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 from each reservoir and absorbed by the system ( if heat from the reservoir is absorbed by the system, and < 0 if heat is leaving from the system to the reservoir) and is the temperature of the reservoir 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, never decrease, 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: where is an infinitesimal heat from the reservoir to the system. In the special case of a reversible process, the equality holds, and the reversible case is used to introduce the state function known as entropy. This is because in a cyclic process the variation of a state function is zero per cycle, so the fact that this integral is equal to zero per cycle in a reversible process implies that there is some function (entropy) whose infinitesimal change is . The generalized "inequality of Clausius" for as an infinitesimal change in entropy of a system (denoted by sys) under consideration applies not only to cyclic processes, but to any process that occurs in a closed system. The Clausius inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer. The Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around. History The Clausius theorem is a mathematical representation of the second law of thermodynamics. It was developed by Rudolf Clausius who intended to explain the