Is there a known relation between the diffusion of a molecule and the diffusion of the reactants?

Question

I have a plasma consisting of elements A and B (which have a similar mass). These elements can react to form molecules AB.

$$ \ce{A + B -> AB} $$

Now if my plasma or gas is expanding based on pressure gradients, I will have some sort of diffusion of these two element concentrations. In fact, this diffusion is the dominant process affecting the concentrations A and B. Based on my spectroscopic analysis, the rate of decay of both elements is similar/the same, so I model them as $$ [\ce{A}](t) = A_0 \exp(-k_0t) \,, \\ [\ce{B}](t) = B_0 \exp(-k_0t) \,. $$ Then the concentration of AB changes with $$ \frac{d[\ce{AB}]}{dt} = k_1 A_0 B_0 e^{-2k_0t}\ - k_2 [\ce{AB}] \,, $$ where $k_2$ is the rate constant for diffusion. The obtained formula for [AB] is then $$ [\ce{AB}](t) = \frac{k_1A_0B_0}{2k_0-k_2} \left(e^{-k_2t} - e^{-2k_0t} \right) \,. $$ This actually fits my data quite nicely, but one thing confuses me: When fitting my data for $[\ce{AB}]$ with this model, the parameter $k_2$ is exactly $2k_0$.

Or rather, they are very close, since the formula has a singularity at $k_2 = 2k_0$. But if the formula is derived with this assumption directly, one obtains $$ [\ce{AB}](t) = (k_1A_0B_0)\, t\, e^{-2k_0t} \,, $$ which also fits my data very nicely.

I don't understand why $k_2 = 2k_0$ and I am worried that I might say something very stupid in my paper when I try to explain this observation. But this is what I am thinking: $k_2$ is the rate of diffusion for AB, which is twice as heavy as A and B. It will have twice the momentum, which could lead to a doubling of the diffusion rate as well. Does this seem like a probable explanation? If not, which other effects will play a role?

Note that the plasma I am looking at is short-lived and changes constantly, which makes it difficult to apply textbook chemistry to it. Basically all signals I am looking at can be modeled as exponential functions or sums of exponential functions, and sometimes multiple processes might be involved. It's a lot of guesswork, and I am worried that I am actually on the wrong track here. But at least intuitively it makes sense to me that the diffusion of a heavier molecule should happen faster.

Since I was asked why I don't use a more complex diffusion/reaction model: Occam's razor, basically. I don't want to assume too much about my plasma. And I fear that a complex model might contain too many free parameters which would lead to overfitting of the data. (If I'm wrong about this approach, feel free to correct me and explain to me how to go about this. What I do not like are downvotes without a comment telling me what I'm doing wrong. None of the downvotes I have received bothered to do this, which is not helpful.)

Answer 1

More a set of queries and comments than an answer.

I don't understand why you make A and B vary with time irrespective of your reaction scheme.

You should really analyse using Fick's diffusion equations with the reaction scheme added and do so for each species. This will have to be a numerical calculation in space and time. (There are well established methods to do this. e.g. Crank-Nicholson). By the symmetry of your experiment you may be able to reduce this to one dimension and time.

You should be able to 'guestimate' the diffusion coefficient from kinetic theory at least their ratio which should be quite accurate.

Your second equation ('new formula') is that for a reaction $\ce{A + B ->[k_1] AB ->[k_1] }$ where both rate constants are the same. In your case there seems to be no reason why the two rate constants are the same, only you know enough about your experiment to explain this :)