The Belousov–Zhabotinsky Reaction: Modelling the Oregonator

Oscillatory chemical reactions attract both researchers and enthusiasts because of their interesting pattern formation and oscillating colour changes, which contrast to conventional chemical reactions. However, their discovery was initially met with strong resistance from the scientific community.

A brief history

By the early 1920s Lotka[1] developed a simple model using two sequential autocatalytic reactions to explain chemical oscillations. Around the same time, Bray[2] discovered the first homogenous chemical oscillator: the iodate-catalysed decomposition of hydrogen peroxide. However, the bast majority of chemist criticized Bray’s work. They thought that chemical oscillation violated the second law of thermodynamics, a sort of perpetual motion in a beaker.

In early 1950s, B.P. Belousov noticed that a mixture consisting of bromate, cerium ions and citric acid in sulfuric oscillated, shifting between colourless and pale yellow. His work faced a lot of resistance. Later, a graduate student named Anatol Zhabotisnky improved the experiment by using malonic acid instead of citric acid and adding ferroin as a redox indicator, which gave a more dramatic red-blue colour change [3]. He also showed that in an unstirred system, the reaction formed patterns or spirals. Giving rise to the so-called Belousov-Zhabotinsky (BZ) reaction.

Later, Prigogine’s group in Brussels developed a simpler and more chemically realistic model than Lotka’s.[4] It shows a variety of interesting spatial and temporal phenomena, which they called dissipative structures. These patterns appear in nonlinear systems kept far from equilibrium.

Field, Körös, and Noyes (FKN) developed a detailed mechanism for the BZ reaction [5], demonstrating that chemical oscillations follow the same principles as ordinary reactions. Later, Field and Noyes created a simplified model called the Oregonator [6]. Several variants of the BZ reaction have been reported[7], but the most interesting is the uncatalyzed version[8] which employed aniline derivatives. This aromatic organic substrate acts as one-electron transfer agent like the metallic catalyst in the BZ system.

The BZ mechanism

In the BZ reaction three processes take place successively to constitute one oscillation:

1. Bromide (Br^-) reacts with bromate (HBrO₃^-) to form hypobromous acid (HOBr) and bromine (Br₂).
2. Autocatalytic production of bromous acid (HBrO₂) happens while the metal catalyst is oxidized.
3. The oxidized catalyst reacts with the organic substrate to produce Br^- again.

In the first step, Br^- is consumed until its concentration drops low, triggering the autocatalytic step. Then, Br^- production in the third step resets the system for a new cycle. The FKN mechanism fully describes the reaction. However, this mechanism involves 20 rate equations, and it is extensive to be covered in this post. Instead, we will use the Oregonator model. While it doesn’t capture every chemical detail, it portraits the essential dynamic behaviour of the BZ reaction. This model consists of five irreversible steps:

\[ \begin{aligned} \ce{BrO3^- + Br^- &->[k_1] HBrO2 + BrMA} \\[6pt] \ce{HBrO2 + Br^- &->[k_2] 2BrMA} \\[6pt] \ce{BrO3^- + HBrO2 &->[k_3] 2HBrO2 + 2Ce^{4+}} \\[6pt] \ce{2HBrO2 &->[k_4] BrO3^- + BrMA} \\[6pt] \ce{MA + Ce^{4+} &->[k_5] fBr^-} \end{aligned} \]

Here, MA represents malonic acid and BrMA denotes a brominated specie of the malonic acid. Steps 3 and 5 are simplified abstractions. Step 3 combines multiple reactions into a single autocatalytic process, while step 5 uses (\(f\)) to approximate the complex reduction of \( \ce{Ce(IV)} \) by the organic substrate. These simplifications capture the essence of the dynamical behaviour of the FKN’s complexity.

Differntial Equations

The following ordinary differential equations model the oscillatory dynamics of key species in the BZ reaction:

\[ \begin{aligned} \frac{d[\ce{BrO3-}]}{dt} = & -k_1[\ce{BrO3-}][\ce{Br-}] - k_3[\ce{BrO3-}][\ce{HBrO2}] + k_4[\ce{HBrO2}]^2\\[6pt] \frac{d[\ce{Br-}]}{dt} = & -k_1[\ce{BrO3-}][\ce{Br-}] - k_2[\ce{HBrO2}][\ce{Br-}] + f k_5[\ce{Ce(IV)}]\\[6pt] \frac{d[\ce{HBrO2}]}{dt} = & k_1[\ce{BrO3-}][\ce{Br-}] - k_2[\ce{HBrO2}][\ce{Br-}] + k_3[\ce{BrO3-}][\ce{HBrO2}] - 2k_4[\ce{HBrO2}]^2\\[6pt] \frac{d[\ce{BrMA}]}{dt} =& k_1[\ce{BrO3-}][\ce{Br-}] + 2k_2[\ce{HBrO2}][\ce{Br-}] + k_4[\ce{HBrO2}]^2\\[6pt] \frac{d[\ce{Ce(IV)}]}{dt} = & 2k_3[\ce{BrO3-}][\ce{HBrO2}] - k_5[\ce{MA}][\ce{Ce(IV)}]\\[6pt] \frac{d[\ce{MA}]}{dt} = & - k_5[\ce{MA}][\ce{Ce(IV)}] \end{aligned} \]

To account the spatial effects, such as pattern or wave propagation, these chemical kinetics are coupled with Fick’s second law. For each specie \( [C_i] \), the partial differential equation is:

\[ \frac{\partial [C_i] }{\partial dt} = R_i + D_i\left(\frac{\partial^2 C_i}{\partial x^2}\right) \]

Here, \( R_i\), represents the chemical rection rate from the corresponding ODE, and \( D_i\) is the species’ diffusion coefficient.

Simulating oscillations

The numerical value of the kinetic parameters used in the simulations are listed in the table below:

K1	K2	K3	K4	K5	f
1.28	2.4×106	33.6	2.4×103	1.0	1.0

The corresponding initial molar concentrations of the chemical species are given in the following table:

BrO3−	Br−	HBrO2	BrMA	Ce(IV)	MA
0.06	2.5×10−3	1.0×10−7	0.0	1.0×10−4	0.1

The plot below shows the oscillatory behaviour of Br₂, HBrO₂, and Ce(IV). Over a period of 30 minutes, thirteen oscillations are observed. As time progresses, the amplitude gradually decreases due to the consumption of the reactants: malonic acid and bromate (BrO₃⁻).

In fact, the oscillatory process occurs only within a limited range of reagent concentrations. For instance, Br₂ exhibits noticeable oscillations when the BrO₃⁻ concentration lies between approximately 0.0316 M and 0.35 M. The animation below showcases the oscillations of Br₂ as a function of BrO₃, alongside the kinetic profile of malonic acid. Clearly, during oscillations, malonic acid decreases in a stepwise manner, and once its concentration approaches zero, the oscillations cease.

Animated concentration profiles of Br₂ and malonic acid as function of BrO₃⁻.

The oscillations also depend on the values of the rate constants. Just as with reagent concentrations, the oscillatory behaviour occurs only within a limited parameter range. Changing these constants dramatically affects both the amplitude and the period of the oscillations. A useful way to visualise this sensitivity is through phase-space plots, where [Br⁻] is plotted against [HBrO₂] or [Ce⁴⁺]. In oscillatory regimes, these plots form closed loops whose shape and size describe the reaction dynamics. The animation below illustrates this behaviour as a function of k₃, the rate constant for the autocatalytic step (A + X → 2X + 2Z).

Phase-space oscillations for Br^-, HBrO₂, and Ce⁴⁺ as a function of k₃.

Sustained Oscillations in Open Systems

In a closed system, BZ oscillations typically persist for 1–2 hours until the organic substrate (e.g., malonic acid) is consumed. However, indefinite oscillations can be achieved by maintaining constant concentrations of bromate and malonic acid through continuous reagent inflow and product outflow in a CSTR (continuously stirred tank reactor) [7]. Remarkably, sustained oscillations are possible even in a minimal system containing only acidic bromate, bromide, and a metal-ion catalyst, known as the minimal bromate oscillator [7]. Mathematically, this steady-state is modelled by setting the time derivatives of the reactants to zero:

\[ \begin{aligned} \frac{d[\ce{BrO3-}]}{dt} = & 0\\[6pt] \frac{d[\ce{MA}]}{dt} = & 0 \end{aligned} \]

As shown earlier, the autocatalytic step, controlled by k₃, remains the dominant parameter even under fixed [BrO₃⁻] and [MA]. Varying k₃ dramatically alters both the amplitude and period of the oscillations. The animation below illustrates this sensitivity, revealing how small numerical changes can modify the dynamic behaviour.

Oscillatory behaviour in a continuously stirred tank reactor (CSTR) for different values of k₃.

Finally, another key parameter is f, the bromide regeneration factor. Under steady-state conditions, oscillations are obtained when f is between approximately 0.3 and 1.5. Small variations in f influence the oscillation amplitude and periodicity, since this factor ultimately controls the regeneration of Br⁻ and the completion of the reaction cycle.

Effect of the bromide regeneration factor (f) on the oscillatory behaviour in a continuously stirred tank reactor (CSTR).

Final Thoughts

The Belousov–Zhabotinsky reaction remains a fascinating example of how simple chemical systems can exhibit non-linear behaviour. The Oregonator captures the essence of these dynamics, showing how small changes in key parameters such as k₃ or f can shift the system from steady to oscillatory states.

References

1. Lotka, A.J. (1920). Undamped oscillations derived from the law of mass action. Journal of American Chemical Society, 42(8), 1595-1599. https://doi.org/10.1021/ja01453a010
2. Bray, W.C. (1921) A periodic reatio in homogeneous solution and its relation to catalysis. Journal of American Chemical Society,43(6), 1262-1267. https://doi.org/10.1021/ja01439a007
3. Zhabotinsky, A.M. (1964) Periodic liquid phase reactions. Proc. Ac. Sci. USSR, 157, 392-395.
4. Prigogine, I.; Lefever, R.J. (1968) Symmetry Breaking Instabilities in Dissipative System. II. The Journal Chemical Physics, 48, 1695-1700. https://doi.org/10.1063/1.1668896
5. Field, R.J., Koros, E. and Noyes, R.M. (1972) Oscillations in Chemical Systems. 2. Thorough Analysis of Temporal Oscillation in Bromate-Cerium-Malonic Acid System. Journal of the American Chemical Society, 94, 8649-8664. https://doi.org/10.1021/ja00780a001
6. Field, R. and Noyes, R.M. (1974) Oscillations in Chemical Systems. IV. Limit Cycle Behavior in a Model of a Real Chemical Reaction. The Journal of Chemical Physics, 60, 1877-1884. https://doi.org/10.1063/1.1681288
7. Epstein, I.R. and Showalter, K. (1996) Nonlinear Chemical Dynamics: Oscillations, Pattern, and Chaos Journal of Physical Chemistry, 100, 13132-13147. https://doi.org/10.1021/jp953547m
8. Robayo, I. and Ágreda, J. (2017) Ultraviolet–Visible Kinetic Study of the Uncatalyzed Bromate Oscillator with Phenol by Multivariate Curve Resolution International Journal of Chemical Kinetics, 60, 1877-1884. https://doi.org/10.1002/kin.21095

If you want to cite this blog post, use:
Robayo, I. (2025). The Belousov–Zhabotinsky Reaction: Modelling the Oregonator. Available at: https://electrochemeisbasics.blogspot.com/2025/11/the-belousovzhabotinsky-reaction.html [Accessed Date Accessed].