Oscillating Chemical Reactions


Table of Contents
History of Oscillating Chemical Reactions
Belousov-Zhabotinsky (BZ) oscillating reaction
The Lotka-Volterra Model
The Brusselator
The Oregonator


Background Information

History of Oscillating Chemical Reactions

Modern chemists are aware that certain chemical reactions can oscillate in time or space. Prior to about 1920 most chemists believed that oscillations in closed homogeneous systems were impossible. The earliest scientific evidence that such reactions can oscillate was met with extreme scepticism. You can read about the early history and the heated debate surrounding oscillating reactions in several references.

The most famous oscillating chemical reaction is the Belousov-Zhabotinsky (BZ) reaction. This is also the first chemical reaction to be found that exhibits spatial and temporal oscillations. You can demonstrate and carry out experiments on this reaction by following recipes to be found in standard references.

Theoretical models of oscillating reactions have been studied by chemists, physicists, and mathematicians. The simplest one may be the Lotka-Volterra model . Some other models are the Brusselator and the Oregonator . The latter was designed to simulate the famous Belousov-Zabotinskii reaction (the BZ reaction for short).


Recipe for the Belousov-Zhabotinsky (BZ) oscillating reaction

source: B.Z. Shakhashiri, "Chemical Demonstrations: A Handbook for Teachers" (University of Wisconsin, Wisconsin, 1985), and R.J. Field, "A Reaction Periodic in Time and Space", J. Chem. Ed. 49, 308 (1972). Also see J.A. Pojman, R. Craven, D.C. Leard, J. Chem. Ed. 71, 84 (1994) for a description of a Physical Chemistry Lab experiment.

Prepare a solution with the following concentrations of reactants. The volume should be large enough for the intendended audience to view. The reaction mixture should be stirred constantly with a magnetic stirring bar. These concentrations will give a system that oscillates with a period of about 30 sec and the oscillations will continue for 50 minutes or more. The reactants can be mixed in any order.

Reactant

Concentration, M

Ce(NH4)2 (NO3)5 (catalyst)

0.002

CH2 (COOH)2

0.275

KBrO3

0.0625

H2SO4

1.5

Ferroin (indicator)

0.0006

Return to contents, see other references .

Chemical Kinetics (under construction)
Differential Equations (under construction)


Models of Oscillating Chemical Reactions

Why Construct Theoretical Models of Oscillating Chemical Reactions?

A model for a chemical reaction consists of the following parts:

The criterion for an acceptable theoretical model is that it agree with experimental observations of measured time varying concentrations. When a theoretical chemist finds an acceptable model he says he "understands" the reaction.

Return to oscillating reactions CONTENTS.



The Lotka-Volterra Model of Oscillating Chemical Reactions

This is the earliest proposed explanation for why a reaction may oscillate.
In 1920 Lotka proposed the following reaction mechanism (with corresponding rate equations). Each reaction step refers to the MOLECULAR mechanism by which the reactant molecules combine to produce intermediates or products. For example, in step 1 a molecule of species A combines with a molecule of species X to yield two molecules of species X. This step depletes molecules A (and adds molecules X) at a rate proportional to the product of the concentrations of A and X.

reaction step

molecular reaction

step contributions to differential rate laws

1

A + X → 2X
2 X + Y → 2Y
3 Y → B

The overall chemical reaction is merely A → B with two transient intermediate compounds X and Y:
The effective rate laws for the reactant A, the product B, and the intermediates X and Y are found by summing the contributions from each step:



Step 1 is called autocatalytic because X accelerates its own production. Likewise step 2 is autocatalytic. Problem: Given the mechanism it is required to solve for [A], [X], [Y], and [B] as functions of time. Lotka obtained oscillating concentrations for both intemediates X and Y when the concentration of reactant [A] is constant (as, for example, A is continuously replaced from an external source as it is consumed in the reaction). An interactive solution is provided in the form of a mathcad file ( lotka.mcd) (you need mathcad to view this file). You can also read a summary of the solution (view with your browser). Lotka's mechanism can be re-interpreted as a model for oscillating populations of predators and preys as was done by Volterra. In this, A represents the ecosystem in which prey X and predator Y live. Step 1 represents pre procreation: prey population doubles at rate k1[A] (typical exponential growth). Then Y is the population of predators that consume the prey in order to sustain (and expand) their population. Step 2 represents this inclination of predators to reproduce in proportion to the availability of prey. Finally (step 3), predators die at a certain natural rate (also exponential) so that they are removed from the ecosystem.

Return to oscillating reactions CONTENTS , Brusselator , Oregonator






Brusselator Model of Oscillating Chemical Reactions

The Brusselator model was proposed by I. Pregogine and his collaborators at the Free University of Brussels. The reaction mechanism and corresponding rates are:

step

molecular reaction

step contribution to rate laws

1

A → X

2

2X+Y → 3X

3

B + X → Y + D

4

X → E

The net reaction is A+B → C+D with transient appearance of intermediates X and Y.

Problem:

  1. Write the system of 2 coupled differential equations for the intermediate concentrations [X] and [Y].
  2. (b) Let each of the rate constants ki = 1 and assume the two reactants A and B have constant concentrations, [A]=1 and [B]=3 (they are added to the system at the same rate as they are consumed in the reactions). Choose initial concentrations for intermediates ([X]0 = 1, [Y]0=1), and perform a numerical integration from time 0 to 50 (arbitrary units).
  3. (c) Make two plots of the results: -log([X]) and -log([Y]) versus time, and -log([Y]) versus -log([X]).

For a solution of this problem, see the interactive mathcad file brussels.mcd .
Return to CONTENTS , Lotka model , Oregonator


Brusselator Model




Oregonator Model of Oscillating Chemical Reactions


Reference : R.J. Field and R.M. Noyes, J. Chem. Phys. 60, 1877 (1974).

A simplified form of this model uses the following mechanism.

step

reaction

contributions to the rate equation

1

A+Y → X

2

X+Y → P

3

B+X → 2X+Z

4

2X → Q

5

Z → Y

The overall reaction, obtained by adding reactions 1, 2, 4 and twice 3 and 5, is A + 2B → P + Q.

Problem: Use the following definitions for dimensionless concentration variables (a, h, and r) and rate constants (q, s) to solve Oregonator model for the BZ reaction:
[HBrO2] = 5.025 x 10-11 a,
[Br-] = 3.0 x 10-7 h,

[Ce(IV)] = 2.412 x 10-8 r,

q = 8.375 x 10-6,

s = 77.27 .
The time variable is also the dimensionless variable t = t/w with w = 0.1610sec. Take the initial concentrations to be a=1, h=1000, and r=1000.

(a )Set up the differential rate equations for a, h, and r.

(b) Solve the d.e. from time 0 to t1=1000, and plot concentrations (better to plot the -log(conc.)) versus time.

(c) Also plot the trajectories of a and r versus h in concentration (or -log(conc.)) space. Be careful, this system of differential equations is "stiff" and requires special treatement.

See an interactive mathcad solution of this problem in the file oregonat.mcd .

Return to oscillating reactions CONTENTS , Lotka model , Brusselator

Oregonator model





Author: Ron Poshusta poshustr@wsu.edu


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