This preview shows pages 1–3. Sign up to view the full content.
57
ON A MATHEMATICAL MODEL FOR THE
BELOUSOVZHABOTINSKII (BZ) REACTION
Min Kyung Kwon, McNair Scholar
Dr. V.S. Manoranjan, Faculty Mentor
Department of Mathematics
ABSTRACT
Many real world problems can be modeled using ordinary differential equations
(ODEs). An example is the BelousovZhabotinskii (BZ) reaction, where malonic
acid is oxidized by acid bromate in the presence of ferroin. Many researchers
have studied the BZ reaction and its ability to generate nonlinear waves. In this
paper, we analyze a mathematical model that describes the BZ reaction in order
to understand the nonlinear phenomena associated with it.
INTRODUCTION
Many real world problems can be modeled and analyzed using ordinary differential equations
(ODEs). An example is the BelousovZhabotinskii (BZ) chemical reaction, in which malonic acid
is oxidized by acid bromate in the presence of ferroin.
The BelousovZhabotinskii reaction was first noted when Belousov discovered the oscillation
of a chemical reaction (Belousov, 1958). However, his study was not proven or published until it
was rediscovered by Zhabotinskii (1964). When Zhabotinskii published his finding, Belousov
was given credit for his original discovery.
Tyson (1976) discusses the BZ reaction in great detail. The chemical reactions that are part of
the BZ reaction can be described as:
.
5
2
4
2
)
(
4
2
2
2
3
2
2
2
2
3
2
2
4
3
2
2
2
4
2
3
3
2
2
3
+
−
+
+
+
−
+
+
−
+
+
−
+
−
−
+
+
+
+
→
+
+
+
+
→
+
+
→
+
+
+
→
+
+
+
→
+
+
H
CO
HCOOH
Br
Ce
O
H
COOH
BrCH
Ce
H
HOBr
BrO
HBrO
O
H
HBrO
Ce
H
HBrO
BrO
Ce
HOBr
H
Br
HBrO
HOBr
HBrO
H
Br
BrO
(1)
Now equations (1) can be simplified into
.
2
2
2
2
hY
Z
P
A
X
Z
X
X
A
P
Y
X
P
X
Y
A
→
+
→
+
→
+
→
+
+
→
+
(2)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document58
by replacing
A
with [
−
3
BrO
], P with [
HOBr
],
X
with [
2
HBrO
],
Y
with [
−
Br
], and
Z
with
[
4
+
Ce
]. Tyson reformulates equations (2) into a mathematical form as given below.
(3)
This system is described only with respect to
X=
[
2
HBrO
]
, Y =
[
−
Br
], and
Z=
[
4
+
Ce
]
,
and the left hand sides of the equations express the rates of change of the chemical substances in
the system. One can seen the terms on the right hand side of the equations (3) are nonlinear.
Therefore, these equations represent a specific example of a nonlinear phenomenon in chemical
studies.
The work presented here focuses on a nondimensioned variation of the mathematical model
given by (3). As noted above, a detailed background on the BZ reaction and on the various
mathematical concepts used in this study can be obtained from Tyson (1976; see also, Field and
Noyes, 1974).
PROBLEM
The BZ reaction model as examined here is as follows:
v
ruv
u
u
dt
du
1
)
1
(
ε
+
−
−
=
(4)
buv
v
dt
dv
−
−
=
2
where
u
and
v
represent the concentrations as [
2
HBrO
],
and
[
−
Br
],
b, r
,
1
, and
2
are
positive constants. Here
u
and
v
are unknown variables and they need to be solved and interpreted.
The above model is a nonlinear system of ODEs and the independent variable of the system
is
t
, the time. It is also a coupled system with the coupling coming via the interaction terms
ruv
and
buv
. Therefore, when solving, both nonlinear ODEs should be solved simultaneously.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '11
 Johnopera
 Differential Equations, Equations

Click to edit the document details