Clausius theorem

The Clausius theorem (1855) states that a system exchanging heat with external reservoirs and undergoing a cyclic process, is one that ultimately returns a system to its original state, The Clausius theorem (1855) states that a system exchanging heat with external reservoirs and undergoing a cyclic process, is one that ultimately returns a system to its original state, where δ Q {displaystyle delta Q} is the infinitesimal amount of heat absorbed by the system from the reservoir and T s u r r {displaystyle T_{surr}} is the temperature of the external reservoir (surroundings) at a particular instant in time. In the special case of a reversible process, the equality holds. The reversible case is used to introduce the entropy state function. This is because in a cyclic process the variation of a state function is zero. In words, 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. The generalized 'inequality of Clausius' for an infinitesimal change in entropy S applies not only to cyclic processes, but to any process that occurs in a closed system. The Clausius theorem is a mathematical explanation of the second law of thermodynamics. It was developed by Rudolf Clausius who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law. Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, 'On the Application of the Theorem of the Equivalence of Transformations to Interior Work'. Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δQ) into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that 'The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing.' In other words, the equation with ?Q being energy flow into the system due to heating and T being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the 'Clausius inequality'.