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You get more work out of a process when it is done slowly because less heat is lost to the surroundings. Since a reversible process is done infinitely slowly, you lose the least possible amount of work. Therefore a reversible process does the maximum work.
The work done by the gas in expansion equals Pex*delta V. During reversible reaction, at each stage of expansion the external pressure must be equal to internal pressure. Since the internal pressure is higher than the external pressure(that's why the gas expands), the Pin will decrease and Pex will increase gradually at each stage. That's why at each stage the Pexis the maximum. On the contrary, during irreversible reaction, the opposing pressure is less than the maximum possible(always the same and never increases) at each stage of the expansion, so that the potential of a system to do work is lost.
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