Clausius inequality
For any thermodynamic cycle, ∮ dQ/T ≤ 0, with equality only for a reversible cycle.
Definition
The Clausius inequality states that for any cyclic process, the integral of the heat exchanged divided by the temperature at which it is exchanged, taken around the complete cycle, is never positive: ∮ dQ/T ≤ 0. The equality ∮ dQ/T = 0 holds if and only if the cycle is reversible.
It is the second law compressed into a single line of calculus. For an irreversible cycle the strict inequality measures exactly how much entropy the process generates. For a reversible cycle the vanishing of the integral means dQ_rev/T is an exact differential — the differential of a state function, which Clausius named entropy.
The inequality is therefore both a statement of the second law and the bridge to the concept of entropy: it is the reason a state function S with dS = dQ_rev/T exists at all.
History
Clausius established the inequality in the 1850s and used it in 1865 to introduce entropy as a state function, completing the mathematical structure of classical thermodynamics.