Maxwell's distribution function for mixture of ideal gases

Question

I came across a question demanding relations between the individual distribution functions $f_n$ for the $n$th gas, and the overall $f$ for the entire mixture.

I believe there is no simple way to express this, and that the mixture will behave as one entity, with a nearly time-independent function inexpressible in terms of the original individual functions and that the individual data of each gas, (however herculeanly obtained) will be time-dependent. Please correct me if I am wrong, any useful reading material is welcomed.

To be absolutely clear, my question is: Do the individual components obey Maxwell's function separately and/or do they obey the function as a mixture with it's own properties such as average molecular mass etc?

Answer 1

On your comment "I believe there is no simple way to express this,...", my review of this educational text, for example, to quote:

Additionally, the function can be written in terms of the scalar quantity speed c instead of the vector quantity velocity. This form of the function defines the distribution of the gas molecules moving at different speeds, between c1 and c2, thus...

where the alternate use of a scalar function approach, as defined by Equation (2) in the referenced text, appears tractable. It also can be integrated to arrive at an average speed, for example, as given by Equation (5) in the cited source.

As such, modifying the average value integration to address a weighted average of say an ideal gas mixture may be relatedly facile.