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The describes the probability distribution of molecular speeds in a sample of an ideal gas. Introduced over 150 years ago, it is based on the work of Scottish physicist and mathematician (1831–1879) and Austrian mathematician and theoretical physicist (1844–1906).

Today, the distribution and its implications are commonly taught to undergraduate students in chemistry and physics, particularly in introductory courses on physical chemistry or statistical mechanics.

In a recent theoretical paper, I introduced a novel formula that extends this well-known distribution to real gases.

Real gases are those that do not obey the ideal gas law, due to significant interactions between particles. These interactions result in a potential energy component in addition to the kinetic energy considered in the ideal case. Most theoretical models currently used to describe gases are valid only for ideal gases. Although several approaches exist to account for non-ideal behavior, none are universally applicable.

My new work applies probability theory to derive a formula that generalizes the Maxwell–Boltzmann distribution. Unlike the original, this extended distribution is not restricted to ideal gases—it can be applied to any gas sample. The core result is a mathematical proof built on a simple yet powerful observation: The physical properties of gases, whether ideal or real, are direction-independent. Although the original derivation of the Maxwell–Boltzmann distribution is grounded in statistical physics and does not explicitly invoke direction-independence, its final form is nonetheless consistent with it.

This new derivation asserts that direction-independence must also apply to non-ideal gases, imposing a strict mathematical constraint on the possible distributions of molecular speeds. An indirect proof shows that the components of a molecule's velocity in any direction must follow a normal distribution. Any alternative would violate the well-established , first formulated in 1733 by (1667–1754).

Using the rules for summing independent random variables, I derived the full three-dimensional speed distribution, which corresponds to a chi distribution with three degrees of freedom.

The resulting generalized formula closely resembles the original one for ideal gases. However, instead of temperature—the sole parameter in the Maxwell–Boltzmann distribution—the new formula involves the product of pressure and molar volume. This product can be related to other thermodynamic parameters if the equation of state for the gas is known.

This new, more general distribution offers straightforward methods for calculating average molecular speeds, collision frequencies, mean free paths, diffusion coefficients, and other key physical properties of real gases—quantities that were previously difficult or impossible to determine directly. These formulas can be systematically derived from textbook expressions for ideal gases by replacing the usual RT (gas constant times temperature) term with pV_m, the product of pressure and molar volume. More broadly, the results highlight the potential of standard probability theory to enhance our understanding of other non-ideal systems as well.

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Gábor Lente obtained his PhD from the University of Debrecen in Hungary in 2001. He is a full professor of chemistry at the Department of Â鶹ÒùÔºical Chemistry and Materials Science at the University of Pécs. His research focuses onmathematical reaction kinetics. He teaches advanced physical chemistry, analysis, geometry, and probability theory.