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let's distinguish between what happens when we _first_ connect
your air column with your copper column, and the _steady_state_.
If the bottoms were already thermally connected ahead of time, then
it is clear that initially thermal energy has to flow from the top of
the copper into the top of the air, warming it, but after enough energy
flows into the air, an isothermal profile in the air will be set up.
As students of atmospheric physics well know, if the environmental
lapse rate is less than the saturated air lapse rate, the column is
absolutely stable
The general principle is that any system in mass transport equilibrium
has a uniform chemical potential. If there are no external forces then
that means the temperature will be uniform. Here, with an imposed
gravitational potential, air at the top of a column will be colder than
at the bottom.