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If there is a process in which a gas is being expanded against a constant pressure (ie the pressure of the atmosphere is constant before the process, during the process and after the process), then is this kind of process always irreversible process or always reversible process?

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It depends on the process. If the gas is initially at a higher pressure than atmospheric, and you suddenly release the piston and allow the gas to expand, then the process is irreversible. If the gas is initially at atmospheric pressure (with the piston unconstrained), and you gradually add heat to the gas (so that the average gas pressure is never significantly greater than atmospheric), then the process is reversible.

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