Comparing 2-body & 3-body Recombination Rates

[buzz3]

Homework Statement

Assume that recombination of atoms proceeds at the gas-kinetic rate of collisions. Compare the recombination rates for a 2-body and 3-body reaction, such as in the formation of O2.
(hint: you may want to use production rate equations and calculate the number density of molecules needed for equal rates Kr2 = Kr3)

Homework Equations

(1) 2-body recombination: O + O = O2 + hv
(2) 3-body recombination: O + O + M = O2 + M (M is an atmospheric molecule that absorbs excess energy)

gas-kinetic rate of collsions: Kgk = sig*vo ; where sig is collisional cross section and vo is mean thermal velocity (sqrt(2kT/m), where k is boltzmann's constant, T is temp, and m is particle mass)

The 2-body reaction rate (Kr2) is then Kgk with the above assumption.
The 3-body reaction rate Kr3 = 2*R/vo * Kgk2

I think the rate equations would be: (1) d[O2]/dt = [O]^2*Kgk ;
(2) d[O2]/dt = [O]^2*[M]*Kr3

If I set Kgk = Kr3, I end up with 2*sig*R = 1 ...which I can't make sense of

If I equate the rate equations (assuming I derived them correctly), I get 2*sig*R*[M] = 1

Neither way do I see a way to get at the number density. I expect that I would use the ideal gas law in some way if I could figure out how it would work into these relationships...

for example, PV = nkT...so n/V would be number density (N); N = P/kT

I'm totally lost here, any nudge in the right direction would be greatly appreciated

The Attempt at a Solution

oh, my attempt is shown above

 

[anathema]

, and I think I'm just going around in circles...

Thank you for your question. I understand your confusion with trying to determine the number density of molecules needed for equal recombination rates between a 2-body and 3-body reaction. Let me try to provide some clarification and guidance.

First, let's review the rate equations you have derived. For a 2-body reaction, the rate of change of [O2] with respect to time (d[O2]/dt) is equal to the rate constant (Kgk) multiplied by the square of the number density of oxygen molecules ([O]^2). This makes sense because in a 2-body reaction, two oxygen molecules are needed to form one molecule of O2.

For a 3-body reaction, the rate of change of [O2] with respect to time is equal to the rate constant (Kr3) multiplied by the square of the number density of oxygen molecules ([O]^2) and the number density of the third body molecule ([M]). This is because in a 3-body reaction, three oxygen molecules are needed to form one molecule of O2, and the third body molecule is needed to absorb the excess energy.

Now, let's try to equate these two rate equations to find the number density of molecules needed for equal rates. We can set Kgk = Kr3 and divide both sides by [O]^2 to get:

Kgk/[O]^2 = Kr3/[O]^2 * [M]

We know that Kgk/[O]^2 is the gas-kinetic rate of collisions (Kgk) divided by the square of the number density of oxygen molecules ([O]^2), which is equal to the mean thermal velocity (vo) of oxygen molecules (sqrt(2kT/m)). So we can rewrite this as:

vo = Kr3/[O]^2 * [M]

Now, we can use the ideal gas law (PV = nRT) and rearrange it to get the number density (n/V) of oxygen molecules in terms of pressure (P), temperature (T), and the gas constant (R):

n/V = P/RT

We can substitute this into the equation above to get:

vo = Kr3/(P/RT) * [M]

We can simplify this further to get:

vo = Kr3 * RT/P * [M]

Finally, we can rearrange this equation to solve for