Here I outline what I believe to be a conclusive argument showing that the "kinetic" temperature is constant with height for the canonical ensemble of an ideal gas in a gravitational field. It is not taken from any "authoritative" source, hence, I make the reservation for errors. Recall the Boltzmann factor
which is the relative probability to find a single particle at height h with speed v when the system has reached equilibrium, that is, maximum Gibbs entropy. (Notice that this kind of factorization can not be done for the micro-canonical ensemble.) Now I define the "kinetic temperature" in the following way:
The reason for this notation is of course that there already exists a temperature T pertaining to the system as a whole. Let N be the total number of particles, using the Boltzmann factor the kinetic temperature can be calculated as follows:
From this point it is very easy to show that Tk is independent of h, which I leave as an exercise. It can also be shown that with this definition we have that
"there already exists a temperature T pertaining to the system as a whole"
SvaraRaderaIt looks to me like you've just ASSUMED a constant temperature T (in eqn. 1 above) throughout the gas.
Reif's "Fundamentals of Statistical and Thermal Physics" (1965), page 211 says,
"...P(z) = P(0) exp(-mgz/kT) 6.3.20
i.e., the probability of finding a molecule at height z decreases exponentially with the height. The result (6.3.20) is sometimes called the 'law of atmospheres,' since it would describe the density variation of the air near the surface of the earth if the atmosphere were at a constant temperature (which it is NOT)."
First of all, we are not (necessarily) talking about the actual atmosphere but a gas assumed to be in equilibrium.
SvaraRaderaI don't think I assume a constant "kinetic" temperature with height by introducing the first formula. There need to be some constant playing the role of T, but if you don't like it we can just change its name to € or something else. Without it the entire enterprise of Gibbs thermodynamics becomes meaningless and the entropy argument would fall with it anyway.