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

If I understand the point of your question properly, you are asking if a tall column of gas can have an adiabatic lapse rate if the top and bottom are held at fixed temperatures.

I would say the situation becomes ill posed. Fixing the temperature at the top of the column at any value other than that consistent with the adiabatic lapse rate (and a given temperature at the base) would cause a heat flux at the top and the situation is no longer adiabatic. The actual lapse rate would have to eventually settle down to zero if the top and bottom temperatures were held at the same value.

Bob at PC

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of Folkerts, Timothy J
Sent: Tuesday, January 17, 2012 9:05 AM
To: Forum for Physics Educators
Subject: [Phys-l] Temperture profile in a graviational field

Here is an interesting question that I have been seeing in the context of
climate science and the "Greenhouse effect". (I may have more points to
make later, since this topic is interesting and important -- for science and for
society.)

Suppose you have a perfectly insulated column of air. Let's minimize
concerns about IR by making the inner walls of the container highly reflect
and making the gas N2 (which emits/absorbs minimal amounts of IR).
Suppose the column is a few km tall, with the base at the surface of the
earth.

1) What will be the temperature profile? Certainly there is a pressure
gradient and a density gradient in the column. I would say there is a
temperature gradient as well. On a microscopic scale, between every
collision, if the molecule gains altitude it will gain PE and lose KE (ie it will
cool). Any molecule moving downward will warm. On a macroscopic level,
this can discussed in terms of the "dry adiabatic lapse rate".
"potential temperature"
http://en.wikipedia.org/wiki/Potential_temperature

In either case, it is clear to me that the equilibrium condition (both in this
ideal column and in the real atmosphere) would be a temperature gradient
(cooling ~ 10K/km). Do others agree?

2) If this is true, how can this best be squared with the 2nd law of
thermodynamics? If the top and bottom of the column were held at a the
same temperature, there would be a continuous flow of energy from top to
bottom, even though they are the same temperature. Even if the top were
slightly cooler than the bottom, there would be a downward flow. This
would violate a standard statement of the 2nd law, since we have
spontaneous heat from cool to warm.

I've been trying to think of a good way to explain that this is not indeed a
violation. I suspect the best explanation will have to involve the more
fundamental statement of the 2nd law -- that entropy tends to a maximum.
The adiabatic lapse rate leads to an isentropic gas and a constant potential
temperature.

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