# Modeling Belousov-Zhabotinsky reaction using ordinary differential equations

Is it possible to model the BZ reaction using first order ODEs? If so what would these equations be?

Or, is there something about the reaction that needs higher orders?

• Yes, I did a project on this once - look up the FKN model, and for more complex schemes, the MBM and GTF models. Along with an understanding of kinetics it's not too difficult. The true mechanism isn't know, these are approximations, and are already quite complicated. – obackhouse May 23 '18 at 15:29

## 1 Answer

Maybe related: I had the chance to perform this very reaction about 3 years ago, for an undergraduate lab course, and there were many things which confounded me back then too. I have a question on the topic here. It contains an experimental procedure as well, in case you are interested.

Is it possible to model the BZ reaction using first order ODEs? If so what would these equations be?

TL;DR: Yes.

Many variants of the reaction exist. The only key chemical is the bromate oxidiser. The catalyst ion is most often cerium, but it can be also manganese, or complexes of iron, ruthenium, cobalt, copper, chromium, silver, nickel and osmium. Many different reductants can be used.

Similarly, there exist many models to describe this reaction. One that is the simplest, and I particular like is called the "Oregonator" (discussed herein). The precise mechanism/model, as far as I know, is still an open question.

We characterise the reaction using the following equations:

$$\ce{A + Y ->[k_1] X } \tag{1}$$ $$\ce{X + Y ->[k_2] P} \tag{2}$$ $$\ce{B + X ->[k_{34}] 2 X + Z} \tag{3}$$ $$\ce{2 X ->[k_5] Q} \tag{4}$$ $$\ce{Z ->[k_6] fY} \tag{5}$$

This model is a simplified version of the FKN model, by applying the steady state approximation, and partial equilibrium approximation to the FKN model.

Overall, $$\ce{fA + 2B -> fP+Q }$$ wherein $f$ is a suitable stoichiometry co-efficient. For any, $f$

$$\begin{cases} (f-1)A + 2B \rightarrow (f+1)P & f > 1 \\ 2B \rightarrow 2fP+(1-f)Q &f < 1 \\ B \rightarrow P &f=1 \end{cases}$$

Now, one can simply write down the rate equations

$$\frac{\mathrm{d}X}{\mathrm{d}t} = k_1 AY - k_2XY + k_{34}BX -2k_5X^2$$ $$\frac{\mathrm{d}Y}{\mathrm{d}t} = -k_1 AY - k_2XY + k_6Z$$ $$\frac{\mathrm{d}Z}{\mathrm{d}t} = k_{34}BX - k_6Z$$

The system evolves with 2 different time scales, and we can expect it to show relaxation oscillations. We can make this more apparent via some mathematical shenanigans, which you may perform as an exercise. I might revisit this later in the day, and write a bit more about them.