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



On 01/17/2012 07:04 AM, Folkerts, Timothy J wrote:
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.

It cannot be squared with the laws of thermodynamics.

I've been trying to think of a good way to explain that this is not
indeed a violation.

It is 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.

In equilibrium, the gas will have constant temperature ... not
constant "potential temperature" but rather constant plain old
temperature. This is quite a fundamental result, flowing directly
from the definition of equilibrium, the definition of temperature,
and the specification of the physical situation (i.e. the fact
that various parcels of air can equilibrate by exchanging energy).
See below(*).

The _stratosphere_ is more-or-less in equilibrium and does indeed
exhibit constant temperature.

Meanwhile, the troposphere is not in equilibrium. It is more nearly
adiabatic than isothermal, because it is being vigorously _stirred_.
Thunderstorms contribute a great deal to the stirring, and it is
no coincidence that the height of the typical thunderstorm is
comparable to the height of the tropopause. This is discussed in
Feynman volume II chapter 9 : "Electricity in the Atmosphere".

=================

(*) Here's the proof that equilibrium is isothermal:

Assume the system as a whole is isolated from the rest of
the universe.

Divide the system in to two parcels, #1 and #2. Assume the
parcels can exchange energy, and have been exchanging energy
long enough to reach thermal equilibrium. Now consider a
fluctuation that moves energy from one parcel to the other:

ΔE1 = - ΔE2 by conservation of energy
ΔS1 = - ΔS2 because (a) the total entropy
can never decrease, and (b) at
equilibrium it cannot increase,
because it is already maximal.

Therefore dS1/dE1 = dS2/dE2 in equilibrium.

This quantity dS/dE is obviously important. It is so important
that if it didn't already have a name, we would need to give
it a name. In fact it is conventionally called the inverse
temperature β. For nonzero β the temperature is defined via
kT = 1/β.