Chromatography Forum

N = 5.545 (Tr/Wh)^2 is a formula given, e.g., by IUPAC, for calculating plate number from a chromatogram resulting from an isocratic (or isothermal) elution. By it, one has but to measure peak width at half height (Wh) and retention time from time zero (Tr).

I am bothered by this formula (and a related one using base peak width). Because the retention time is measured with time zero being the injection time, rather than with time zero being the elution time of a completely nonretained compound. N therefore will not be independent of the extracolumn volume of the system.

One implication of this is that a column manufacturer could artificially inflate their reported number of theoretical plates, for instance in liquid chromatography by performing their chromatography on a system having a longer length of tubing between the injector and the column head. OK, sure, such a system might also cause an increase in peak width Wh, which would have an opposite effect on (would reduce) plate number, but I'd hardly expect these two effects to cancel each other out completely in all cases. For instance, the longer tube could be made a highly curved (sinusoidally or tortuously curled) one, which configurations are reported to induce less dispersion per length of tubing. Or the sample could be dissolved in a much weaker solvent, allowing it to become refocussed (stack) at the column head.

Am I missing something? Why is plate number such a gold standard for column efficiency?

Am I missing something?

Yes; quite a lot, actually.

First off, ignoring the science for a moment, there is very little incentive for a vendor to cheat because a good fraction of customers test columns when they come in the door (I've been on both sides of the aisle; I've always done it as a customer, and I've had interactions as a vendor). The last thing a vendor wants is columns being rejected as out of specification.

On a practical level, the length of tubing required is unrealistic. Taking a "workhorse" 150 x 4.6 mm column as an example, the dead volume will be somewhere around 1.5 mL (i.e., 1,500 microliters). Standard 0.010" id tubing (1/16" od) has about half a microliter of volume per centimeter of length. To double t0 would require 3,000 cm (30 meters!) of tubing. Even going up to 0.030" id (which is as big as you can get in 1/16" od) would require more than 3 meters of tubing, and I can guarantee that you're not going to be able to coil that tightly enough to avoid band broadening and mixing with the mobile phase.

You can actually measure the effective extra-column contribution to band broadening (and the "true" plate number of the column) by running an isocratic separation of some well-behaved peaks and plotting the variance of the peaks (σ^2; σ is 1/4 of the baseline width for a Gaussian peak) as a function of the retention time squared. The intercept should equal the extra-column σ^2; take the square root of that to find the extra-column contribution to σ); the slope should be the reciprocal of the column plate number, N.
(By way of “truth in advertising”: the numbers are only approximate because they are based on the simplifying assumption that all peaks have the same plate number, which is not exactly true but is "close enough" for practical purposes.)

Here's the math behind it:

Why is plate number such a gold standard for column efficiency

The dispersion of a peak in chromatography is essentially a linear function of retention time, so the efficiency of a column could well be characterized by the simple ratio of retention time to dispersion (width) or vice versa. Why don't we just do it that way? It's a mix of history and convenience. An early model of chromatography dynamics (late 40's - early 50's by A.J.P. Martin, I think -- someone correct me if I'm wrong on that) treated the distribution of molecules in a band as if it were the result of some finite number of stages of equilibration between the mobile and the stationary phase, much like what happens in a distillation column. The word "plates" is borrowed from distillation theory. The convenience part comes in because the plate number is a linear function of the column length, so plates/meter is a good measure of the quality of a column, independent of column length.