Clausius theorem

The Clausius theorem (1855), also known as the Clausius inequality, states that for a thermodynamic system (e.g. heat engine or heat pump) exchanging heat with external thermal reservoirs and undergoing a thermodynamic cycle, the following inequality holds. where is the total entropy change in the external thermal reservoirs (surroundings), is an infinitesimal amount of heat that is taken from the reservoirs and absorbed by the system ( if heat from the reservoirs is absorbed by the system, and < 0 if heat is leaving from the system to the reservoirs) and is the common temperature of the reservoirs at a particular instant in time. The closed integral is carried out along a thermodynamic process path from the initial/final state to the same initial/final state (thermodynamic cycle). In principle, the closed integral can start and end at an arbitrary point along the path. The Clausius theorem or inequality obviously implies per thermodynamic cycle, meaning that the entropy of the reservoirs increases or does not change, and never decreases, per cycle. For multiple thermal reservoirs with different temperatures interacting a thermodynamic system undergoing a thermodynamic cycle, the Clausius inequality can be written as the following for expression clarity: