Clausius inequality
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Suppose you carefully tally up all the heat that flows into and out of a system over one complete cycle, dividing each little dab of heat by the temperature at which it crossed the boundary. The Clausius inequality says that when you add up that ledger around the whole loop, the total can never come out positive — at best it lands at exactly zero, and only for a perfectly reversible cycle.
In symbols it is written as the cyclic sum of (heat in ÷ temperature) being less than or equal to zero. The equality holds for a reversible cycle; the strict 'less than' holds for any real, irreversible one. This single compact statement is the mathematical seed from which the whole idea of entropy grew — Clausius noticed that the reversible version, equaling zero, meant the quantity 'heat over temperature' defines a true state function.
Why it matters: the inequality is the second law in working clothes, the form engineers and chemists actually compute with. It quantifies exactly how much entropy a real process generates beyond the bare minimum. The caveat to hold onto: the temperature in the formula is the temperature of the reservoir or boundary where the heat crosses, not necessarily the messy, possibly non-uniform interior of the system.
Run a real engine through one cycle and the ledger of heat-over-temperature comes out negative — a measurable shortfall. That deficit is exactly the entropy the engine spilled into the universe through friction and finite-speed heat flow. The closer the engine to ideal, the closer the ledger creeps to zero.
Around any real cycle the heat-over-temperature ledger is negative; reversible makes it zero.
The inequality is the bridge between the engine-language of the second law (Clausius and Kelvin's statements) and the entropy-language we use today. Setting the reversible cycle's sum to zero is precisely what lets entropy be defined as a state function.