The uniqueness of Clausius' integrating factor

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' well-known l/T), 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 [xi](T,V) =constant or [xi](T,P)=constant. 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' integrating factor 1/T. The existence of other integrating factors is rarely mentioned in textbooks; instead, the integrating factor 1/T is usually taken for granted relying on the second law or, occasionally, one finds it 'derived' incorrectly from the first law of thermodynamics alone.