Kinetic Plots Made Easy - - Chromatography Online

Kinetic Plots Made Easy

Nov 1, 2009
By: Uwe D. Neue
LCGC North America

In recent years, kinetic plots have become the tool by which different liquid chromatography (LC) columns, particles, and particle sizes have been compared with each other. Also, questions about performance, column length, and the merits of very high pressures have been tackled using the same tools. However, the discussion of kinetic plots has been limited to only a few authors, and even there one can see a multitude of different versions of the kinetic plots, together with often very different interpretations. The understanding has remained limited to a few experts. However, this does not need to be so. In this month's "LC Troubleshooting," we will develop the principle of these kinetic plots from scratch such that they become accessible to everyone. The content of this article is based upon a presentation at PittCon 2009 (1).

Varied Applications

The very first example of a kinetic plot — although not called by this name — was shown by Giddings in 1965 in a "comparison of theoretical limit of separation speed in gas and liquid chromatography" (2). A paper in German by Halász and Görlitz (3), unread by later investigators, outlined the complete theory of kinetic plots, without the use of this expression, which was created by Desmet and coworkers around 2005 (4,5). Later, and apparently independently, Poppe used kinetic plots to explore the compromises between speed and efficiency in LC (6). The credit for the modern resurrection of these "Poppe plots" probably goes equally to Schoenmakers for the use of Poppe plots in size exclusion chromatography (7) and to Desmet and coworkers, who published two important papers (4,5) that laid the foundation for the modern use of kinetic plots. Wang expanded the Poppe plot to gradient conditions (8). Desmet followed up with many papers using the kinetic plots for a range of comparisons between different devices or chromatographic conditions, from monoliths (9) to elevated temperature (10,11), to very small particles (12–14). His views have been summarized in reference 15 with an outlook on the future technology and designs of column support formats. A recent publication by Carr elaborated the math of kinetic plots and related optimizations (16). In more practically oriented papers, the conditions required to reach 100,000 plates (17), the validity of kinetic plots under ultrahigh-pressure liquid chromatography (UHPLC) conditions (18), the influence of analyte properties (19), and the properties of different packings (20) have been examined using the kinetic plot tools.

Many chromatographers are not very familiar with kinetic plots, so we decided to present a simple, but accurate, description of this technique. We have chosen to minimize the mathematical complexity; however, the underlying math can be found in a sidebar. To describe the technique, we will start with plots of plate number N versus analysis time, which have been used frequently to explain column performance (21–24). The limiting form of these plots is one basic version of a kinetic plot. We then show how the basic kinetic plot changes as we change the particle size. Finally, we will briefly cover other versions of these plots and explain why and under what circumstances they are useful.

A Simple Kinetic Plot

Figure 1

Figure 1 is a plot of the plate number as a function of the analysis time for a 100-mm-long column packed with 3.5-µm particles. The curve was calculated for a fixed retention factor of k + 1 = 10. At short analysis times (high flow rates), the plate number is low and increases as we reduce the flow rate, that is, as the analysis time increases. A maximum plate number is reached as we further increase the analysis time (by reduction of the flow rate). Then the column performance declines as we get into the diffusion-controlled range of the van Deemter curve for this particle size at very low flow rates. The curve stops at the left at the point where the pressure limit of the LC instrument is reached. We have chosen a value of 400 bar (6000 psi), corresponding to the upper pressure limit of conventional LC equipment. The point where the limiting pressure is reached is one point of the kinetic plot curve.

Figure 2

We can generate the same curve for a range of different column lengths, all packed with the same particle size. This is shown in Figure 2. Here, the column length is varied from 20 to 6000 mm (6 m). The pressure limit is kept at 400 bar for all column lengths. This means that we can run the fastest analysis on the shortest column: with a 20-mm column, the analysis time is 2.5 s, with a plate number of 175 plates. The speed of the analysis changes quickly with the column length (remember that the pressure is fixed at 400 bar).The 50-mm column reaches about 1000 plates in 15 s at 400 bar, whereas the 150- and 250-mm-long columns generate 8000 and 19,000 plates, respectively, at the pressure limit. At the other end of the graph, approximately 400,000 plates can be achieved in 1.5 days with a 4500-mm-long column. The upper limit of the time axis was arbitrarily set to 10,000 min, or about 7 days.

A kinetic plot is simply the line drawn through the individual column graphs at the point where each column reaches the pressure limit that we have selected. This corresponds to the heavy line on the left side of the graph. It tells us what plate number is achievable at a particular analysis time for the selected particle size at the preselected pressure. In the kinetic plot view of the world, the column length is freely adjustable. Thus, it shows directly the compromise between the analysis time and the plate number that can be reached with the given particle size and pressure limit. This pressure limit is often that of the LC instrument; in most cases 400 bar, as selected here.

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