# Has any work been done to describe the transient behaviour of a non-uniform ideal gas in an adiabatic process?

Recently, I've come across a discussion where a person believed classical thermodynamics predicted loss of energy in free expansion. After discussing for a while, I found out that that person made a mistake by applying the ordinary adiabatic equation that requires pressure and temperature equilibrium to free expansion, a situation where such equilibrium cannot be assumed. The problem can be solved easily by acknowledging that no collision happens during the expansion and the wall stays stationary after that. However, I also want to study the more general case where adiabatic expansion is not instantly but it happens within $$\Delta t > 0$$. This requires calculating diffusion and the work that the gas does on the moving wall of the container. Unfortunately, this approach faces a lot difficulties:

1. Molecules with different kinetic energy diffuse at different speed.
2. When the wall is moving, it reacts with molecules at different velocity differently. For example, low speed particles won't collide, and particles are more likely to collide and also lose more energy.

As a result, this seems to break every single commonly used equilibrium, including Maxwell-Boltzmann distribution. Well if I let $$\Delta t = 0$$ (the free expansion), the second problem can be resolved, and I keep the Maxwell-Boltzmann distribution, with the caveat that I have to keep track of volumetric concentration distribution at each kinetic energy level.

I've also tried to search for online articles but most do not consider the transient behaviour at all, perhaps due to its sheer complexity. So my question is: Is there any work done on giving a general description of the transient properties of this kind of adiabatic process?

• Have you researched non equilibrium thermodynamics? Commented Apr 29 at 15:10
• @BobD thanks for the suggestion! I've looked into some of the equations regarding mass diffusion and heat diffusion, like Fick's law for example, but I'm unsure if they are equivalent to the kinetic theory in such extreme condition. I noticed that the free expansion is quite pathological though since temperature is undefined for the vacuum so I'm not sure how to treat it. Any help is appreciated. Commented Apr 29 at 15:43
• I'm sorry but i am not conversant in non-equilibrium thermodynamics. For the non quasi-static adiabatic expansion I can only analyze the initial and final equilibrium states. good luck Commented Apr 29 at 15:55
• Are you looking for something like “What lies between a free adiabatic expansion and a quasi-static one?”? See also its references and the work citing it. Commented Apr 29 at 16:59
• @Chemomechanics Thanks. It's very close to what I'm looking for! Thouh they did assume local equilibrium but I think this might be close enough Commented Apr 30 at 3:56