Boltzmann Equation for Entropy:
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In irreversible expansion, the outer pressure will suddenly drop from say 2 to 1 ATM and so the piston will constantly be pushing against 1 ATM of pressure. In reversible expansion, the pressure outside slowly drops bit by bit from 2 to 1.9 to 1.8 to 1.7 (not actually by .1 but by an infitesimally small amount) so the piston is pressing against a little bit less pressure each time. The reversible expansion does more work because it doesn't press by 1 ATM constantly but by 1.9 to 1.8 to 1.7 (again not exact values) and so on and so forth.
Reversible can be thought of as if the system and surroundings are at equilibrium, where any change in volume can be seen as a infinitesimally close to the original thermodynamic state; irreversible tends towards one direction and no small change can restore initial conditions.
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