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

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?

• You oversimplify things greatly. It is very unlikely that only one reaction will suffice to describe the kinetics in your plasma. Also the reaction you mention must involve a third scattering body to take away the binding energy or requires a very fast radiative recombination rate. It is also likely that the electrons in your plasma are responsible for exciting (part of) your molecules. You'll have to consider a more complex mechanism to predict the production of the excited states of CaCl and observed fluorescence. – Paul Nov 22 '18 at 10:13
• The points you mention (third scattering body, fast radiative recombination) are exactly the kind of thing that I wanted to know more about. This is why I am asking the question in the first place. As for the excitation, I would have thought that this can be regarded as a separate step. First, I need to know how many molecules even form before I can look at their emission rate. – PoorYorick Nov 22 '18 at 11:40
• The point about kinetics is that - in general - it involves coupled differential equations, so you should look at all relevant processes simultaneously. I think you're best of consulting a text book on the topic such as Laidler's Chemical Kinetics. Articles that might be of interest are J. Photochem. 25, 389 (1984) or Phys. Rev. E 57, 4684 (1998). – Paul Nov 22 '18 at 12:34
• Thank you for the literature recommendations. In the beginning of the first article, three reaction equations are presented. Then they say that based on these, the intensity of an O2 emission should be I = k1[O]^2[M]/(1+k2/k3 [N2]). This is very interesting, can you maybe help me understand what they are doing there? I mean, this seems to be the type of derivation that I have in mind - start with something complex, but end up at something simple. – PoorYorick Nov 22 '18 at 15:47
• They assume equilibrium conditions so that the reaction rate equals zero. Taking the expression for the rate of formation of excited oxygen molecules and setting the rate to zero allows to write the equilibrium concentration of O2 in terms of the other species. – Paul Nov 22 '18 at 17:43

$$\ce{A + B ->[k1] AB^{*}} \\ \ce{AB^{*} ->[k2] A + B} \\ \ce{AB^{*} + M ->[k3] AB + M} \\ \ce{AB^{*} + M ->[\gamma k3] AB(A) + M} \\ \ce{AB(A) + M ->[k4] AB(X) + M} \\ \ce{AB(A) ->[k5] AB(x) + h\nu} \\$$ 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: $$I \propto [A] [B]$$