Gas with Temperature Gradient

Source: thermodynamics – Is there a pressure gradient in a stationary gas with a temperature gradient? – Physics Stack Exchange

The answer to the posed question is indeed a simple “no”, but to establish this does not require any analysis of internal gas dynamics.

If there is ever an overall pressure gradient, then in the absence of external forces the container will accelerate. (By conservation of momentum the gas within will accelerate in the opposite direction until the pressure gradient is eliminated and eventually reversed, and in the absence of friction this would result in an oscillation, but in any case it won’t be an equilibrium until the forces on the walls are in balance.)

In the equilibrium situation the constant pressure means that the density will (in the ideal gas approximation) be inversely proportional to temperature, and I think that any student familiar with the gas laws would accept that each layer of gas can thus remain in balance with its neighbours (one being cooler and denser and the other hotter and more rarified).

The analysis of how this is derived from kinetic theory (with molecules not being confined to layers etc) is more interesting, but does not appear necessary for an answer to the original question.

With regard to what was probably the really intended question, namely how to reconcile the $n \sqrt{T}$ flux out of each layer with the constancy of $nT$, it might be sufficient to tell a student who is not ready for the full analysis just that the flux of those incoming molecules which interact with the layer does not come just from the neighbouring layers but is a mix from various distances which turns out to give zero net flux when $nT$ is constant.

Source: thermodynamics – Is there a pressure gradient in a stationary gas with a temperature gradient? – Physics Stack Exchange

Light Mill applet — Greg Egan