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

[John Mallinckrodt <ajm@csupomona.edu>]

Historically you will see this referred to as the Loschmidt Gravito-Thermal effect.  Loschmidt argued precisely as you have while Maxwell and Boltzmann argued as John Denker has and as I would as well.  As far as I can tell there are respectable people who still consider this an unresolved controversy.  You can find plenty of material on the web by searching undder some combination of loschmidt boltzmann maxwell gravity gas etc.  One example:

http://bit.ly/ywdmz3

I think it's at least pretty clear that the situation is not as simple as it might seem from your (and Loschmidt's) arguments.  The question is not whether individual molecules moving to lower altitudes (without collisions, of course) will have higher kinetic energy, the question is what does the *distribution* of energies look like at different altitudes.  It's not particularly hard to imagine that molecules moving to higher altitudes simply take their place as a lower velocity component of the *same* distribution that simply has a lower overall density.
 
John Mallinckrodt
Cal Poly Pomona

On Jan 17, 2012, at 6:04 AM, Folkerts, Timothy J wrote:

> 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.
>
>
> _______________________________________________
> Forum for Physics Educators
> Phys-l@carnot.physics.buffalo.edu
> https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l