The physics and mathematics of the second law of thermodynamics

Contents1.Introduction41.1.The basic questions41.2.Other approaches81.3.Outline of the paper112.Adiabatic accessibility and construction of entropy122.1.Basic concepts132.2.The entropy principle192.3.Assumptions about the order relation212.4.The construction of entropy for a single system242.5.Construction of a universal entropy in the absence of mixing292.6.Concavity of entropy322.7.Irreversibility and Carathéodory's principle352.8.Some further results on uniqueness363.Simple systems383.1.Coordinates for simple systems403.2.Assumptions about simple systems423.3.The geometry of forward sectors454.Thermal equilibrium544.1.Assumptions about thermal contact544.2.The comparison principle in compound systems594.3.The role of transversality645.Temperature and its properties675.1.Differentiability of entropy and the existence of temperature675.2.Geometry of isotherms and adiabats735.3.Thermal equilibrium and uniqueness of entropy756.Mixing and chemical reactions776.1.The difficulty in fixing entropy constants776.2.Determination of additive entropy constants797.Summary and conclusions887.1.General axioms887.2.Axioms for simple systems887.3.Axioms for thermal equilibrium887.4.Axiom for mixtures and reactions89Acknowledgements92Appendix A92A.1.List of symbols92A.2.Index of technical terms93References94 The essential postulates of classical thermodynamics are formulated, from which the second law is deduced as the principle of increase of entropy in irreversible adiabatic processes that take one equilibrium state to another. The entropy constructed here is defined only for equilibrium states and no attempt is made to define it otherwise. Statistical mechanics does not enter these considerations. One of the main concepts that makes everything work is the comparison principle (which, in essence, states that given any two states of the same chemical composition at least one is adiabatically accessible from the other) and we show that it can be derived from some assumptions about the pressure and thermal equilibrium. Temperature is derived from entropy, but at the start not even the concept of 'hotness' is assumed. Our formulation offers a certain clarity and rigor that goes beyond most textbook discussions of the second law.