The ideal gas law: derivations and intellectual background

Abstract A detailed overview of the origins and possible derivations of the ideal gas law are given in this lecture text. The tacit assumptions in the usual textbook derivations are identified and ways of thought are proposed to justify their use. It is shown that the solution of the Schrödinger equation for the particle in a box model is very closely connected to the ideal gas law. A detailed derivation is presented on how equipartition and the Boltzmann distribution lead to the velocity and speed distributions observable in ideal gases. Major emphasis is given to the mathematical consequences of the lack of direction dependence in these properties. General formulas are introduced and proved to describe the thermodynamics of ideal gases. During the derivations, a method for students to employ computer-aided symbolic calculations using the Wolfram Alpha web-based service is also described.