On the ideal gas law

Abstract

The air density on earth decays as a function of altitude $z$ approximately according to an $\exp(-w\,z/θ)$-law, where $w$ denotes the weight of a nitrogen molecule and $θ=\kB T$ where $k_B$ is a constant and $T$ the thermodynamic temperature. To derive this law one usually invokes the Boltzmann factor, itself derived from statistical considerations. We show that this (barometric) law may be derived solely from the democritian concept of corpuscles moving in vacuum. We employ a principle of simplicity, namely that this law is \emph{independent} of the law of corpuscle motion. This view-point puts aside restrictive assumptions that are source of confusion. Similar observations apply to the ideal-gas law. In the absence of gravity, when a cylinder terminated by a piston, containing a single corpuscle and with height $h$ has temperature $θ$, the average force that the corpuscle exerts on the piston is: $\ave{F}=θ/h$. This law is valid at any temperature, except at very low temperatures when quantum effects are significant and at very high temperatures because the corpuscle may then split into smaller parts. It is usually derived under the assumption that the temperature is proportional to the corpuscle kinetic energy, or else, from a form of the quantum theory. In contradistinction, we show that it follows solely from the postulate this it is independent of the law of corpuscle motion. On the physical side we employ only the concept of potential energy. A consistent picture is offered leading to the barometric law when $w\,h\ggθ$, and to the usual ideal-gas law when $w\,h\llθ$. The mathematics is elementary. The present paper should accordingly facilitate the understanding of the physical meaning of the barometric and ideal-gas laws.