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Quasistatic process are almost always in equilibrium. We know that equilibrium implies zero entropy change. And zero entropy change implies that the process is reversible. So why quasistatic doesn't imply reversible?

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All reversible processes are quasistatic process because they're all slow enough for the system to remain in equilibrium. But all quasistatic processes aren't reversible this is because some involve entropy production. Change of entropy of a cyclic process is only zero.

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Consider the case of a gas expanding slowly against a piston, which has friction.

The friction will dissipate some energy into heat, and this will happen no matter now slowly you move the piston; the dissipation is always going to be at least $\mu N \, \Delta x$. This dissipation increases the entropy, so quasistatic doesn't imply reversible.

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