Computer Algebra Recipes

40 CHAPTER 1. PHASE-PLANE PORTRAITS
1.2.2 The Oregonator
Everybody's youth is a dream, a form of chemical madness.
F. Scott Fitzgerald, American writer (1896{1940)
The Belousov{Zhabotinski (BZ) chemical reaction is now probably one of the
best known of the chemical oscillators. However, because it was contrary to
the then current belief that all solutions of reacting chemicals must go monotonically
to equilibrium, Belousov could not initially get his chemical oscillator
discovery published in any Soviet journal. Only years later, when his results
were con¯rmed by Zhabotinski, was he given due recognition for his discovery.
For his pioneering research work he was awarded, along with Zhabotinski, the
Soviet Union's highest medal, but unfortunately for him, he had passed away
10 years earlier.
The BZ reaction may be achieved [Tys76] by dissolving 4.292 g of 0.28 M
malonic acid 7 and 0.175 g of 0.002 M cerium ammonium nitrate in 150 ml of 1
M sulfuric acid and stirring well. The solution will initially be yellow, then turn
clear after a few minutes. Then, on adding 1.415 g of 0.063 M sodium bromate,
the solution will oscillate between yellow and clear with a period of about one
minute. A more dramatic color change between red and blue can be achieved
by adding a few ml of 0.025 M ferroin.
Field, K}orÄos, and Noyes [FKN72][FN74] were able to measure the periodic
oscillations in the Br ¡ concentration and the Ce 4+ /Ce 3+ ratio in the BZ reaction.
They then isolated the ¯ve most important reactions in the complicated
chemistry that was taking place and created a kinetic model called the Oregonator,
the name re°ecting the location of where the research was carried out.
Labeling the concentration of BrO ¡
3 as A, thatofHBrO2 as X, thatofBr ¡
as Y ,thatofCe 4+ as Z, and all other less-important chemical species as ¤, the
relevant chemical reactions in the Oregonator model are as follows:
A + Y + ¤
X + Y + ¤
A + X + ¤
k1
! X + ¤
k2
! ¤
k3
! 2 X +2Z + ¤ (1.13)
2 X k4
! A + ¤
Z + ¤
k5
! hY + ¤
Here the ki denote the rates of reaction and h is a numerical \fudge factor"
introduced because of the severe truncation of the full set of equations describing
the complicated chemistry. How sensitive the results are to the value chosen
for h will be left as a problem.
Ignoring the less-important chemical species and noting that the depletion
of A can be neglected, i.e., A is constant, the rate equations for the production
7 Belousov [Bel58] used citric acid, but malonic acid is now commonly substituted.