Deriving the Ideal Gas Law: A Statistical Story

The ideal gas law is $PV=NkT$, where $P$ is the pressure, $V$ is the volume, $N$ is the number of particles, $k=1.38\times10^{-23}\,\mathrm{m}^2\,\mathrm{kg}\,\mathrm{s}^{-2}\,\mathrm{K}^{-1}$, and $T$ is the temperature. It constitutes one of the simplest and most applied “equations of states” in all of physics, and is (or will become) incredibly familiar to any student of not only physics but also related fields such as chemistry and various engineering disciplines.

Recently, I’ve come to appreciate the ideal gas law as a very good illustrative example of what statistical mechanics is capable of. In the eighteenth and nineteenth (during the industrial revolution), the field of thermodynamics arose in pursuit of answers to how much energy can be extracted from systems. At the time, the notion of temperature was hotly debated, with one ostensibly reasonable theory being that the flow of a fluid, called “caloric,” facilitated heat transfer. It should be appreciated just how many important results could be derived by our predecessors without precise knowledge of the microscopic physics involved.

Statistical mechanics arises from a desire to understand how thermodynamics arises from microscopic physics, either classical or quantum. While classical thermodynamics is able to achieve a great deal, it is philosophically difficult to reconcile the microscopic and macroscopic worlds without some statistical framework that can translate the former to the latter. Moreover, it allows us to determine the thermodynamic behavior of new systems that are hard to reason about a priori—how, for example, should we expect the stars in a star cluster to behave? What about a complicated, novel quantum field theory?

In this article, I will present a number of different derivations of the ideal gas law within the framework of statistical mechanics. In the process, I hope to motivate the different ensembles of statistical mechanics (which ultimately just encode the behavior of possibly large systems for which only a small amount of bulk information is known). In the process, I hope that an overview of these derivations will be a useful introduction for current physics students (or other interested people) to a notoriously (but arguably needlessly) opaque field.