Chromatographic Peak Shape from Stochastic Model: Analytic Time-Domain Expression in Terms of Physical Parameters and Conditions under which Heterogeneity Reduces Tailing

Abstract

A time-domain representation of chromatographic peak shapes is presented as an analytic expression designed for high computational efficiency, which can be used for direct time-domain peak fitting with parameters that represent physical quantities. The underlying model integrates the effects of axial diffusion (molecular and multipath/eddy), finite initial spatial variance, and two distinct retention mechanisms: one characterized by a high rate of short-duration events (fast kinetics), and another by a low rate of long-duration events (slow kinetics). Fits to experimental chromatograms yield substantially smaller residual standard error (RSE) than the standard EMG and the lowest average normalised RSE among 12 established peak-shape functions in the examined cases. The stochastic approach is reformulated using single-particle probability laws, providing a rigorous basis for future theoretical extensions. The validity of the foundational Poisson assumption is critically examined by deriving expressions for the excess variance caused by correlated microscopic retention rate fluctuations. A statistical interpretation of the HETP is presented and used to determine a lower bound on the number of microscopic retention events from chromatogram-derived macroscopic observables. This, in turn, justifies the applicability of the Gaussian limit for the mobile-plus-fast component, as established by analysis of the cumulant generating function of the closed-form benchmark derived herein. The contribution from the slow-kinetic mechanism is incorporated via a decoupling approximation, whose validity is established through a cumulant-based analysis that explicitly bounds the decoupling-induced error. Finally, the notion that mechanistic heterogeneity necessarily exacerbates peak tailing is qualified, analytically delineating parameter regions in which it leads to a reduction in peak asymmetry.