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Well ... I fear that much of the confusion in this thread is the
result of differing hidden assumptions about what is given so let's
try to be very specific. First, a few facts:
1. For a given gas at thermal equilibrium, both the energy and the
entropy depend only on the temperature and the volume and, for a
given volume, both the energy and the entropy increase monotonically
with temperature.
2. All of the above is also true for an ideal gas with the slight
simplification that the energy does *not* depend on the volume.
Now some cases. In each, I assume a) that the gas is thermally
insulated, b) that it starts and ends in thermal equilibrium, and c)
that the initial state is given.
CASE 1: An irreversible compression to a given final volume, (i.e.
Carl's case):
1. Stirring has occurred and the final entropy is greater than the
initial entropy. (Because we are explicitly told that the process is
irreversible.)
2. The final energy is greater than it would have been had the
process not been irreversible. (Because higher entropy => higher
temperature => higher energy for a given volume.)
3. More work was done on the gas than would have been had the process
not been irreversible. (Because the extra energy was entirely the
result of extra work done.)
CASE 2: The energy input is specified:
1. We know the work done. (Because the energy input was entirely the
result of work.)
2. We don't know the final temperature. (Because we don't know the
final volume.)
3. We don't know the final entropy. (Because we don't know the final
temperature OR the final volume.)
4. We don't know if stirring occurred. (Because we don't know if the
entropy increased.)
CASE 3: The energy input is specified and the gas is ideal:
1. We know the work done. (Because the energy input was entirely the
result of work.)
2. We know the final temperature. (Because we know the final energy.)
3. We don't know the final entropy. (Because we don't know the final
volume.)
4. We don't know if stirring occurred. (Because we don't know if the
entropy increased.)
CASE 4: The energy input and the final volume are specified.
1. We know the work done. (Because the energy input was entirely the
result of work.)
2. We know the final temperature. (Because we know the final energy
and the final volume.)
3. We know the final entropy. (Because we know the final temperature
and the final volume.)
4. We know if stirring has occurred. (Because we know if the entropy
increased.)
5. We know if the problem specification was unphysical. (Because it
is possible to specify an energy input and final volume that result
in a final entropy that is less than the original entropy and *that*
is physically impossible.)
John Mallinckrodt
Cal Poly Pomona
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