Mean Free Path for a Mixture of Gases

[Egret]

So this is a fairly simple conceptual question: can you estimate/compare the mean free paths for individual components of a mixture of gases?

I'm primarily looking at the equation given here and the information accompanying it.

Consider the case where we have several individual, separate pure samples of different gases, all at the same pressure and temperature and containing the same number of moles. We can estimate the relative values of their mean free paths by comparing their molecular sizes. In this situation it's easy to recognize that the gas particles in a sample of CO₂ will have a shorter mean free path than those in a sample of helium.

However, if we mix the gases together, can we still make such a comparison when they're now interacting with particles of different size and velocity?

I ask because the equation derived at HyperPhysics (linked above) appears to be based on the Maxwell speed distribution, which I'm not sure is valid for a mixture of gases.

So if I have a chamber containing one mole each of several gases in a mixture, can I still compare the gases' individual mean free paths and conclude that, even in a mixture, CO₂ would have a shorter mean free path than He?

I apologize if this is obvious or something, I'm a bit out of my league trying to examine the math and concepts involved in determining mean free path.

Thanks for any replies!

 

[Khashishi]

There's no fundamental difference in the physics for a mixed gas. It's just messier. You have multiple different kinds of collisions to consider, and you just add up the collision rates (not their mean free paths!) If you consider the situation with 50% CO2 and 50% He (by molarity), then the possible collisions for CO2 are
CO2 + CO2
CO2 + He
and the possible collisions for He are
He + He
He + CO2
CO2 + He and He + CO2 are the same thing, and have the same rate. But CO2 + CO2 is slower than He + He, so the total collision rate for CO2 will be slower than He.
The mean free path is approximately the mean velocity divided by the collision rate. Of course, the mean velocity of CO2 is slower than He. Overall, the mean free path of CO2 will be smaller than for He, which is the intuitive answer.

 

[Egret]

Thanks Kashishi!

Laying out the collision possibilities like that really helped, and thanks for clarifying the theory too. I was just concerned that maybe the explanations I'd seen wouldn't apply to a mixture due to some factor I'm unaware of.

 

[cjl]

Without going back and reviewing my textbooks, I seem to remember that mean free path isn't really species-dependent in a mixed gas unless it is very rarefied, and instead, its the collision rate that varies by molecular type (higher collision rates for lighter species, and vice versa).

 

[Egret]

Without going back and reviewing my textbooks, I seem to remember that mean free path isn't really species-dependent in a mixed gas unless it is very rarefied, and instead, its the collision rate that varies by molecular type (higher collision rates for lighter species, and vice versa).

Interesting, this is the kind of nuance I was wondering about. Could you check? I don't have access to any books that actually address this topic.

 

[cjl]

Sorry it took so long to reply again - I've been busy and didn't have time to dig into my references. The more complete answer is somewhat in between. If all the molecules have very similar collisional cross sections, then the mean free path will be basically species independent and the collision rate is indeed what varies. If the cross sections do vary somewhat though, the collision rate will slightly change due to the different molecular sizes. The collision rate still varies significantly between species though - probably more than the mean free path does, but the mean free path will show a slight species-dependence as well (with a longer mean free path for smaller molecules/atoms).

Note that size and molecular weight aren't necessarily correlated - a He/H2 mixture is a good example of this. He is smaller, but H2 is lighter weight, so the mean free path of the hydrogen will be shorter despite the higher average molecular velocity (resulting in a significantly higher collision rate).

For normal gases, molecular diameter only varies by about a factor of 3 or so at most between different species, so there's still much more possible variability in molecular velocity, but it's still interesting to think about. Id you wanted to treat the problem in detail, you would need to examine each possible collision separately - if you had 3 species, you'd have to look at the collision cross section for species A with itself, for species A with species B, and for species A with C, and then repeat the process for B and C (and then sum up all the results to get the final answer). In each case, the effective collision cross section will be proportional to the square of the sum of the diameters of the molecules in the collision, and the collision rate for any given collision type will depend on the collision cross section, the velocity of the molecules of that type, and the number of molecules present in the volume. I can try to go into the more detailed math if you'd like, but hopefully this gives a good overview at least...