Period
Dynamics of the B-Z Reaction
Julia
Lenzi & Martin Miller
University
of Houston, Physics
Abstract:
The Belousov-Zhabotinsky (BZ) chemical reaction is a system that exhibits periodic behavior and complex patterns. Changes in the forces acting on the system can measurably alter the dynamics of the system, especially when dealing with the light-sensitive form of the reaction [[1]]. Models of the reaction typically involve a large number of constants, depend on chemical concentrations, and describe only the constant flow reaction [5]. The dynamics of the periodic oscillations, however, can be used to describe general information about the energy of the system.
The system we are studying, a
forced non-linear chemical reaction, has many analogous systems occurring in
nature such as the growth pattern of fungi, fibrillation of heart tissue and
circadian rhythms [2].
Background:
The Belousov-Zhabotinskii
reaction is a non-linear oscillatory reaction that involves the production of
molecular bromine from bromate and bromide ions in the presence of an acid.
By the 1970s, many people
were working on some aspect or other of the BZ reaction and many more had heard
of it. Biochemists and
biophysicists and mathematicians had all embraced the BZ reaction as a model
for something they were more interested in. Biology-minded scientists had long seen similar oscillatory
patterns in multitudes of natural phenomena from fungi growth to the
fibrillation of heart tissue.
Mathematicians saw the oscillatory nature of the BZ reaction as arising
from problems of differential equations yet to be explored. Research and activity surrounded the BZ
reaction, especially as the field of chaos emerged from under the rugs of every
natural science.
In 1958 however, when B. P.
Belousov first discovered that the oxidation of citric acid by bromate in the
presence of cerium ions did not proceed uniformly to equilibrium, there was no
rush to understand and use this intriguing reaction, instead it was quietly
accepted and then ignored. In
fact, in the western hemisphere it was largely unknown despite a 1967 Nature
article by H. Degn. Belousov
reported that instead of proceeding uniformly to equilibrium, this reaction
maintained oscillations between a yellow and a colorless state with astonishing
regularity, and yet apparently no one save A. M. Zhabotinskii saw such behavior
as anything beyond an academic curiosity.
From a chemists’ perspective at the time, before the emergence of chaos
as a field and the analytical tools which emerged with it, this reaction was
merely a curiosity. However, when
Zhabotinskii and colleagues went to the 1968 international conference on
biological and biochemical oscillators in Prague they found an audience waiting
to pounce on their results and the BZ reaction.
Methodology:
In our experiment we hoped
to determine the base frequency of oscillation and change in frequency for a
given BZ reaction and then to determine the effect of changing the initial concentration
of bromide ions, as well as the effect of adding a catalyst, on the frequency
of oscillation or the change in frequency.
Our initial reaction was
created from [[2]]
1. Dissolve 3 ml concentrated sulphuric
acid and 10 g potassium bromate in 134 ml water.
2. Dissolve 1 g sodium bromide in 10 ml
water.
3. Dissolve 2 g malonic acid in 20 ml
water.
(Note that the Cl, Br, and
Na ions are not active participants in the reaction. [[3]])
In a small glass beaker,
add 1 ml of solution 2 to 12 ml of solution 1. Then add 2 ml of solution 3 and wait a few minutes for the
solution to become clear. This
should be done in a well-ventilated area as it may put off a small amount of
fumes. Then, add 2 ml of 0.025M
(standard) Ferroin indicator. Mix
well and pour into a 90 mm petri dish and cover it. The solution is uniformly orange but in a minute or two the
solution will turn clear and then go back to orange. This is the base oscillation.
Soon after the base oscillation begins, clear/white dots will
appear and begin to spread out in rings as well as perhaps forming labyrinthine
patterns or spirals (Figure 1). These
depend very much on the initial conditions to an extent that cannot be
controlled in our laboratory. The
appearance of target patterns or spirals at all is not so much due to the
solution as it is due to dust contamination and other impurities in the
solution. These points of
contamination are points of nucleation for a phase difference that radiates outwards. These patterns fall into the category
of Trigger Waves; one of two general categories for pattern formations in any
given BZ reaction. Trigger waves
are waves of excitation and are dependent on diffusion. The other category is Kinematic Waves –
independent of diffusion and only occurring in self-oscillatory reagent; our
experiment did not produce any of these.
The base oscillation,
flushing clear and then again orange, however, is what is measured for this
experiment. The data collection
was achieved using a laptop and a computer program designed by Martin Miller
which acts as a stopwatch. Hitting
the space bar at each oscillation recorded the time between oscillations. However, given that the oscillations
are on the order of approximately half a minute, a standard stopwatch would
suffice if the data were needed to be taken manually.
Our experiment was
originally derived from a Nature article [[4]] which discussed varying the
oscillation frequency with the aid of a light-sensitive catalyst Tris (2,2’
bipyridyl) dichlororuthenium (II) hexahydrate. This chemical both catalyzes and slows the reaction rate
when exposed to light in the range 430-470nm. We attempted to include this into our general BZ reaction
and measure the effect this catalyst had on the oscillation frequency when
exposed to diffused light from LEDs emitting light at approximately 466nm. The diffusing was done using waxed
weigh-paper.
Results/Conclusions:
We performed nine
experiments in which time data was taken until the base pattern became
unrecognizable, which produced approximately twenty minutes for each trial, and
about 30-40 data points, which was enough points to allow statistical analysis.
·
Three trials used the standard recipe
listed in the methods. (Figure 3)
·
One trial changed the amount of sodium
bromide to 1.5 grams. (Figure 4)
·
Five trials added 0.5, 1.0 (twice),
1.5, and 2.0 mL of 1 mM Ru(bpy)3+2Cl2 . (Figure
5)
In order to analyze the results,
a model is needed to relate the observables of the experiment to physical
processes. The current models for
the B-Z reaction represent the reaction as a system of coupled partial
differential equations based on ion concentrations through time and arbitrary
constants [5].
However, measurement of the ion concentrations is beyond the capability
of the laboratory, and the PDE system is difficult to calculate. As a result, a simplified model is
needed for this experiment.
To generate the model,
several key factors contributed.
·
It can be observed qualitatively that
the intensity of the solution color reduces with time.
·
The period of oscillation becomes
shorter with time.
·
The reaction will eventually go to
extinction.
·
The decay of some ions is exponential [[5]].
If one assumes that the ion
concentration is related to the indicator intensity, analogies can be made with
a non-chaotic oscillatory system.
Recall from mechanics the partially elastic bouncing ball. As the ball bounces towards rest in a
finite time, the height of the bounces reduces exponentially with time and the
period of each bounce is shorter as the height decreases.
The attached calculations
show how this model was related to the B-Z reaction that was observed. The model predicts a linearly decreasing
period with time (Figure
2). This
line has two attributes: a y-intercept and a slope. The y-intercept, labeled t0 to represent an
imaginary initial period, is proportional to the square root of initial energy,
and inversely proportional to the driving forces of the reaction by some
function. The slope is related to
the fraction of energy lost per oscillation, ε, through a constant, α, which describes
the dynamics of the reaction rate.
While t0 is an
interesting observation, it is not very useful. Not only does it lump all of the unknowns into a single
number, but it is also highly dependent on what time in the reaction that the
measurement begins. Since the
solutions may sit for different lengths of time during stirring, the
relationships between the t0 of the trials depends more on procedure
than the reaction itself. Looking
at Figure 9, the t0 varies dramatically for even
identical trials, and in (b) the error bars of most of the trials still
overlap, indicating that t0 does not vary significantly. Although trials from separate batches
were more disparate than trials mixed from the same batch (cf. Ru 1.0 vs. Ru
1.0 (2)). Also, although the t0
measurements are inconclusive, the slightly higher values for the catalyzed
trials is consistent with more energy being available to the reaction because
of the catalyst.
The slope of the line is
the focal point of this experiment.
The three trials using the original recipe were used as a control. Figure
10 shows the fraction of energy that was lost per
oscillation in each trial, ε, derived from the slope of the regression
line. The 0.5 and 1.0 mL trials
appear to lose energy at a much lower rate than the control trials. This is consistent with the use of a
catalyst. The 1.5 and 2.0 mL
trials do not conform to this trend as strongly. However, Figure
7 shows one point in the 1.5 trial which appears to be
erroneous. When that point is
removed, the ε-value for the 1.5 trial drops to a value approximately equal to
the 0.5 trial. Looking at Figure 8 shows that the high variance was primarily caused by
points toward the end of the trial.
These points also made the slope of the line steeper than the initial
points would suggest, and a steeper slope produces a higher ε-value. The most probably cause of this is that
the depth of solution in the petri dish for this trial was slightly too deep,
and three-dimensional patterns had overcome the base frequency well before the
trial was ended. Additional 2.0
trials should be tested to determine if the results are consistent.
Although there was some
deviation, the simplified model seems consistent with the action of a catalyst
on the B-Z reaction.
The one trial with altered
bromide concentration failed to produce promising results, despite the fact
that bromide is one of the key reaction components. The bromide trial fell in line with the three control trials,
indicating that the bromide concentration does not impact the time dynamics of
the experiment.
We would like to thank Dr.
Swinney’s group at UT for their help in getting started with our experiment.
