Temperature Variations of Ideal Gas in Gravitational Field

[mma]

Could somebody tell me, how temperature of an ideal gas varies on height in homogeneous gravitational field in equilibrium?
I mean a gas column perfectly isolated from its environment.

 

[mma]

The interesting in this question is, that a thermodynamic system in equilibrium has only one temperature: T=dU/dS, so our gas column has only one temperature.
But, if we divide our gas column into horizontal layers, then considering these layers as thermodynamic systems, they will have different temperatures, because the average speed of the molecules is greater on the lower layers as in the upper ones. It is a necessity, because every molecule moving upward loses from its speed.
On the other hand, neighboring layers are connected with each other thermally, and therefore they must have equal temperature in equilibrium, so we come again to the other consequence, that our system has only one temperature.
This is a contradiction. What is the solution?

 

[pervect]

Could somebody tell me, how temperature of an ideal gas varies on height in homogeneous gravitational field in equilibrium?
I mean a gas column perfectly isolated from its environment.

In non-extreme situations (i.e. planetary atmospheres), where there is also very little heat conduction, you can use the approximation of the adiabatic atmosphere.

This uses the "adiabatic gas law" for the rate of cooling as the gas expands, plus the usual hydrodynamic equilibrium equations.

See for instance

http://farside.ph.utexas.edu/teaching/sm1/lectures/node56.html
http://daphne.palomar.edu/jthorngren/adiabatic_processes.htm

If you are interested in exotic situations, relativistic effects may become important.

 

[mma]

If you are interested in exotic situations, relativistic effects may become important.

No, thank you, the non-relativistic approach is enough for me. But this adiabatic approximation seems too heuristic for my taste.
I've found an article for the single-particle distribution for an ideal gas in a gravitational field:
http://gita.grainger.uiuc.edu/IOPText/0143-0807/16/2/008/ej950208.pdf"
But how can I calculate the temperature distribution from this?

 

[mma]

OK. Then what about this? :
http://www.iop.org/EJ/abstract/0143-0807/17/1/008"

If a vertical column
of an adiabatically enclosed ideal gas is in thermal
equilibrium, is the temperature the same throughout
the column or is there a temperature gradient along
the direction of the gravitational field? According to
Coombes and Laue, there are two conflicting answers
to the above question:
(1) The temperature is the same throughout because the
system is in equilibrium.
(2) The temperature decreases with the height because
of the following two reasons.
(a) Energy conservation implies that every
molecule loses kinetic energy as it travels
upward, so that the average kinetic energy of
all molecules decreases with height.
(b) Temperature is proportional to the average
molecular kinetic energy.
Coombes and Laue concluded that answer (1) is the
correct one and answer (2) is wrong.

What is surprising, that concrete mathematical analysis also shows that (1) is the correct answer in the thermodynamical limit.
Isn't it interesting?