Clausius theorem

The Clausius theorem states that the cyclic integral of heat transfer divided by temperature is nonpositive, becoming zero for a reversible cycle.

Clausius inequality: For a cyclic process, the integral of heat transfer divided by boundary temperature is nonpositive. This common formulation makes the theorem’s sign constraint explicit.
Thermodynamic cycle: A sequence of thermodynamic processes that returns a system to its initial state. The theorem applies to processes that close into such a cycle.
Rudolf Clausius: A German physicist who helped establish the modern science of thermodynamics. He formulated the theorem and introduced the concept of entropy.
Carnot theorem: The result that no engine operating between two reservoirs can outperform a reversible engine. It follows from applying the cyclic inequality to competing engines.
Kelvin–Planck statement: A second-law formulation that forbids a cyclic device from converting heat from one reservoir entirely into work. It gives an equivalent restriction without using the cyclic integral.
Reversible process: An ideal process that can be reversed while leaving no net change in system and surroundings. Its cycles attain equality in the theorem.
Heat: Energy transferred between systems because of a temperature difference. Its signed transfer is the quantity integrated around the cycle.
Second law of thermodynamics: The principle that physical processes have a preferred direction and impose limits on energy conversion. The theorem gives a quantitative expression of this law for cycles.
Thermodynamic efficiency: The fraction of supplied energy converted into useful work by a thermodynamic device. The theorem constrains the maximum efficiency of cyclic engines.
Clausius postulate: A statement that no cyclic process can transfer heat from a colder body to a hotter body without other effects. This verbal form is often confused with Clausius’s mathematical theorem.

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Thermodynamic equationsRelated: It gives a mathematical expression of the second law for cycles.