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

Sethna has this problem in his "Entropy, Order Parameters, and Complexity" book (recapitulating some of Feynman's reasoning). You can access it here: http://pages.physics.cornell.edu/~sethna/StatMech/EntropyOrderParametersComplexity.pdf ; the particular problem (6.1) is called "Exponential atmosphere", and appears on page 124. I haven't looked at his simulation yet, but the statement of the problem is a nice way of working through some of these issues.


 
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________________________________
From: "Folkerts, Timothy J" <FolkertsT@bartonccc.edu>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Tuesday, January 17, 2012 7:04 AM
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".
http://en.wikipedia.org/wiki/Lapse_rate#Dry_adiabatic_lapse_rate ; and
the "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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