Oscillating Chemical Reactions
 Table of Contents
 History of Oscillating Chemical Reactions
 BelousovZhabotinsky (BZ) oscillating reaction
 The LotkaVolterra 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 BelousovZhabotinsky (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 LotkaVolterra model . Some other models are the Brusselator and the Oregonator . The latter was designed to simulate the famous BelousovZabotinskii reaction (the BZ reaction for short).
Recipe for the BelousovZhabotinsky (BZ) oscillating reaction
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(NH_{4})_{2} (NO_{3})_{5} (catalyst) 
0.002 
CH_{2} (COOH)_{2} 
0.275 
KBrO_{3} 
0.0625 
H_{2}SO_{4} 
1.5 
Ferroin (indicator) 
0.0006 
Return to contents, see other
references .
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:
 A mechanism. This is a set of elementary chemical reactions to describe how reactants form intermediates, intermediates combine with one another and reactants, and ultimately products are produce.
 A set of Rate equations. These are differential equations corresponding to the reaction mechanism and giving the rates of change of all reactants, intermediates, and products.
 A set of Integrated rate equations. These show the concentrations as functions of time for reactants, intermediates, and products. They are obtained by integrating the rate (differential) equations.
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 LotkaVolterra 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:
step 1: A + X → 2X
step 2: X + Y → 2Y
step 3: Y → B
sum of steps: A → B
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 reinterpreted 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 k_{1}[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:
 Write the system of 2 coupled differential equations for the intermediate concentrations [X] and [Y].
 (b) Let each of the rate constants k_{i} = 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).
 (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:
[HBrO_{2}] = 5.025 x 10^{11} a,
[Br^{}] = 3.0 x 107 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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