kinetics - What is the probability of 3 possible products, between two chemical species?

I am not a chemist. I hope I will be specific enough.

Suppose there are two chemical species $\ce{A}$, $\ce{B}$ with the following properties:

• at temperature $t < T_r$, no reaction occurs between $\ce{A}$ and $\ce{B}$ (in any combination).


• at $t\ge T_r$, $\ce{A}$ interacts with itself to create $\ce{A_2}$, $\ce{B}$ reacts with itself to create $\ce{B_2}$, and $\ce{A}$ and $\ce{B}$ are reacting to create $\ce{AB}$.


• $\ce{A_2}$, $\ce{B_2}$ and $\ce{AB}$ are never reacting.

In experiment, we first mix $\ce{A}$ and $\ce{B}$ in temperature $t

1. What amounts of $\ce{A_2}$, $\ce{B_2}$, and $\ce{AB}$ can be expected to be produced?


2. To obtain the amounts, should probability theory be used? E.g., amount of $\ce{AB}$ equals to probability that species $\ce{A}$, $\ce{B}$ will interact ("collide" or similar interpretation).

Assume the rates of the reactions are equal.

Answer

Well, if you assume the rates are known and the reactions' order follows from stoechiometry (e.g. if they are elementary reactions), you can put the chemical kinetics into simple equations:

$$\frac{\mathrm da}{\mathrm dt} = -k_1 a(t)^2 - k_3 a(t)b(t)$$ $$\frac{\mathrm db}{\mathrm dt} = -k_2 b(t)^2 - k_3 a(t)b(t)$$

($t$ here being time, not temperature).

Knowing initial amounts or concentrations $a_0=a(t=0)$ and $b_0=b(t=0)$, you can pretty much integrate the system to find out what happens.

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Edit — solving this system for $k_1=k_2=k_3$ yields the quantities of AA, AB and BB at infinite time to be as follows:

$$aa = \frac{a_0^2}{a_0+b_0}$$ $$ab = \frac{a_0 b_0}{a_0+b_0}$$ $$bb = \frac{b_0^2}{a_0+b_0}$$