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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.)