Physical Chemistry Third Edition

588 13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics

reverse steps are included, the overall reaction corresponds to the conversion of grass

into dead wolves:

A→P (13.3-12)

The differential equations for this mechanism are

d[A]

dt

k 1 [A][X] (13.3-13a)

d[X]

dt

 2 k 1 [A][X]−k 2 [X][Y] (13.3-13b)

d[Y]

dt

k 2 [X][Y]−k 3 [Y] (13.3-13c)

For the case in which [A] is held fixed, we have only Eq. (13.3-13b) and Eq. (13.3-13c)

to solve simultaneously. Since we expect an oscillatory solution, we cannot apply the

quasi-equilibrium approximation or the quasi-steady-state approximation. The equa-

tions must be solved numerically, using standard methods of numerical analysis to

obtain the time dependence of [X] and [Y].^19 The concentrations of X and Y must

satisfy the expression

k 2 ([X]+[Y])−k 3 ln([X])−k 1 [A]ln([Y])C (13.3-14)

whereCis a constant.

Exercise 13.14

Show that the foregoing expression satisfies the differential equations in Eq. (13.3-13).Hint:

Differentiate this expression with respect to time and substitute the differential equations for [X]

and [Y] into this equation.

The results can be displayed by a plot of [Y] as a function of [X]. A mathematical

space with time-dependent variables plotted on the axis is called aphase space.If[Y]

is plotted as a function of [X] for a constant value of [A], there a closed curve that is

retraced over and over again as time passes, exhibiting periodic behavior. However,

there are different curves for different initial states, but the oscillations predicted by the

mechanism resemble the actual fluctuations in predator and prey populations in actual

ecosystems.

The known mechanisms that produce oscillatory behavior have two characteristics

in common. The first is autocatalysis. The product of a step must catalyze that step,

as in steps 1 and 2 of the Lotka–Volterra mechanism. The second is that nonlinear

differential equations occur. That is, the variables must occur with powers greater than

unity or as products. A mechanism has been proposed for the BZ reaction that has

18 steps and involves 21 different chemical species.^20 A computer simulation of the

18 simultaneous rate differential equations for the mechanism has been carried out

and does produce oscillatory behavior. It also exhibits the interesting behavior that

all curves in phase space corresponding to different initial states eventually approach

(^19) As of 2006, the website http://tu-dresden.de/Members/thomas.petzoldt provides a program that you can
run to carry out the solution to the Lotka–Volterra mechanism.
(^20) R. J. Field, E. Körös, and R. M. Noyes,J. Am. Chem. Soc., 94 , 8649 (1972).