For a closed system that contains an arbitrary pure substance, which can exchange energy as heat and as expansion∕compression work, but no particles, with its surroundings, the inexact differential of the reversibly exchanged heat is a differential in two variables. This inexact differential can be turned into an exact one by an integrating factor that, in general, depends on both variables. We identify the general form of the integrating factor as the reciprocal temperature (Clausius’s well-known ), which is guaranteed to be a valid integrating factor by the second law of thermodynamics, multiplied by an arbitrary function of the implicit adiabat equation or . In general, we cannot expect that two different equations of state (corresponding to two different substances) predict identical equations for the adiabats. The requirement of having a universal integrating factor thus eliminates the volume-dependent or pressure-dependent integrating factors and leaves only a function of temperature alone: Clausius’s integrating factor . The existence of other integrating factors is rarely mentioned in textbooks; instead, the integrating factor is usually taken for granted relying on the second law or, occasionally, one finds it “derived” incorrectly from the first law of thermodynamics alone.
The transfers of heat and of work are two ways in which the system can exchange energy with its surroundings. “Heat” denotes the transfer of energy due to a (possibly infinitesimally small) temperature difference, while “work” is energy exchanged due to a (possibly infinitesimally small) pressure difference between the system and its surroundings. The system itself possesses neither heat nor work, just energy.
The term “reversible” denotes a special way of conducting the transfer of heat or work in a given process. It ensures that, at all times, the differences in temperature and pressure between the system and its surroundings are only infinitesimal; that is, the temperature and pressure are matched in such a way that heat and∕or work are just transferred in the desired direction (to or from the system), but the values of the temperature and pressure are equal for the system and for its surroundings for all practical purposes. The total entropy of the supersystem consisting of the system and its surroundings remains constant in a reversible process, implying that the change in entropy of the system is exactly equal, but opposite in sign, to the change in entropy of the surroundings.