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Theory of the probability of total resolution in chromatograms with systematic variation of average peak spacing and peak width

Davis, Joe M.
Journal of chromatography 2019 v.1588 pp. 150-158
arithmetics, chromatography, equations, geometry, models, prediction, probability
An equation is proposed for the probability that all mixture constituents are separated, when the density (i.e., average number of eluting constituents per time) and width of single-component peaks (SCPs) vary systematically. The probability Pr that m SCPs are separated is modeled as the product of the m – 1 probabilities that adjacent pairs of SCPs are separated. Pr is then expressed as the geometric mean of the probability product raised to the power of m – 1. This geometric mean is approximated by an arithmetic mean equaling the probability that adjacent SCPs are separated, as calculated from previously developed statistical overlap theory (SOT) for variable SCP density and width. The theory is tested using previously reported and current in-house simulations of isocratic chromatograms of SCPs with random differences in standard chemical potential. In such chromatograms, more SCPs elute at short times than long times, and their widths are less at short times than long times. The average difference between 179 previously reported and currently predicted values of 100 x Pr is about 0.6, when 100 x Pr > 5. The theory requires numerical computation of one integral but can be approximated by an analytic equation for SOT probabilities close to one. For SCPs having retention times exceeding twice the void time, this equation simplifies to a previous SOT expression, with the gradient peak capacity replaced by the isocratic peak capacity. The versatility of the Pr theory is tested using three other models of chromatograms, in which the density and width of SCPs vary. The Pr predictions agree with simulation for all three models.