If one inserts a thin and stationary horizontal adiabatic wall (well... ok, «insulated wall») at any height L within a gas
column at equilibrium (no net diffusive, radiative or convective heat flows within this column) then the pressure on both sides of the wall integrated over its surface match the weight of the column above.
There is however vigorous random mixing of the molecules up and down the ideal gas
column at equilibrium in a gravity field leading to non-zero pressure and non-zero temperature gradients at equilibrium proven by reasonable experiment and theory of published physicists.
specifically deny that there is a vertical temperature gradient in
the column at equilibrium for any number of particles» is based on confusing two different usages of «temperature.»
• The gas
column at equilibrium is isothermal (as demanded by the First and Second Laws, when they are properly understood and applied).
Not exact matches
At some point in the vertical atmospheric
column KE will equal PE and that is the point of
equilibrium for the AAL.
For the sake of his proof, Dr. Brown assumes (as he and most of us believe, contrary to fact) that
at equilibrium, i.e., in the highest - probability, highest - entropy state, the isolated gas
column exhibits the dry adiabatic lapse rate.
It only proves that the
column is
at equilibrium as defined by dG = 0 (or equivalently dA = 0) throughout.
However, if you measure the temperature of the
column bolometrically from any given height, what you see will always be isothermal
at equilibrium.
Even though I can't imagine gravity functioning as a Maxwell Demon, even though Caballero in section 2.17 both states and leaves as a student exercise the proof that the thermodynamic
equilibrium state of a vertical
column of gas is isothermal, there has been a lot of confusion and strange assertions about a gas arriving
at a state because of bulk transport that sorts out temperature differences approximately adiabatically (neglecting conduction), but that is somehow thermodynamically stable without transport and with conduction in the end.
The
column could come to
equilibrium only
at the point in which the earth surface is the same temperature as the lapse rate effected temperature of the air immediately above it.
This ideal gas mixing continues throughout the
column to
equilibrium at max.
I agree that a
column of Ideal Gas in the Dr. Brown's
column above will be isothermal
at equilibrium.
The saturated partial pressure of water vapor
at the surface pv (Ts)(4) is determined by surface temperature and, as it is in hydrostatic
equilibrium, equals the weight of water vapor in the static
column.»
In hydrostatic
equilibrium, air pressure
at any height is equal to the weight of air in the atmospheric
column above that height.