Chemistry 030.204
Handout 1:
Autocatalytic and Oscillatory Reactions


Fig. 27.5 The concentration of product during the autocataysed A-->P reaction discussed in the example (using b=0.1).

27.7 Autocatalysis


The phenomenon or autocatalysis is the catalysis of a reaction by the products. For example, in a reaction A --> P it may be found that the rate law is

v = k[A][P]

(17)

and so the reaction rate increases as products are formed. (The reaction gets started because there are usually other reaction routes for the formation of some P initially, which then takes part in the autocatalytic reaction proper.) An example of autocatalysis is provided by two steps in the Belousov-Zhabotinskii reaction (BZ reaction) which will figure in discussions later in the section:

BrO3- + HBrO2 + H3O+ --> 2BrO2 + 2H2O

2BrO2 + 2Ce(III) + 2H3O+ --> 2HBrO2 + 2Ce(IV) + 2H2O

The product HBrO2 is a reactant in the first step.

Example 27.5: Calculating concentrations in an autocatalytic reaction
Set up and solve the rate equation for the A --> P autocatalytic reaction in eqn 17.
Answer. Write [A] = [A]0 - x, [P] = [P]0 + x, when the rate law becomes

Integration by partial fractions, using

Comment. The solution is plotted in Fig. 27.5. The rate of reaction is slow initially (little P present), then fast (when P and A are both present), and finally slow again (when A has disappeared).
Exercise. At what time is the reaction rate a maximum?

[tmax = -(1/ka) ln b]

  The industrial importance of autocatalysis (which occurs in a number of reactions, such as oxidations) is that the rate of the reaction can be maximized by ensuring that the optimum concentrations of reactant and product are always present.

27.8 Oscillating reactions


One consequence of autocatalysis is the possibility that the concentrations of reactants, intermediates, and products will vary periodically either in space or in time (Fig. 27.6). Chemical oscillation is the analogue of electrical oscillation, with autocatalysis playing the role of positive feedback. Oscillating reactions are much more than a laboratory curiosity: they are known to occur in some industrial processes, and there are many examples in biochemical systems where a cell plays the role of a chemical reactor. Oscillating reactions, for example, maintain the rhythm of the heartbeat. They are also known to occur in the glycolytic cycle, in which one molecule of glucose is used to produce (through enzyme catalysed reactions involving ATP) two molecules of ATP. All the metabolites in the chain oscillate under some conditions, and do so with the same period but with different phases.

Fig. 27.6 Some reactions show oscillations in time; some show spatially periodic variations. This sequence of photographs shows the emergence of a spatial pattern.

Fig 27.7 The periodic variation of the concentrations of the intermediates X and Y in a Lotka-Volterra reaction. The system is in a steady-state, but not at equilibrium.


Fig. 27.8 An alternative representation of the periodic variation of [X] and [Y] is to plot one against the other. Then the system describes closed orbits, different ones being obtained with different starting conditions.

 The Lotka-Volterra mechanism
We shall use a model autocatalytic reaction of a particularly simple form that illustrates how these oscillations may occur. The actual chemical examples that have been discovered so far have a different mechanism, as we shall see. The Lotka-Volterra mechanism is as follows:

Steps (a) and (b) are autocatalytic. The concentration of A is held constant by supplying it to the reaction vessel as needed. (B plays no part in the reaction once it has been produced, and so it is unnecessary to to remove it; in practice, though, it would normally be removed.) These constraints leave [X] and [Y], the concentrations of the intermediates, as variables. Note that we are considering a steady-state conhition, which is maintained by the flow of A into the reactor. This steady-state condition must not be confused with the steady-state approximation made earlier: in the present case we shall solve the rate equations exactly for the variable concentrations of X and Y, but hold [A] at an arbitrary but constant value.

  The Lotka-Volterra equations can be solved numerically using the techniques explained in Further information 1, and the results can be depicted in two ways. One way is to plot [X] and [Y] against time (Fig. 27.7). The same information can be displayed more succinctly by plotting one concentration against the other: this representation gives the series of closed curves shown in Fig. 27.8.

The periodic variation of the concentrations of the intermediates can be explained as follows. At some stage there may be only a little X present, but


Fig. 27.9 Some oscillating reactions approach a closed trajectory whatever their starting conditions. The closed trajectory is called a 'limit cycle'.		 reaction (a) provides more, and the production of X autocatalyses the production of even more X. There is therefore a surge of X. However, as X is formed, reaction (b) can begin. It occurs slowly initially because [Y] is small, but autocatalysis leads to a surge of Y. This surge, though, removes X, and so reaction (a) slows, and less X is produced. Since less X is now available reaction (b) slows. As less Y becomes available to remove X, X has a chance to surge forward again, and so on.

The brusselator
Another very interesting set of equations is called the brusselator (because it is an oscillator that originated with Ilya Prigogine's group in Brussels):

Since the reactants (A and B) are maintained at constant concentration, the two variables are the concentrations of X and Y. These two concentrations may be calculated by solving the rate equations numerically, and the results are plotted in Fig. 27.9. The interesting feature is that whatever the initial concentrations of X and Y, the system settles down into the same periodic variation of concentrations. The common trajectory is called a limit cycle, and its period depends on the values of the rate coefficients.

The oregonator
One of the first oscillating reactions to be reported and studied systematically was the BZ reaction mentioned earlier, which takes place in a mixture of potassium bromate, malonic acid, and a cerium(IV) salt in an acidic solution. The mechanism has been elucidated by Richard Noyes, and involves 18 elementary steps and 21 different chemical species. The main features of this awesomely complex mechanism can be reproduced by the following oregonator (so called because Noyes and his group work in Oregon), where A stands for HBrO2, B for Br-, and Z for Ce4+:

(a) A + Y --> X
(b) X + Y --> C
(c) B + X --> 2X + Z
(d) 2X ---> D
(e) Z --> Y

A, B, C, D are held constant (in the model). The oscillations arise in a similar way to the brusselator, and can be traced to the autocatalysis in step 3 and the linkage between the reactions provided by the other steps.


Fig. 27.10 A system showing bistability. As the concentration of X is increased (by adding is to a reactor) the concentration of Y decreases along the upper curve but at A drops sharply to a low value. If X is then decreased, the concentration increases along the lower curve, but rises sharply to a high value at B.


Fig. 27.11 Chemical oscillation in a bistable system occurs as a result of the effect of a third substance Z which can react with X to produce Y and with Y to produce X. As a result, the system switches periodically between the upper and lower curves.


Fig. 27.12 The leaping from one branch to another of a bistable system appears to the observer as a periodic surge and depletion of concentration.

 Bistability
Attempts have been made to discover the underlying causes of oscillation more deeply than simply recognizing the role of autocatalysis. It appears that three conditions must be fulfilled in order to obtain oscillations:
  1. The reactions must be far from equiIibrium.
  2. The reactions must have autocatalytic steps.
  3. The system must be able to exist in two steady states.
The last criterion is called bistability, and is a property that takes us well beyond what is familiar from the equilibrium properties ofsystems. The property of bistability is a chemical version of supercooling, in which the temperature of a liquid may be lowered beneath its freezing point without it solidifying.

  Consider a reaction in which there are two intermediates X and Y. If the concentration of Y is at some high value in a reactor, and X is added, then the concentration of Y might decrease as shown by the upper line in Fig. 27.10. If X is at some high value, then as Y is added the reaction might result in the slow increase of Y as shown by the lower line. However, in each case, a concentration may be reached at which the concentration will jump from one curve to the other (just as a supercooled liquid might suddenly solidify). The two curves represent the two stable states of the bistable system. Neither is an equilibrium state in the thermodynamic sense: they occur in steady states that are well removed from equilibrium, and the concentrations of X and Y represent the consequences of reactants continuously flowing into and of products flowing out of the reactor.

  Now consider what happens when a third type of intermediate, Z, is present. Suppose Z reacts with both X and Y. In the absence of Z the flows of material might correspond to a certain state on the upper curve of Fig. 27.11. However, as Z reacts with Y to produce X the state of the system moves along the curve (to the right, as Y decreases and X increases) until the sudden transition occurs to the lower curve. Then Z reacts with X and produces Y, which means that the composition moves to the left along the lower curve. There comes a point, however, when the concentration of X has been reduced so much, and that of Y has risen so much, that there is a sudden transition to the upper curve, when the process begins again. It is the leaping from one stable state to another that we see as the sudden surge or depletion of the concentration of a species (Fig. 27.12).

   By studying the regions of concentrations and the rate coefficients for the individual steps of a reaction it is now becoming possible to predict the occurrence of bistable chemical systems and to anticipate the occurrence of oscillations. We are still far, however, from being able to use these ideas to account for gene expression, the patterns on tigers and butterflies, and the oscillation of cool flames, in all of which it is thought these processes play a part.

Further information 1: integration of rate equations


The general method of solving differential equations numerically is to approximate differentials by small but finite differences. Suppose we have the differential equation df/dx=g(x), then instead of df=g(x) dx we write f (x + dx) - f(x) = g(x)dx, with dx a smaIl increment of x. Then

f(x + dx) = f(x) + g(x)dx

This equation is exact when dx is infinitesimal. Now we just move along the x coordinate in steps of magnitude dx, using the known initial value of f for the first step, and evaluating gdx at that initial value of x, which gives the value of f at x + dx; then we use this new value of f as the initial value of f for the next step together with the value of gdx, at rhe new value of x, to calculate f two steps away from the origin, and so on, until we have reached the value of x of interest. For example, to integrate the HBr rate law in Example 27.1 we write [H2] = a - x, [Br2] = a - x (assuming equal concentrations initially), and [HBr] = 2x (assuminp that the intermediates are at such low concentrations that they can be neglected). Then, with y=x/a, the rate law becomes

dy/dt = c1(1 - y)5/2/(1 + c2y)

where c1=1/2ka1/2 and c2 = 2k' - 1. We integrate this expression by stepping along the t axis and calculating successive values of y.

  The integration procedure is easy to implement on a microcomputer, and may be extended to the case where the values to use in one equation ars determined by another (i.e. when the differential equations are coupled). A simple program would have the form:

  • Input step size
  • Input initial and final values of x
  • Evaluate f(initial x)
  • Loop from initial x to final x
    • Evaluate f(x + step = f(x) + g(x) * step
    • Print or plot f(x)
  • Repeat loop

  A set of coupled differential equations can be integrated similarly. Thus, if the two equations are

df/dx = af(x) + g (x)
dg/dx = bg(x) +f(x)

we express them as finite difference equations:

f(x + dx) = f(x) + {af(x) + g(x)}dx
g(x +dx) = g(x) + {bg(x) + f(x)}dx

and integrate them as before. There are, however, several problems with simple integration procedures. One is that it may be inaccurate to use the value of g(x) at the start of the interval for which we are calculating f(x + dx), and similarly for f in the equation for g. Inaccuracy is avoided by predictor-corrector techniques that make predictions of the later value of g and h on the basis of their recent values.