Rob Ellison: “Molecules have a kinetic energy far in excess of the gravitational potential at 273K.”
When the molecule travels a vertical distance dz, the variation in kinetic energy just is the variation in potential energy. They’re identical. You aren’t entitled to consider only the effect from one and neglect the effect from the other on the velocity distribution at given heights.
“A *factoid* that the environmental lapse rate refutes.”
The environmental lapse rate is affected by radiation, convection, horizontal winds, and latent heat releases. The tropospheric columns are very much out of equilibrium. So, the ELR doesn’t have any bearing on the present problem.
“So unless you have an actual treatment [...]”
Pekka provides an “actual treatment.” He demonstrates that the stationary density gradient of the barometric formula *derives* from — and hence is consistent with — the conditions of local thermodynamic equilibrium (Maxwell speed distributions depending only on temperature) for the isothermal case. The mathematics is very simple and only involves partial derivatives of simple products and exponential functions.
The conclusion is: “The equations (1), (2) and (3) represent the stationarity requirement that particles located in certain volume with certain velocities will at a later moment be replaced by an equal number of other particles which have the same velocities when the influence of gravity on velocity is taken into account. It’s shown that the isothermal atmosphere with Maxwell-Boltzmann velocity distribution and barometric vertical density profile
satisfies this requirement.”
The American Journal of Physics article provides three more derivations of the same result. You can’t just hand-wave away mathematical results that you don’t like.