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The objective of this work is to establish a means of correcting the theoretical maximum peak capacity of comprehensive two-dimensional (2D) separations to account for the deleterious effect of undersampling first-dimension peaks. Simulations of comprehensive 2D separations of hundreds of randomly distributed sample constituents were carried out, and 2D statistical overlap theory was used to calculate an effective first-dimension peak width based on the number of observed peaks in the simulated separations. The distinguishing feature of this work is the determination of the effective first-dimension peak width using the number of observed peaks in the entire 2D separation as the defining metric of performance. We find that the ratio of the average effective first-dimension peak width after sampling to its width prior to sampling (defined as ) is a simple function of the ratio of the first-dimension sampling time (t(s)) to the first-dimension peak standard deviation prior to sampling (1sigma): = square root1+0.21(t /(s)(1) sigma(2) This is valid for 2D separations of constituents having either randomly distributed or weakly correlated retention times, over the range of 0.2 on t(s)/1 sigma from this expression is in qualitative agreement with previous work based on the effect of undersampling on the effective width of a single first-dimension peak, but predicts up to 35% more broadening of first-dimension peaks than is predicted by previous models. This simple expression and accurate estimation of the effect of undersampling first-dimension peaks should be very useful in making realistic corrections to theoretical 2D peak capacities, and in guiding the optimization of 2D separations.

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The low prediction by statistical-overlap theory of the numbers of singlets and peaks in two-dimensional separations containing zones represented by either circles of small number or eccentric ellipses of any number is shown to result from use of probability expressions for unbound spaces of infinite extent. An exact theory is derived for the probability of singlet formation in a reduced two-dimensional space of unit length, width, and area. The probability is a weighted sum of the probabilities of singlet formation in the interior, edge, and corner regions of the space, which depend only on saturation. The weighting factors are the fractions of area associated with each region and depend on the number of zones, the aspect ratio, the saturation, and the ellipse's spatial orientation. The average numbers of doublets, triplets, and peaks in the space are approximated by combining these results with Roach's equations describing the clustering of circles in an unbound two-dimensional space. Simulations show that theory predicts the number of singlets, doublets, triplets, and peaks, when the number of zones is 25 or more, the aspect ratio is 100 or less, and the saturation is 2 or less. The relationship is derived between the aspect ratios of ellipses in the reduced space and actual separation space. Calculations are presented for comprehensive two-dimensional gas chromatography.