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I have an ideal gas in a thermally insulated cylinder and piston. I
suddenly compress the piston. To be specific, suppose the piston's
speed profile starts from zero, rapidly (practically stepwise) rises
up to the speed of sound, then drops rapidly back to zero once the
gas is compressed by dV (which is negative). In practice I might
accomplish this by having a huge weight sitting on the piston which
is at the top end of the cylinder and held in place by a pin. I pull
out the pin and let the piston fall a distance dx until it slams into
a stop.
I think we have T dS > 0 (because the process is certainly
irreversible), dQ = 0, dE > 0 (because work -P dV was done on the
gas) and so it doesn't look like dQ can be equal to either T dS or
dE. I "computed" dQ by noting that the cylinder (including the
piston) is thermally insulated (and I'm further helped that the
process is so fast there isn't time for heat transfer even if it
weren't insulated, noting that no thermal insulation is perfect in
the real world).