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Clausius Inequality is not only for reversible process

Since overall (system + surrounding) entropy always increase.

(∆s) system + surrounding ≥ 0

For a cyclic process (∆s) system = 0

So (∆s) surrounding ≥ 0

Note that in cyclic integral of (dQ/T) dQ is positive for heat entering system form surrounding.

So for surrounding change in entropy after a cyclic process is given by

cyclic integral of (-dQ/T) which is ≥ 0

Thus cyclic integral of (dQ/T) ≤ 0

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Q: Proof of Clausius inequality for reversible process?
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