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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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