I cannot wrap my head around a supposed temperature gradient versus total energy gradient paradox for thermodynamic equilibrium of (open space ) gas in gravitational field.
For simplicity, consider ideal monoatomic gas like ideal argon. There is supposed the zero temperature gradient in equilibrium. But how to deal with the constant total energy and altitude dependent kinetic energy of molecules between collisions with nonzero vertical velocity projection?
$$\frac{d(E_\mathrm{k} + E_\mathrm{p})}{\mathrm{d}h} = \frac{\mathrm{d}(\frac 12 mv^2 + mgh)}{\mathrm{d}h} = 0$$
With reversible exchange $E_\mathrm{k}$ and $E_\mathrm{p}$ and for the statistical means, it should be like
$$\frac{\mathrm{d}(\frac 32k_\mathrm{B}T + mgh)}{\mathrm{d}h}=0$$
In case of zero temperature gradient, how comes descending molecules do not convert their potential energy to kinetic one and inject thermal energy to lower layers (and vice versa)? How comes it does not cause temperature gradient until the total mean molecular energy gradient is zero?
I feel I am missing something and that density gradient somehow compensates the effect of molecular energy gradient at zero temperature gradient, but I do not see how.
A detailed kinetic theory analysis is very probably above my abilities. I have discussed in in chats in both CH and PH SE sites at:
CH SE: density-gradient-vs-entropy-of-mixing
chat discussion-between-poutnik-and-theorist
and
PH SE: what-is-the-reason-of-dt-dh-0-in-the-gas-column
chat discussion-between-poutnik-and-giorgiop
I have also searched site:stackexchange.com for related Q/A about gas equilibrium and gravitationalfield, but I have not found a topic addressing it unless I have missed it.
PH Se: in-a-gravitational-field-will-the-temperature-of-an-ideal-gas-will-be-lower-at considers Earth atmopsphere, which is not at equilibrium ( I have meteorological background from my days of an enlisted airfield meteorologist so I am aware of dry-adiabatic gradient 0.0098 K/m.)
Best Answer
In equilibrium, there's no temperature gradient, no kinetic energy gradient, and no heat transfer. But like most results in kinetic theory, it's unintuitive unless you follow what each particle in detail.
First, let's explain why there's no kinetic energy gradient. Think about the particles that start low and end up high. Since it costs energy to go up, doesn't that mean that the particles that end up high should be moving slower? No, because particles that were originally moving slowly don't have enough energy to get up high in the first place. The only particles that get high are those that got an unusually high kinetic energy through a lucky collision. As they go up, they lose that extra kinetic energy to potential energy, arriving at the top with the typical amount of kinetic energy.
(Of course, in reality there's some distribution of kinetic energies, but this logic holds for each part of the distribution. Suppose you had some mix of particles with kinetic energy $0$, $1$, $2$, $3$, ... at the bottom. The particles with kinetic energy $0$ don't make it up. The particles with kinetic energy $1$ arrive with kinetic energy $0$. If you work through it quantitatively, you end up with exactly the same distribution.)
Second, let's explain why there's no heat flow. The point is that the density at each level stays the same in equilibrium. The particles falling from the "high" level to the "low" level pick up a lot of kinetic energy, so they arrive at the "low" level with more kinetic energy than most of the particles already there. But at the same time, particles are leaving the "low" level to go up to the "high" level, and as we just argued, the only particles that can do this are the most energetic ones. So in equilibrium, you predominantly have particles with unusually high total energy going in each direction, but since the flow of particles balances, there is no net heat flow from up to down.
By the way, as you suspected, the existence of a density gradient is essential to maintain equilibrium. That's because all of the particles at the high level can fall to the low level, but only the highest energy particles at the low level can go up to the high level. For the rates to balance, there need to be more particles at the low level, which is precisely what happens in equilibrium.