# What coefficients can I use in a Reaction-Diffusion Equation?

I am qualitatively analyzing an abstract chemical reaction

$A + B \rightarrow 2A$

where $A$ and $B$ are some liquid substances that undergo Reaction-Diffusion with a second order reaction (actually my model is much more complicated than that, but this is for simplicity). I have Reaction-Diffusion Equation that models this reaction:

$u = \left(\begin{array}{c}a\\b\end{array}\right), \quad u(0) = u_0, \quad d \in \mathbb{R}_+, k \in \mathbb{R}_+$,

$a_t = d \Delta a + k a b$,

$b_t = d \Delta b - k a b$.

However, I need some specific coefficients $d$ and $k$, and starting densities for $u$ to do my simulations on.

What would be a decent Fermi estimate for diffusion coefficient $d$, reaction rate $k$ and density $u_0$ for reactions present in reality?

In terms of speed, I'm interested in reactions like one of reactions present in the Belousov–Zhabotinsky Reaction-Diffusion model and I just need a very, very rough estimate of values (with physical units) to put in my model so that I keep studying behavior of reactions that might plausibly happen in reality for some specific substances. So far I used $k \approx 1 \frac{dm^3}{kg \cdotp s}$, $d \approx 1 \frac{dm^2}{s}$, and $u_0 \approx 1 \frac{kg}{dm^3}$, but I'm particularly unsure if this ratio of $k$ to $d$ is anywhere near realistic. I know I could just scale time, space, and mass in this equation, but I'm interested in specific value ranges or examples.

• Too little information. For example diffusion coefficients tend to depend on if the system being studied is a gas, liquid or solid. – MaxW Jul 28 '18 at 4:10
• Thanks for the comment, I updated the question. So, let it be liquids. But is that of the order of 1 or more like 1 million? Would you be able to give an example of concrete coefficients for any R-D equation like that? – Xilexio Jul 28 '18 at 4:37
• What you use depends on the specific reaction and then whether or not you use 'dimensionless' values. As an example of generating numerical version of equations see 'From Calculus to Chaos' by D. Acheson (publ. OUP) or for pattern generation 'Essential Mathematical Biology' by N. Britton (publ Springer) . As a start you might consider using concentration is units of, say, millimolar and times in millsec. – porphyrin Jul 28 '18 at 7:04
• Diffusion coefficients for small molecules in water or other mobile solvents are around $10^{-9} \mathrm{dm^2s^{-1}}$, rate constants can vary hugely but say no more than $10^6\mathrm{dm^3/mol/s}$ would probably be big enough and concentrations in the millimolar range. – porphyrin Aug 2 '18 at 20:42
• Thank you for your comments, @porphyrin. If you could write an answer out of them, I'd accept it. – Xilexio Aug 3 '18 at 0:10