On Classical Ideal Gases

TL;DR

This paper presents a novel, elementary derivation of the ideal gas laws based on Democritian corpuscles and a principle of simplicity, enhancing understanding of classical thermodynamics.

Contribution

It introduces a new method for deriving ideal gas laws using corpuscle models, independent of laws of motion, and extends to general potentials and entropy definitions.

Findings

Stability of ideal gases proven from corpuscle force expressions

Entropy remains unchanged when separating and rejoining cylinders

Entropy can be defined via the ratio of heat to temperature

Abstract

The ideal gas laws are derived from the democritian concept of corpuscles moving in vacuum plus a principle of simplicity, namely that these laws are independent of the laws of motion aside from the law of energy conservation. A single corpuscle in contact with a heat bath and submitted to a $z$ and $t$-invariant force $-w$ is considered, in which case corpuscle distinguishability is irrelevant. The non-relativistic approximation is made only in examples. Some of the end results are known but the method appears to be novel. The mathematics being elementary the present paper should facilitate the understanding of the ideal-gas law and more generally of classical thermodynamics. It supplements importantly a previously published paper: The stability of ideal gases is proven from the expressions obtained for the force exerted by the corpuscle on the two end pistons of a cylinder, and the internal energy. We evaluate the entropy increase that occurs when the wall separating two cylinders is removed and show that the entropy remains the same when the separation is restored. The entropy increment may be defined at the ratio of heat entering into the system and temperature when the number of corpuscles (0 or 1) is fixed. In general the entropy is defined as the average value of $\ln(p)$ where $p$ denotes the probability of a given state. Generalization to $z$-dependent weights, or equivalently to arbitrary static potentials, is made.