My LC Blog: The Resolution Equation

As established in the very beginning - the goal of any chromatographic separation is adequate resolution of the components of interest in reasonable time. Method development aims at achieving this goal, which is where the Resolution Equation comes to play. It is shown in the title picture and expresses the dependency of Resolution (Rs) on Efficiency (N), Selectivity (α) and Retention (k). So, in simple terms, it can show us how changing chromatographic operating parameters can have a dramatic effect on the performance of a method. You see why understanding and remembering this equation would be quite important when looking to develop a chromatographic separation method.

Let’s start with a glossary:

• Resolution (Rs): Measure of the separation of two peaks in a chromatogram - based on differences in retention and peak width.
• Efficiency (N): Is expressed in "theoretical plates" (N) and indicates how many peaks can - in theory - be separated by the column in the given chromatographic system. The narrower the peaks, the larger N.
• Selectivity (α): Indicator of the relative retention of two components in a mixture, meaning the ability of the chromatographic system to differentiate between them.

• Retention (k): Measure of how well an analyte is retained.

And this is the point where we can no longer avoid looking at some equations to learn of the dependency of these parameters on different factors that we can influence when developing a new HPLC method. We'll do it one at a time and start with Resolution. In most textbooks, or documents, listing method requirements you'll find that "baseline resolution" is equivalent to a calculated value of 1.5 - personally I think that's cutting it close, but who am I to argue with the authorities. Anyhow, 1.5, 1.7 or 2 - how do we get to that number? There are actually different ways of calculating it. I'll give you only one, and that is the one stated in the European Pharmacopoeia (EP) (figure 1). Just remember, that there are different ways that may result in slightly different values - so if you're comparing data, make sure it was calculated in the same way.

This is the mathematical expression of the definition of resolution, showing that it is related to the retention time and peak width. The value of resolution can therefore be increased by increasing the spacing between the two peaks, or by decreasing peak widths to have narrower peaks. And THAT brings us right back to the Resolution Equation, seeing that "Selectivity" is another term for "the spacing between two peaks" and "Efficiency" is a measure for how sharp the eluting peaks are. But both can only be achieved once the compounds are sufficiently retained on the column to begin with. Figure 2 highlights the effect of increasing values of Retention, Selectivity and Efficiency on Resolution.

As can be seen in the graph, once retention reaches a k value > 2 the curve flattens out and increasing retention does not strongly affect resolution. After that, it is selectivity that has the greatest effect on resolution, which is why the focus in method development should centre on changing selectivity. The effect of increasing column efficiency is less marked, and can be used to optimize a method or speed it up, but it won't be very useful to achieve separation of overlapping peaks in the first place. Figure 3 gives a more visual example of the graph in figure 2.

For further information, have a look at this nice and detailed explanation that was prepared by my colleagues in the UK a while back: Back to Basics - The Resolution Equation

I think that is enough for today, next time I'll explain more about Efficiency (N), how to calculate and how to improve it. Until then - please don't hesitate to contact me with any questions you may have.