was the iodate-

catalyzed decomposition of hydrogen peroxide, discovered in 1920. In this reaction,

the color of the solution (due to iodine) and the evolution of oxygen can vary in

an oscillatory and nearly periodic way. The most famous oscillatory reaction is the

Belousov–Zhabotinskii reaction(abbreviated BZ), which is the reaction of citric acid,

bromate ion, and ceric ion in acidic solution. This reaction not only produces oscilla-

tions in time, but can also produce chemical waves, consisting of roughly concentric or

spiral rings of different colors that move outward from various centers. The first article

describing this reaction was rejected for publication, because of the common belief that

chemical oscillations could not occur. However, since the 1960s chemical oscillations

have been widely studied and have even become common lecture demonstrations.

We can illustrate oscillatory behavior with theLotka–Volterra mechanism:

(1) A+X→2X

(2) X+Y→2Y

(3) Y→P

(13.3-11)

This mechanism can show oscillatory behavior if A is continually replenished so that

[A] remains constant: This mechanism has been used in ecology as a simplepredator–

prey model, in which A represents the food supply (grass) for prey animals (hares),

represented by X. Predators (wolves) are represented by Y, and dead wolves are repre-

sented by P. The consumption of grass (A) by the hares (X) allows them to reproduce as

in step 1, and the consumption of hares by wolves (Y) allows the wolves to reproduce

as in step 2. Step 3 corresponds to the death of wolves by natural causes. Since no