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Re: [Phys-l] Temperture profile in a graviational field



Date: Tue, 17 Jan 2012 16:43:07 -0700
From: John Denker <jsd@av8n.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Subject: Re: [Phys-l] Temperture profile in a graviational field
>
> On 01/17/2012 03:46 PM, Christopher M. Gould wrote:
> > mass transport equilibrium .... Here, with an imposed
> > gravitational potential, air at the top of a column will be
> > colder than at the bottom.
>
> <hypothesis>
> Let's explore that idea.
>
> To make it more interesting, set up a copper rod next to the
> air column. There will be no mass transport in the copper,
> just thermal conductivity, so it will be isothermal.

Well, no, not quite. There _is_ mass transport in the copper. You
know them as electrons and in principle I would assume that there is
a lapse rate for them as well as air molecules, but owing to their
degeneracy, my guess is that the purely electronic lapse rate would
be insignificant compared to the lapse rate in air. This does not
affect your point which was:

> The rod and the air column have the same temperature at the
> bottom, since they are both in contact with the heat bath
> at that point.
>
> If they have a different temperature at the top, you can
> run a heat engine using the difference. Voilà! Perpetual
> motion machine.
> </hypothesis>

No, 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, and hence there will be no further need for heat to
flow. End of engine.

The same thing happens if you take two (tall) columns of different
gasses and open the top and bottom to allow flow between them. If
before connecting them equilibrium was reached so that there was a
temperature difference between the tops and bottoms -- but different
between the two -- then when connecting them there will initially be a
flow of gasses to equilibrate pressures at the top and bottom (a
one-shot effect, not steady state) and then there will be thermal
diffusion (again, one-shot, not steady state) until in steady state
there will be a new temperature difference between the top and bottom.
End of engine.

_________________
Christopher M. Gould 213-740-1101 Dept. of Physics & Astronomy
gould@usc.edu Univ. of Southern California
http://physics.usc.edu/~gould/ Los Angeles, CA 90089-0484