Which reaction rate formula is suited for diatomic molecules in a plasma?

Question

I am an experimental physicist, so I unfortunately do not know much about chemistry except the basics. For my current study, I am investigating diatomic molecule formation in plasma. These diatomic molecules are often not stable, but they get excited and emit light, which makes them interesting for spectroscopy.

What I want to know is: If I have atoms, ions and electrons in my plasma, and I have some temperature T, which model for the molecular reactions would best describe this system? What kind of reaction rate formula would I need?

To give a specific example and to explain my current assumptions: In plasma of samples containing Ca and Cl (among other elements), I get emission from the diatomic molecule CaCl. I assume that the concentrations correlate with the stoichiometry of the sample. Now I want to know the amount of CaCl that forms in the plasma, and how it relates to the concentrations in the sample. I currently assume that the reaction can be described like this: $$\ce{Ca + Cl -> CaCl}$$ So it should be very simple, as every element is already present either as an atom or as an ion. (Note that I do not really understand how the ions play into this, I'm assuming that the reaction is the same but that an electron is captured during the reaction - if someone could elaborate on this, I would also be grateful.)

As for the reaction rate, I have found the following rate law: $$ r = k [A]^m [B]^n $$ Where m and n are integers and [A] and [B] are the concentrations of two reacting species A and B. What I don't understand is the physical interpretation of m and n - I have a feeling that for single-element reactants like the ones in my case, they should both be 1. Is that correct, or is there a way these could be wildly different, and maybe even non-integers? Because my experimental results can best be described by this formula if m and n are non-integers (m = 0.6 and n = 1.3, for example). But that confuses me a lot and makes me think that maybe this type of reaction rate formula might not be suited for reactions of atoms/ions in a plasma.

I have found also another formula for the bimolecular reaction on a surface (called the Langmuir–Hinshelwood mechanism): $$ r = k \frac{K_1[A]K_2[B]}{(1+K_1[A]+K_2[B])^2} $$ While I have no idea how this would relate to my situation, the results I get from this formula are quite nice, and I do not have to change anything from integers to non-integers. So maybe this could be a suitable explanation instead?

I should note that, if I say "it fits my experiments", I mean that I have done several experiments with different concentrations of Ca and Cl in my sample, and I try to fit them all by saying that the intensity of my CaCl emission should be proportional to the reaction rate, and the reaction rate parameters have to be the same in each case. There are a lot of assumptions here, but since the plasma itself is very complex, this is the best I can do for now.

Thank you very much for your help!

EDIT: After reading some of Paul's suggestions, my understanding is now that I have to think about every possible reaction that might take place, and that I have to distinguish between the excited states CaCl(A) and CaCl(B) and the ground state CaCl(X): $$ \ce{Cl + e- <=> Cl-} \\ \ce{Cl <=> Cl+ + e-} \\ \ce{Ca <=> Ca+ + e-} \\ \ce{Ca + Cl <=> CaCl(A)} \\ \ce{Ca + Cl <=> CaCl(B)} \\ \ce{Ca+ + Cl- <=> CaCl(A)} \\ \ce{Ca+ + Cl- <=> CaCl(B)} \\ \ce{CaCl(A) + M <=> CaCl(X) + M} \\ \ce{CaCl(B) + M <=> CaCl(A) + M} \\ \ce{CaCl(B) + M <=> CaCl(X) + M} \\ \ce{Ca+ + Cl + M <=> CaCl+ + M} \\ \ce{Ca + Cl+ + M <=> CaCl+ + M} \\ \ce{CaCl+ + e- <=> CaCl(A)} \\ \ce{CaCl+ + e- <=> CaCl(B)} \\ \ce{CaCl(A) -> CaCl(X) + h\nu} \\ \ce{CaCl(B) -> CaCl(X) + h\nu} \\ $$ This just seems ridiculously complicated, if you ask me. (And I have not even started taking into account that Ca also reacts with O to CaO.) Surely this can be simplified to some extent? Some of these reactions are probably very unlikely?

But so for each species, I would then set up a reaction rate based on the equations above: $$ \frac{d[\ce{Cl}]}{dt} = - k_1[\ce{Cl}][\ce{e-}] + k_1'[\ce{Cl-}] - k_2[Cl] + k_2'[\ce{Cl+}][\ce{e-}] + \dots \\ \text{etc.} $$ Then I can solve this set of equations either numerically or by simplifying something.

Am I missing something here? Are the equations correct? Do you think they can be simplified in some way?

Answer 1

After some further research, I have decided on the following solution. First, let us define the reactions that we assume are involved in the formation of diatomic heteronuclear molecule AB:

$$ \ce{A + B ->[k1] AB^{*}} \\ \ce{AB^{*} ->[k2] A + B} \\ \ce{AB^{*} + M ->[k3] AB + M} \\ \ce{AB^{*} + M ->[\gamma k3] AB(A) + M} \\ \ce{AB(A) + M ->[k4] AB(X) + M} \\ \ce{AB(A) ->[k5] AB(x) + h\nu} \\ $$ Here $\ce{AB^{*}}$ is a precursor of molecule AB that dissociates easily and needs to relax by collision with M to some form of AB, which may be excited or not. A fraction $\gamma$ of these collisions will produce AB(A), which is molecule AB in the electronic state A. This can either collide with another species M again, or it can emit a photon. In both cases it transitions to the ground state AB(X). Only the last transition is the one that produces my molecular emission, so that is the one that interests me. Using the defined reaction constants, I can describe the formation of $\ce{AB^{*}}$ and AB(A) this way: $$ \frac{d[AB^*]}{dt} = k_1 [A] [B] - k_2 [AB^*] - k_3[AB^*]\\ \frac{d[AB(A)]}{dt} = \gamma k_3 [AB^*] [M] - k_4 [AB(A)] [M] - k_5 [AB(A)] $$

(Note that [M] represents different concentrations depending on which species is most dominant in each of these terms, i.e. [M] cannot be factored out.) Finally, I can make the assumption that these reactions are very fast in comparison to the changes in my plasma, so that I have a quasistatic process. This needs to be validated by actual experiments, but let's assume it for now. Then we can say that the concentrations do not change, i.e. $\frac{d[AB^*]}{dt} = \frac{d[AB(A)]}{dt} = 0$, and we get: $$ [AB^*] = \frac{k_1 [A] [B]}{k_2+k_3} \\ [AB(A)] = \frac{\gamma k_3 [AB^*] [M]}{k_4 [M] + k_5} = \frac{\gamma k_1 k_3 [M]}{(k_2 + k_3)(k_4 [M] + k_5)} [A] [B] $$ Now, since my intensity is proportional to $k_5 [AB(A)]$, I can write my intensity as: $$ I \propto [A] [B] $$

Lo and behold, it's the formula I started with. The difference is that I now know that it's not a single reaction rate that is involved here, but multiple ones, which intertwine with one another.

What this exercise also taught me is that the reaction orders m and n are really based on the stoichiometry of the reactions, i.e. if I need 1 Ca and 1 Cl then m and n will be 1. This has been something that has been very opaque in the literature I've read (in which it was often emphasized that m and n can be fractions as well, but it was not mentioned under which circumstances).

Other than that, I guess I'm a bit bummed out that nobody answered my question.