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
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).
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
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.
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
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
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 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).
- (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
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