How can resolution be determined when peak width cannot be measured?

Figure 1: An example of poorly resolved peak pairs A, B, and C.

I have had several reader inquiries lately regarding how to estimate resolution between two peaks in a liquid chromatography
(LC) separation when the traditional calculation doesn't work. An example of this is shown for peak pairs A, B, and C in the
chromatogram of Figure 1. In each case, the valley between the peaks does not dip below 50% of the height of the smaller peak,
making it impossible to measure the peak width at the baseline or half-height. In this month's "LC Troubleshooting" instalment,
I would like to share a simple technique to estimate resolution that has been in use for many years (for example, see reference
1), but may not be well known because of our dependence on automatic data processing systems today.

Traditional Measurements

Most of us use the method of equation 1 or 2 to calculate the resolution, R_{s}, of a pair of peaks with retention times t_{1} and t_{2}:

where w_{b1} and w_{b2} are the baseline peak widths between tangents drawn to the sides of the peaks, and w_{h1} and w_{h2} are the corresponding peak widths measured at half the peak height. That is, the resolution is the difference in retention
times divided by the average baseline peak width (thus the factor of 2 in equation 1). The peak width at the baseline for
a Gaussian peak is 4 σ (4 standard deviations), whereas at the half-height, it is 2.354 σ, so the factor in equation 2 is
(2 × 2.354/4) = 1.18. Because the half-height peak width is easier to measure (no tangent drawing involved), most data systems
use the half-height method (equation 2) to calculate resolution.

Figure 2: Simulated chromatograms for peak-height ratios of 2:1. (a) R_{s} = 1.3; (b) R_{s} = 0.9, h_{1}, h_{2}, and h_{v} are heights of the first peak, second peak, and valley, respectively; (c) R_{s} = 0.75.

The technique of equations 1 and 2 works well when the peaks are well separated, as with Figure 2(a), where R_{s} = 1.3. When the resolution drops much below this, it will be difficult to measure the baseline peak width, but it may be
possible to measure the half-height width. However, as the resolution drops, the valley between the two peaks rises, and at
some point it becomes no longer possible to measure the half-width, either (for example, Figure 2[b]). Unfortunately, as the
amount of peak overlap increases, measurement of resolution becomes more important. This is because it is more difficult to
accurately measure peak areas when resolution drops below ~1.2, so system suitability tests often have minimum resolution
requirements for partially separated peaks.

MS — The Practical Art: Kate Yu brings her expertise in the field of mass spectrometry and hyphenated techniques to the pages of LCGC. In this column she examines the mass spectrometric side of coupled liquid and gas-phase systems. Troubleshooting-style articles provide readers with invaluable advice for getting the most from their mass spectrometers. LATEST: Radical Mass Spectrometry as a New Frontier for Bioanalysis