Which theorem states that the cyclic integral of δQ/T is never positive?

The Clausius theorem states that the cyclic integral of δQ/T is never positive.

For a thermodynamic cycle, the Clausius inequality is written ∮δQ/T ≤ 0 when δQ is heat absorbed by the system and T is the temperature of the surrounding reservoir during the transfer. Equality holds for a reversible cycle; an irreversible cycle makes the integral negative. The result is a mathematical expression of the second law.

The theorem is named for Rudolf Clausius, who developed key parts of thermodynamics in the 19th century and introduced the concept of entropy. Because a system returns to its initial state after a cycle, its net entropy change is zero. Irreversibility, however, generates entropy in the combined system and surroundings, constraining the heat transfers so their temperature-weighted sum cannot be positive.

A frequent mix-up is treating δQ/T as the system’s entropy change in every process. The equality dS = δQ_rev/T applies to a reversible path. For an irreversible process, entropy generation means the heat integral alone does not directly equal the system’s entropy change. The theorem also underlies the definition of entropy as a state function.