Appendix A - Ideal Gas Law from the Equipartition Theorem

Index

1. Introduction

While we're usually introduced to the Ideal Gas Law as an empirical result in its own right. The lucky among us may seen a loose derivation of it as a combination of Charles's law, Boyle's law, the Guy-Lussac law, and Avogadro's rule somewhere around highschool chemistry. Ultimately, it is often taught early enough that the student is expected to absorb it and take it for granted.

Interestingly, though, we can actually find a derivation of the ideal gas law from the equipartition theorem and Hamiltonian mechanics. It is, in fact, a theorem in the mathematical sense — a statement, based upon definitions, that can be proven. We will see that there is good reason that the equipartition theorem is not derived in first thermodynamics. To derive it from scratch requires the development of Hamiltonian mechanics from symplectic manifolds. From there, we will be needing an understanding of phase spaces, ergodic processes, and the canonical and microcanonical ensembles.