The Physics and Mathematics of the Second Law of Thermodynamics

TL;DR

This paper rigorously derives the second law of thermodynamics and entropy for equilibrium states using fundamental postulates, emphasizing the comparison principle and avoiding statistical mechanics.

Contribution

It provides a clear, axiomatic formulation of the second law based solely on classical thermodynamics principles, without relying on statistical mechanics.

Findings

Entropy is defined only for equilibrium states.

The comparison principle is derived from assumptions about pressure and thermal equilibrium.

The second law is deduced as the increase of entropy in irreversible processes.

Abstract

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.