Velocity persistence in Chapman-Enskog theory

[Dank matter]

Hi guys!

My question is about the diffusion coefficient of the mixture of gases. Consider two gases in thermal equilibrium (Maxwell velocity distribution) having different densities n₁ and n₂. Let's suppose that the molecules are rigid spheres with diameters d₁ and d₂ and masses M and m respectively. As far as I am concerned, the most exact expression for the diffusion coefficient can be obtained via Chapman-Enskog approach and for the case described above the first approximation gives [Chapman, Cowling, 1970] $$ D_{12} = \frac{3\pi}{32(n_1 + n_2)\sigma_{12}} \sqrt{\frac{8kT}{\pi\mu}}\, ,$$ where $\sigma_{12} = \pi(d_1+d_2)^2/4$ is the cross section of the collision and $\mu = mM/(m+M)$ is the reduced mass.

If $M \gg m$ the diffusion coefficient becomes independent of M. However, in Meyer's approach the diffusion coefficient depends on the persistence factor $$D_{12} \sim \frac{1}{1-\theta} \frac{1}{(n_1+n_2)\sigma_{12}}\sqrt{\frac{8kT}{\pi M}}. $$ For rigid spheres the average persistence factor is [Jeans, 1967] $$\frac{1}{1-\theta} = \frac{3}{4}\frac{M}{\mu},$$ so the diffusion coefficient grows as $\sqrt{M}$ with the increase of M.

I would (naively) expect that Meyer's corrected formula yields a better description, because particles with an enormous mass will travel through the second gas almost without any interruption. On the other hand, of course, their average thermal velocity goes to zero with the growth of M, but I am not sure whether those two effects compensate each other according to Chapman-Enskog formula. Thus, I wonder if the effect of velocity persistence is accounted for in Chapman-Enskog approach? I know that there are next-order corrections to this formula, but at least the second correction for the case $n_1 \ll n_2$ is also independent of M when $M \gg m$ and according to [Chapman,Cowling, 1970] the third correction is very hard to compute (and I do not expect it to influence the result much). I tried to follow the derivation of the Chapman-Enskog formula for the diffusion coefficient, but this is not that easy

I hope my question is clear. I would be very grateful if you could help me with that.

 

[eljose79]

I would like to clarify some points regarding the diffusion coefficient in the case of a mixture of gases with different densities and masses.

Firstly, in the Chapman-Enskog approach, the diffusion coefficient is derived from the Boltzmann equation, which takes into account the molecular collisions between the two gases. This means that the effect of velocity persistence is already accounted for in this approach.

Secondly, in the Chapman-Enskog formula, the diffusion coefficient depends on the average thermal velocity of the molecules, which is inversely proportional to the square root of the molecular mass. This means that as the mass of one of the gases increases, the average thermal velocity decreases, leading to a decrease in the diffusion coefficient.

However, in Meyer's approach, the dependence of the diffusion coefficient on the persistence factor leads to an increase in the diffusion coefficient with increasing mass. This is because the persistence factor takes into account the fact that heavier molecules will travel through the second gas with less interruption.

In conclusion, both the Chapman-Enskog approach and Meyer's approach take into account different factors in determining the diffusion coefficient. While the first approach considers the molecular collisions, the second approach takes into account the effect of velocity persistence. It is difficult to say which approach yields a better description, as it depends on the specific conditions and assumptions made. It would be best to compare the results from both approaches with experimental data to determine which one is more accurate in a given scenario.