Re: [Phys-l] T dS versus dQ

[John Denker <jsd@av8n.com>]

On 01/13/2010 04:40 PM, LaMontagne, Bob wrote:

> I have seen lots of statements so far that the final temperature
> "must" be higher, and the final entropy "must" be greater, but no
> actual clear argument as to why this is so. I would love to see an
> argument that would make it as clear to me as it appears to be to the
> rest of you - take it as a little teaching challenge.

It's very simple really.

In the real world, friction exists.  In the real
world, dissipative processes occur.  Dissipation 
creates entropy from scratch.  Entropy is not a 
conserved quantity.  

There are many examples of dissipative processes,
including cannon-boring, sound waves, shocks, et
cetera.  They all have the property of being
irreversible.  They create new entropy from scratch.
What we put in is low-entropy "non-thermal" energy, 
and what we get out is high-entropy "thermal" energy.

For example, stirring a gas is a dissipative process.  
Sudden movements of a piston stir the gas;  the 
equations of fluid dynamics, even in their simplest 
form, leave no doubt about this.

If you want to see a detailed example of how this 
works, let's consider a simplified version of 
Carl's scenario.

Consider the case of an _oscillating_ piston that
starts out at rest, oscillates for a while, and
then comes to rest again at the original position.

Here's an apparatus for doing this.  It's a 
loudspeaker in an unusual _full_ enclosure.  
(Loudspeakers are normally only half-enclosed.)

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That is sorta like two unported speaker enclosures 
face to face, completely enclosing the speaker driver 
that sits in the center.  When the driver (aka piston)
moves to the right, it compresses the right volume and 
expands the left volume.  The box as a whole is thermally 
isolated / insulated / whatever.  That is to say, no 
entropy crosses the boundary.  No energy crosses the
boundary except for the electricity feeding the speaker.

You could build this apparatus for a few dollars.  
It is considerably easier to build and operate than 
Rumford's cannon-boring apparatus.

The recent assertions that "the state of the gas 
depends only on the final position of the piston, 
no matter whether it moved quickly or slowly" are
clearly untenable in this version of the scenario.
The oscillating piston radiates sound into the gas
at a rate that depends on the square of its velocity
among other things.  The piston does P dV work not
just against the average pressure of the gas, but
also against the pressure in the radiation field.
The work done against the average pressure averages
to zero over the course of one cycle of the oscillatory
motion ... but the work against the radiation field
does *not* average to zero.  The dV is oscillatory
but the field pressure is oscillatory too, and the
product is positive on average.  Positive by a lot.

The energy radiated into the gas is in the short
term not in thermal equilibrium with the gas.
In the longer term, the sound waves are damped
i.e. dissipated by internal friction and also
by thermal conductivity, at a rate that depends
on the frequency and wavelength.

What we put in is P dV (call it "work" if you wish)
and what we get out in the long run is an increase
in the energy and entropy of the gas (call it "heat"
if you wish).

At no time is any entropy transferred across the
boundary of the region.  The increase in entropy
of the region is due to new entropy, created from
scratch in the interior of the region.

The kinetic energy of the piston is of no importance
in any version of this scenario.  Consider a very
low-mass piston if that helps.  Besides, whatever
KE goes into the piston is recovered.  Furthermore,
it is trivial to calculate the F dot dx of the piston
_excluding_ whatever force is necessary to accelerate
the piston.  Let's assume the experimenter is clever
enough to apply this trivial correction, so that we
know, moment by moment, how much P dV "work" is
being done on the gas.  This is entirely conventional;
the conventional pressure is associated with the force 
F1 on the side of the piston _facing the gas_, not the 
force F2 that is driving the piston.  To relate F1 to 
F2 you need to consider the mass of the piston, but 
if you formulate the problem in terms of F1 dot dV, 
as you should, questions of piston mass and piston 
KE should not even arise.

Carl's version of the scenario is more complicated.
It is farther from equilibrium, which makes it a
bit harder to analyze.  He used a single example to
make a number of points, including some sophisticated
points far beyond the basic notion of dissipation
and irreversibility -- valuable points that addressed
the hard questions asked at the beginning of this
thread -- but there is no value in discussing the 
hard questions version until we reach consensus about 
the basic facts of dissipation and irreversibility.

One detail that is shared between the simple version
and the fancy version of the scenario is the idea
that the piston does work not just against the
average pressure but also against the pressure of
the radiation field.  The work done against the
radiation field depends on the velocity, not just
on the initial and final positions of the piston.