I am starting to see some interesting frame-of-reference issues coming
into play.
Here is my next thought experiment:
Consider the tube-in-a-spaceship version. Assume the spaceship does not
have the engines on, so the initial distribution for the gas would be
isobaric and isothermal. Imagine adding some massless, thermally
conducting partitions every 1 m (ie 1000 sections thermally connected to
the regions on either side). Now turn on the engines so the ship
accelerates (at say 9.8 m/s^2). The gas will start collecting at the
back, which will compress the gas. The back section will have net work
done on it as it compresses, causing the temperature to rise. The
farther forward you go, the less compression there is -- the regions
toward the front will expand and be cooler than before.
Of course, the temperature gradient will cause heat flow, which will
tend to equalize the temperatures. (Without the partitions there might
be convection, but now we can old have conduction). The question is
then "will the temperatures become the same throughout, or could some
gradient remain because the systems is constantly being 'disturbed' by
the accelerating ends?" It is interesting to consider the solution from
two different frames. Ultimately, I come to the same conclusion for
each -- that the acceleration of the ends does not matter and the result
would be isothermal.
IN THE INERTIAL FRAME (not accelerating with the spaceship) the back
piston is constantly doing positive work on the gas, and the forward
piston is constantly doing negative work. They both move the same
distance, but the back piston applies more force (because the pressure
is higher), so there is net work being done. One obvious effect is that
the center of mass will accelerate and the gas as a whole will gain
kinetic energy. Since there is a pressure gradient, each section will
have a slightly larger pressure on the back than the front, so each
section will have net work being done. However, this net work should
be the same on each section, so each section is gaining the same CoM KE.
( If the some sections did gain more KE than others, then those gas
molecules would collide with the partitions extra hard and would thereby
warm up, which could set up a permanent temperature gradient.)
IN THE CENTER-OF-MASS FRAME OF THE GAS, neither end is moving, so
neither end is doing any work. In fact, each of the 1000 sections will
have no net work being done, so no section is deviating from a simple
equilibrium condition. So eventually, conduction will simply drive the
whole situation to thermal equilibrium (ie a uniform temperature).
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Making one end a thermal reservoir would simply cause the final
temperature to be equal to that temperature. Making both ends
reservoirs at different temperatures would create a thermal gradient and
a steady heat conduction.
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John said " I was discussing the /equilibrium/ state. Thermally
insulating the two ends of the system from each other and then
disturbing things pretty much guarantees a non-equilibrium situation."
As discussed here, I don't think this current situation IS much
different from the previous situation. Either BOTH an insulated
cylinder on earth and an insulated cylinder in the space ship are
"disturbed and non-equlibrium" and could have a temperature gradient
(because the zeroth law doesn't guarantee equal temperatures in
non-equilibrium situations), or both will be in equilibrium and be
isothermal.