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



In keeping with the phys-l tradition of pursuing answers far beyond
what the original question was asking for, let me nitpick a couple
of my previous messages:

On 01/17/2012 04:43 PM, I wrote:

I'm sticking with thermal equilibrium being isothermal, even for a
gas in the presence of the gravitational field.

Correction: If you could look reeeally closely, in thermal equilibrium,
the gas at the top of the troposphere would be colder than at the bottom.
I reckon it would be colder by about 345 picoKelvin, because of the
gravitational red shift, in accordance with general relativity.

This relativistic lapse rate is many orders of magnitude smaller than
the adiabatic lapse rate contemplated in the original question.

The /physics/ of my previous explanation remains correct: if you have
two parcels that reach thermodynamic equilibrium by exchanging energy,
you must have the same ∂S/∂E | N,V in both parcels. The trick is,
when you measure the local E, you must account for the fact that E got
red-shifted during the exchange.

Alas, way back on 12/06/2010 09:49 PM, I wrote:

In a curved spacetime, the thing that is the same everywhere is
Tolman's redshifted temperature T √(-g00).

In the weak-field limit, g00 = -(1 + 2Φ) where Φ is the depth of
the gravitational potential.

Please cross out the word "depth" there. The correct physics is:
Here Φ represents the _gravitational potential_ ... as usual.

Note that Φ increases upward (by definition of "up"). Near the surface
of the earth Φ is negative, and becomes less negative as we move upward.
(In contrast, depth is usually taken to be a positive number that decreases
as we move upward.)

Student's get confused by "depth" all the time. I'm supposed to know
better. Sorry.

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

Here's an amusing tangent, splicing together some recent threads: As
we have discussed a few times, it is a Bad Idea to write equations
resembling:

☠ dQ = T dS ☠ [3]
☠ dW = P dV ☠

That's bad because (except in trivial cases) there is no potential Q
such that d(Q) = T dS. It is not possible for d(anything) to be equal
to T dS.

It is also quite unnecessary to write anything resembling equation [1].
By way of example, Dan Schroeder called such equations a "crime" against
the laws of mathematics, and wrote an entire thermodynamics book without
them. You would never notice their absence, if he didn't point it out,
which is important evidence that doing things right is no harder than
doing things wrong.

As it turns out, this fact that T dS is not generally d(anything) has
been understood for quite a while. I mention this because it connects
to the history question that came up recently. As early as 1909, Max
Planck was writing things like

q = dU + P dV [4]

I have changed the capitalization, but the point remains that there is
no "d" operator on the LHS of equation [4]. Planck didn't call attention
to the missing "d", but it is not missing by accident; it is clear that
he knew exactly what he was doing. He knew that q existed but was not
equal to d(anything).

Reference: _Eight Lectures on Theoretical Physics_ page 22ff.
Available for free from Google Books.

If somebody is looking for a nice little "history of science" project:
It might be amusing to see how far back a correct treatment of this
issue can be traced.
++ I wonder if Planck was influenced by differential topology, which
was just getting started in the early 1900s.
++ I wonder how Planck handled such issues in his dissertation, way
back in 1879.
-- In contrast: Circa 1900, Gibbs and Boltzmann were writing dQ and
dW with gay abandon.

On 01/09/2012 02:52 PM, John Mallinckrodt wrote:

1) You can doggedly reject most of them them and cling to the idea
that ONLY one of them is "correct"

We agree that is unwise.

2) You can accept the fact that people simply WILL use many different
and not unreasonable definitions of work (usually without thinking
carefully enough about it) and then try to educate them about the
differences between those definitions and what energy changes they
turn out to be equal to.

I agree with that as stated, both parts together. The second half provides
important context, explaining what the first half means by "accept". That
is, we should not let the inmates run the asylum. Specifically, we should
anticipate a certain amount of sloppiness and confusion, but we should not
give it our approval, tacitly or otherwise. We should tolerate it only long
enough to transmute it into something better.

As a lower bound on the magnitude of the problem:
-- Sometimes work means pseudowork, i.e. the work done on the center of mass.
-- Sometimes work means macroscopic work done at the points of application.
-- Sometimes work includes ultra-microscopic work, whereupon there is little
if any meaningful distinction between work and heat.
-- Sometimes work means P dV, which is a function of state but not a scalar.
-- Sometimes work means ∫ P dV, which is a scalar but not a function of state.

Rather than waging holy wars about "the" definition of work, we can use
adjectives and/or dependent clauses to make sense of (some of) these notions.
That's one option. Another option is to avoid problematic concepts and use
better concepts instead. Let's be clear: Part of the confusion is due to
bad terminology, but part of it is due to defective concepts, which is a much
deeper problem.

Here are some examples of the "replacement" option: Once upon a time there
was long-running confusion about "vis viva"; folks could not figure out
whether it was mv or mv^2. Eventually they gave up on vis viva, and replaced
it with *two* well-defined concepts, momentum and energy. Similarly, once
upon a time, there was long-running confusion about phlogiston. Eventually
folks gave up on phlogiston and replaced it with *two* well-defined concepts,
oxygen and energy. Similarly, we need to stop hunting for a viable concept
of "heat", and to replace "heat" with *two* well-defined concepts, entropy
and energy.

-- Sometimes "heat" means enthalpy ("heat of reaction").
-- Sometimes "heat" means energy.
-- Sometimes "heat" means energy *and* entropy ("heat exchanger").
-- Sometimes "heat" means T dS, which is a function of state but not a scalar.
-- Sometimes "heat" means ∫ T dS, which is a scalar but not a function of state.
-- Sometimes "heat" means heat flow across a boundary.
-- Sometimes "heat" includes dissipative contributions to T dS.

I'm pretty sure some of those should be considered misnomers. On some days
I think all of them should be considered misnomers ... but mostly I just don't
care anymore. If I want to talk about T dS, I just call it "T dS", pronounced
"tee dee ess". Similarly if I want to talk about P dV, I just call it "P dV",
pronounced "pee dee vee".

=======

All this ties together as follows:

There are a tremendous number of practical situations where ∫ T dS is can
be considered a function of state, to a perfectly reasonable approximation.
Students are often firmly wedded to this assumption, whether or not they
understand its limitations. Indeed there are plenty of physics teachers
who do not fully understand this issue. This is a problem, because
thermodynamics is pretty much all about the cases where ∫ T dS is *not*
function of state, not even approximately, grossly not.

This is analogous to the prevalent misconception that voltage is synonymous
with electric potential (or potential difference_ ... i.e. that every electric
field is the gradient of some potential.

The idea that T dS is a vector field -- often with (roughly speaking) nonzero
curl -- is quite a deep idea. There are ways of getting the point across, but
fussing with the terminology is not one of them. Some edificational diagrams
and discussion can be found here:
http://www.av8n.com/physics/non-grady.htm

It just astonishes me that anyone would think that superficial fussing with
the terminology would improve the situation ... yet we see papers (editorials!)
in AJP telling us "Heat is not a noun". I keep expecting the companion papers
"work is not a noun" and "electric field is not a noun".

Why do we tolerate this?

Planck had this figured out 100 years ago. Why are we still treating this
as an unsolved problem?

I saw another AJP article that said:

Much of the previous [educational] research has involved processes
in which heat is the only means of energy transfer. Several of
these studies indicate that students tend to treat heat as a
substance residing in a body.

Uhhhhh, situations where "heat" is the only means of energy transfer are
generally the situations where you *can* treat ∫ T dS as a function of
state. In such a situation, modeling "heat" as an abstract conserved
fluid is perfectly reasonable. (Just don't try to extrapolate to other
situations.)

It just astonishes me when "experts" berate the students for lack of
critical thinking, when the students have got the physics right and the
"experts" have got it wrong ... and when the "experts" are evidently not
able to think critically about their own work ... and when they take
credit for getting the students to go along with them, which means they
have forced the students to not think critically either.

This makes contact with the important points Joe Bellina posted yesterday:
We all know that students are capable of abstraction and symbolism and
multi-step planning and imagination and creativity, because we see it in
their sports and games and other out-of-class activities. The reason they
don't exhibit much (if any) of that in class is that they've been taught
it doesn't bring any advantage. Usually it just gets them into trouble.

Gaaaack!