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) from x 0 to x x(t ) x gives
1
[A] 0 + [F] 0
ln
[A] 0
[F] 0 + x
[F] 0 ([A] 0 − x )
k app t
(13.3-9)
where x x(t ). This expression can be solved for x :
x [F] 0
e βt − 1
1 +
[F] 0
[A] 0
e βt
(13.3-10)
where β ([A] 0 + [F] 0 )k app and 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 concentrations 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 iodatecatalyzed 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 oscillations 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 the Lotka–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 simple predator–
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 represented 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
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) from x 0 to x x(t ) x gives
1
[A] 0 + [F] 0
ln
[A] 0
[F] 0 + x
[F] 0 ([A] 0 − x )
k app t
(13.3-9)
where x x(t ). This expression can be solved for x :
x [F] 0
e βt − 1
1 +
[F] 0
[A] 0
e βt
(13.3-10)
where β ([A] 0 + [F] 0 )k app and 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 concentrations 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 iodatecatalyzed 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 oscillations 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 the Lotka–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 simple predator–
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 represented 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