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The way works for me is to simply think that reversible expansion is a slow, equilibrium process, and the changes of pressure happen in a lot of small steps. For the irreversible expansion, the pressure changes in one step and stays there for the rest of the process.
Another way to think about it is, for a reversible process, the system is always in equilibrium with its surroundings. For an irreversible process, on the other hand, there will always be an entropy change.
For a reversible process, pressure inside and outside of the system will be equal. For an irreversible process, there will be a significant difference between the pressure inside and outside of the system.
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