Modeling Light Transmission in Absorbing Media

Abstract:This paper presents an elegant alternative method for deriving the fundamental Beer–Lambert–Bouguer law based on a discrete microscopic model of a disperse medium. Unlike the classical phenomenological approach, which introduces the absorption coefficient via differential equations, this model utilizes the geometric cross-section of spherical particles within a uniform suspension. By partitioning the medium into single-particle layers and applying the second remarkable limit for the transition to a continuous medium, the empirical absorption constant is derived purely theoretically and given a concrete physical definition. The paper outlines the boundary conditions of the model, including geometric optics limitations and the extinction efficiency factor (extinction paradox). The proposed mathematical framework offers a highly intuitive and physically justified tool for engineering applications, such as calculating the transmission coefficients of smoke, dust, industrial suspensions, and atmospheric precipitation. Keywords: Beer-Lambert law; absorption coefficient; disperse systems; geometric optics; light attenuation; suspension transmittance; Zenodo open science.