Is the van Deemter (Knox) equations applicable for gradients

[Rob Burgess]

Forgive me if this sounds a bit simplistic, but are the fundamental parameters of the van Deemter equation really applicable to gradient HPLc separations. I haven't had any luck so far in pinpointing the answer in various texts? Are van Deemter plots only really true and relevant for isocratic separations?

Are there any other thoughts on this question?

[Uwe Neue]

The van-Deemter equation applies equally well to gradients and isocratic separations. The plates change in exactly the same way as in isocratic separations. The problem is that you can not measure the plate count from a gradient chromatogram. I'll explain this below.

You can test the validity of this statement by running the same gradient at different flow rates, and plot the peak width multiplied by the flow rate versus the flow rate. If you run this over a sufficiently wide range of flow rates, the plot will look exactly like a van-Deemter curve. When you do this, you need to scale the gradient such that the gradient volume remains constant. This means when you increase the flow 2 times, you need to reduce the gradient run time 2 times.

The reason that you can not measure the plate count from a gradient chromatogram is as follows: the plate count is really defined as the squared ratio of the column length divided by the standard deviation (= 1/4 of the peak width) inside the column. When you run isocratic chromatography, this ratio simply multiplies out to the commonly used equation. This is because the same factors apply to the retention time and the peak width. When you do a gradient, the multiplication factor is not known, and therefore you can not do the calculation.

There is a trick that relies on fairly good assumptions that allow you to get at the plate count. This trick assumes that the Snyder theories about gradients hold. You would need to run two gradients with a different slope. From this you can calculate the real retention factor of the peak as it leaves the column (k at point of elution). Once you know that, you can calculate the plate count in a gradient.

All of this stuff is only of interest for theoreticians, not for the practical chromatography most people do...

Oh, the trick how to get plates from gradients is described in my book on page 77. There is an error in the equation 3.21, but those of you who really want to get into this will see it immediately...

[Rob Burgess]

Hi Uwe,
Many thanks for your most informative answer, it is much appreciated.

The reason I asked this question originaly, is that many instrument manufacturer's (and column come to that) insist on showing us van deemeter plots for sub 2µ particle packed columns at high pressure conditions. I wonder if the data they are showing is for isocratic separations (i.e. std. reduced h vs. v plots) or whether it really is for gradient separations under these conditions (which I presume would be peak width x. flow rate vs. flow rate).

Can anyone enlighten us here on this forum?

PS. For many of us performing impurity separations (on pharmaceuticals) I would hazard a guess that > 90% of methods are still gradient based! Therefore we need to truly figure out what would the real advantage be for running under UPLC conditions? For any methods I develop, resolution (read robustness) and sensitivity (0.05% and below of main peak) would be more important parameters than simply time savings on a single run.

[Uwe Neue]

If you got the examples from Waters, they were real van-Deemter curves obtained under isocratic conditions. The same is true for plate count statements. I can not guarantee that all manufacturers play by the rules of science.

The measure that is used for the assessment of the quality of a gradient separation is the peak capacity. While it is a good measure of the performance of a gradient separation, it also depends on the gradient conditions, especially the gradient span. The gradient span is the ratio of gradient volume to column volume. I can pack two times as many peaks in a gradient span with double the gradient run time as I can with the simple run time at the same flow rate. When you play the game with a fixed gradient run time, but different flow rates, the story gets complicated: you increase the peak capacity due to the higher flow rate, but the higher flow rate also decreases the plate count. Depending on the details of your gradient conditions, this may sometimes get you into a better spot for your separation, sometimes not.

For your practical purposes, I would ask for van-Deemter curves under isocratic conditions to get a feeling for the quality of the column(s), and/or the quality of UPLC vs. HPLC, and then use the information provided above to optimize my own separations. With respect to the parameters that you asked for for specific separations (robustness, sensitivity etc.), you can ask you sales rep for application examples, or go to the website and look for examples.

[4ugirish]

i have one doubt in relation with the topic under discussion.
In gradient run we get sharp(more N) peaks than that of the isocratic run , how can it be explained on the ground of Van Demeter equation?

please quote

[tom jupille]

As Uwe pointed out in the second post, the commonly used isocratic formula is meaningless in a gradient system.