Validation of Extended Statistical Overlap Theory for Application to Small Second Dimension Peak Capacity

Approximate equations for the numbers of singlet peaks and total peaks in two-dimensional (2D) separations of randomly distributed component zones predict values of zero as the second-dimension peak capacity 2nc goes to zero. This is the result of deriving the equations for an unbounded 2D space. More rigorous equations are needed to assess how large 2nc must be before the error in the approximate equations is acceptably small. An exact equation for the number of singlet peaks in a 2D separation with a bounded second dimension is derived in Online Resource 1. The equation accounts for the reduced likelihood of overlap of zones represented as circles or ellipses near the second-dimension boundaries and is the result of the lengthy derivation of different expressions of probability, with the appropriate expression depending on the value of 2nc. In this paper, the equation is used to modify the prediction of the Roach equation for the number of total peaks in the 2D separation. Computer simulations of the clustering of circles confirm the equation for singlet peaks over a wide range of first-dimension saturations α1D (0.05 ≤  α1D ≤  5) and 2nc values (0.1 ≤ 2nc  ≤ 15). The saturation α1D, equal to the ratio of the number of components requiring separation to the first-dimension peak capacity available for separation, is a metric of chromatographic crowdedness. Over the same α1D range, the equation for total peaks slightly underestimates the simulations when 2nc  < 2.5 but agrees with them at larger 2nc. It is shown that the extended theory differs most from the approximate one when α1D is large and 2nc is small.