Direction independence as a key property to derive a particle speed distribution in real gases

Abstract

Starting from the probability theory of continuous random variables and the central limit theorem, a rigorous mathematical proof is presented to show that the one-dimensional velocity components of particles in gas phase at thermal equilibrium can only be normally distributed if the physical properties are independent of direction. As such direction-independence is true for all gases, no matter whether they are ideal or not, a general distribution can be introduced. It is also shown that the particle speeds, which are the Euclidean norms of the velocity vectors, are always described by a chi distribution with three degrees of freedom, which converts into the Maxwell-Boltzmann speed distribution if the ideal gas law is valid. Furthermore, many of the formulas derived for ideal gases have analogs for real gases, which can be constructed by replacing RT (gas constant multiplied by temperature) terms by pVm (pressure multiplied by molar volume).