Carathéodory's principle and the existence of global integrating factors

Abstract

A proof is given of a theorem on the integrability of Pfaffian forms which is used in Carathéodory's approach to thermodynamics. It is pointed out that Carathéodory's original proof of the existence of entropy and of absolute temperature is incomplete, since it fails to take into account the local nature of this theorem. By combining the theorem with the results ofBuchdahl andGreve on the existence of continuous empirical entropy functions, it is shown that the First and Second Laws of Thermodynamics imply the existence of a globally defined differentiable empirical entropy function for every simple thermodynamic system. This result supplies the missing step in Carathéodory's argument and makes a separate proof of the principle of increase of entropy unnecessary.