The Ideal Gas Tautology

Two Measurements of Molecular Motion

Abstract

Temperature and pressure are two statistical measurements of the same phenomenon: molecular motion. Temperature measures average translational kinetic energy per molecule; pressure measures momentum flux from molecules to boundaries. Both emerge from mean squared velocity. Under ideal gas assumptions (dilute, negligible interactions, elastic collisions, isotropy, equilibrium), this shared dependence makes the ideal gas law PV = NkT a definitional necessity rather than an empirical discovery.

Contribution: This paper presents no new physics. Kinetic theory established these connections in the 19th century (Maxwell 1860, Boltzmann 1872). The contribution is pedagogical synthesis: making explicit the definitional relationship between temperature and pressure that standard empirical presentation (Boyle's law, Charles's law, Gay-Lussac's law as separate discoveries) obscures.