Essentials of LC Troubleshooting, VIII: A Deeper Look Into Column Efficiency

Column efficiency is a major determinant of resolution in both isocratic and gradient liquid chromatography (LC) separations. Developing expectations for the column efficiency we should see under a given set of conditions is important for effective troubleshooting because this can help us decide when to initiate a troubleshooting exercise. In this installment, we explore the key relationships between efficiency and mobile phase flow rate and particle size, as well as secondary determinants of efficiency, including analyte retention, size, and chemistry, and instrumental effects.

This installment of “LC Troubleshooting” is a continuation of the “Essentials of Troubleshooting” series I started a few years ago. As I have touched on several aspects encountered in troubleshooting, including variations in retention times, peaks widths, pressure, and flow, I have emphasized the importance of developing and having expectations related to our chromatographic experiments. Simply put, if we do not have a solid view of what to expect from our experiment (for example, what should I expect the retention time/peak width/pressure to be?), then we have no basis to observe that is something is not right, and deserves a closer look­—maybe even a troubleshooting effort. Expectations around peak width are of primary importance because it contributes directly to resolution, which is one of our primary metrics of chromatographic success or failure, and peak width is, in turn, heavily influenced by the chromatographic efficiency of the column we are working with. In my conversations with people from all corners of the chromatographic world, I find that many practitioners do not have a complete grasp of the factors that dictate efficiency, and thus their ability to develop solid expectations for efficiency and peak width can be limited. In a related vein, digesting information about chromatographic efficiency from research articles and technical materials from column suppliers can be confusing, or even misleading in some cases. Here again, developing a solid sense for what is expected and reasonable with respect to chromatographic efficiency is a powerful asset in any troubleshooter’s toolkit.

Efficiency: What It Is, and How We Measure It

In the chromatographic literature and community, a handful of terms are used more or less interchangeably when referring to chromatographic efficiency, including “efficiency,” “plates,” “theoretical plates,” and “plate number.” For our purposes here, they refer to the same thing, and so I will simply use the term “efficiency” or “plate number,” and the symbol N, throughout the rest of this installment. The significance of actual, physical plates comes from the design of distillation columns, particularly as they are used in chemical processing and manufacturing (for example, distillation towers used for refining petroleum) (1). In this context, the more plates added to a distillation column, the more resolution can be obtained between fractions of the distillate that are similar in boiling point. In modern columns used for liquid chromatography, there are no physical plates, thus the continued use of the term is a legacy that has held on from the early 20^th century. Nevertheless, the characterization of a column’s performance as consistent with having a certain number of plates is practically useful, and relatively easily measured. As stated by W.A. Peters, Jr. in 1922 in his article introducing the idea, “We have found it very convenient to measure the reciprocal of this efficiency, which will be called the ‘height equivalent to a theoretical plate’, or HETP.” (2). Given that the column length and the HETP (most people today just use H) both have units of distance, the idea that the number of plates that will “fit” in a column is intuitive. Thus, the relationship between the efficiency, or number of plates exhibited by a column, is straightforwardly given by equation 1:

Since there are no physical plates inside of our columns, we cannot directly measure their heights. Rather, plate heights (H) are either:

• calculated from the number of plates exhibited by the column based on experimental data; or
• predicted using a van Deemter Equation, or similar equation

For the relationship between N and the characteristics of peaks in the chromatograms we work with, I have two strong preferences. Conceptually, I suggest that people remember equation 2, where t_R is the mean retention time of a peak in a chromatogram, and σ_t is the standard deviation of the distribution of retention times that contribute to the peak—that is, the distribution of retention times, with some eluting from the column faster than an average analyte, and some eluting slower, is what gives the peak a width. Figure 1 shows the physical meaning of σ_t, where we assume here that the peak shape is well described by the Gaussian Distribution. Stated in words, N is the square of the ratio of retention time to the standard deviation of the distribution of retention times that give rise to the peak width. This formulation is simple, and easy to remember. To a first approximation, N for any peak in a chromatogram, regardless of retention time, should be roughly constant. There are important theoretical and practical reasons this is never exactly observed in practice, but I encourage people to view the plate number as a roughly constant property of the column (operated under a particular set of fixed conditions), which is very different from viewing the plate number as a property of an analyte.

But, unfortunately σ_t is not as easy to measure from experimental chromatograms. So, when it comes to actually making measurements to calculate a plate number from a chromatogram, I prefer equation 3, which uses the peak width at half its height, w_1/2. The half-height width is simple to measure. With the chromatogram in hand, either on paper or on a screen, we find the halfway point between the baseline and the peak apex, and draw a horizontal line across the peak. The time difference between the intersection points of the horizontal line and the leading and trailing parts of the peak is w_1/2, as shown in Figure 1.

Now, this is all fine when the peaks we are dealing with are symmetrical, and well-described by a Gaussian distribution. But, in practice, peaks are very often asymmetric (3), and these cases call for other estimates of the peak width. We don’t have the space to get into this here, but interested readers can take a look at the compilation of about 90 different functions used to describe chromatographic peak shapes reviewed by Marco and Bombi (4). The 2019 article by Howard Barth in this magazine on these topics is also a useful reference (5).

Primary Determinants of Efficiency: Velocity and Particle Size

Most introductions to chromatography—whether through a bachelors-level analytical chemistry course or an intensive short course at a conference, for example—teach that efficiency depends on mobile phase flow rate. An example of this dependence is shown in Figure 2. Here, we see that when the flow is low, efficiency suffers, and when the flow is high, efficiency suffers. There is an intermediate flow rate that maximizes the efficiency, and in many practical applications of liquid chromatography (LC), this is where we prefer to work, because high efficiencies increase the likelihood of achieving satisfactory resolution of the sample components of interest. Instrument and column manufacturers offer products that make it possible to work in this range. For example, the optimal flow rates for small molecule separations with 4.6-mm-i.d. columns and 3.5-µm-diameter particles are in the 1 mL/min to 2 mL/min range, which is achievable at high pressure with modern LC systems.

More advanced courses in chromatography discuss the effect of flow rate on efficiency differently. Flow rate is replaced with mobile phase linear velocity (u_m), and efficiency is replaced with HETP, or just H, and the resulting plot of H vs. u_m is known as a van Deemter curve (6). The relationship between N and H was already shown in equation 1. The relationship between flow rate and velocity is shown in equation 4, where t_m is the column dead time, F is the flow rate, r is the column inner radius, and ε_T is the total porosity of the column, including spaces between the stationary phase particles, and in the pores inside the particles. The first ratio in equation 4 (L/t_m) is convenient when an experimental value for t_m is available; the second ratio is convenient when we want to estimate the velocity (u_m) without making any measurements. Typical values of ε_T for porous particles used in LC are in the range of 0.5 to 0.7.

The switch to u_m and H is more of an engineer’s perspective, but it is very useful because it enables us to fairly compare data from different experiments where one or more variables such as particle size and column diameter have been changed. Figure 3a shows three curves that exhibit the dependence of H on F, for different combinations of particle size and column diameter. Suppose that these are for different technologies, and we want to compare them to see which might be best for our application. Interpreting this result is very difficult because of the changes in particle size and column diameter at the same time (that is, comparing the blue and black traces). Of course, in the best case, we would make such comparisons changing only one variable at time, but this is often very expensive if we need to buy columns just for this purpose, impossible due to limitations in commercial offerings, or because we are trying to use data from the literature where we don’t have control over the experimental details. A simple view of the trends in Panel A is that the 3.5-µm particle in the 4.6-mm i.d. column will provide the best result for fast separations, because it provides lowest plate height (and thus the largest efficiency) at the highest flow rate (and thus the shortest analysis time). However, this view is misleading. The first thing we need to fix is to account for the difference in column diameters, and a good way of doing this is to plot the plate heights versus mobile phase velocity instead of flow rate. The effect of this change is shown in Figure 3b where we have the same plate heights as in Figure 3a, but plotted against the velocity axis instead of the flow rate axis. Now our view of the data is very different: This plot suggests that the 2.5-µm particle in the 2.1-mm-i.d. column is the clear winner, because it yields the smallest plate height at highest mobile phase velocity. This view, that the 2.5-µm particle is the clear winner, is correct, but with one caveat—that this performance comes at the cost of higher pressure, and realizing this level of performance requires that our LC instrument can provide that pressure. Fortunately, most modern LC instruments are able to provide the pressure needed to successfully operate many columns with particles in the 2-µm range.

To complete this story, we need to take one more step, and deal with the fact that in Figure 3b we still have variations in particle size. Here, I think an analogy to boxing is instructive. Comparing the performance of different particle sizes in Figure 3b is a bit like comparing boxers from different weight classes. Putting a heavyweight boxer in the ring with a featherweight boxer will not result in a fair fight. Instead, sports analysts talk about “the best athlete, pound-for-pound”. Similarly, comparing the performance of a column with 5-µm particles to one with 2.5-µm particles based on plate height alone is not fair. And for LC columns, the equivalent of the “pound-for-pound” comparison is to plot the data in Figure 3 in “reduced terms” (another concept from engineering). We plot reduced plate height versus reduced velocity (Figure 3c), where the reduced parameters are given by equation 6. When we do this, we see that the plate height versus velocity curves that look very different in Figure 3b now collapse to a single unified curve in Figure 3c. In other words, the three columns discussed here perform identically on a “pound-for-pound” basis.

To explain the difference between the dependence of H on u_m in Figure 3b, and the dependence of h (reduced plate height) on v (reduced velocity) in Figure 3c, we start with a version of a van Deemter-like equation shown in equation 5. This relationship nicely describes the dependence of H on u_m, where d_p is the particle size, and D_m is the diffusion coefficient of the analyte in the mobile phase.

Then, we define the reduced parameters h and v:

Substituting the reduced parameters into equation 5 gives us a reduced form of the van Deemter equation (equation 7), which describes the set of three overlapping curves in Figure 3c:

The rigorously correct version of equation 7 would specify a specific type of reduced velocity referred to as the interstitial reduced velocity, but I’m not making that distinction in this installment because it is inconsequential for our discussion here.

The bottom line from this is that a fair comparison of columns really requires that we look at the dependence of efficiency on flow rate through this framework, where we convert flow rate to reduced velocity, and efficiency to reduced plate height. This effectively levels the playing field when comparing columns having different diameters and particles. The framework also enables us to quickly assess what column efficiency would should observe (and thus, the peak widths we should see) under a given set of conditions, so that we can recognize situations when a column is underperforming, or more happily, “punching above its weight.” It is generally understood in the field today that a “good” column should produce a minimum reduced plate height (that is, the value of h around the minima of the curves in Figures 3 and 4) of around 2 (7). In cases where the column seems to be underperforming, a troubleshooting exercise is warranted.

Secondary Determinants of Efficiency: Retention, Analyte Size, Analyte Chemistry, and Instrument Factors

In addition to the major influences of flow rate and particle size on efficiency, several other factors can influence observed efficiencies, and we must be aware of these when trying to assess whether or not a column is producing the efficiency we expect it to provide. Each of these factors deserves a much longer discussion; what follows here is a brief description of each effect, along with references to resources where readers can learn more about each of them.

Effect of Retention on Efficiency

A simple view of the effect of retention efficiency is that the more retained an analyte is (that is, with a higher retention factor), the more time it has to diffuse along the long axis of the column, broadening the peak and decreasing efficiency. This is particularly pronounced at low velocities where diffusion already increases peak broadening. Figure 4a shows an example of this. At a flow rate of 2 mL/min, the plate height is nearly independent of retention factor. However, at 0.5 mL/min. we see that there is a strong increase in the plate height with increasing retention factor. Readers interested in learning more about these effects are referred to a recent, detailed article in this area (8).

Effect of Analyte Size on Efficiency

In general, plate heights obtained at high mobile phase flow rates increase with increasing analyte size. This is due to the fact that larger molecules diffuse slower than smaller ones, which is characterized by a decrease in diffusion coefficient (D_m). Since D_m appears in the denominator of the C ^. u_m ^. d_p / D_m term of equation 5, decreasing D_m increases H, particularly at high mobile phase velocities to the right of the minima in the curves in Figure 3. There is not a whole lot we can do about this in LC, aside from increasing the column temperature (9) or decreasing particle size (10), but it is important to recognize this effect, particularly when working with molecules of intermediate size or larger.

Effect of Analyte Chemistry on Efficiency

Whereas the preceding discussion assumes that all analytes behave ideally, unfortunately it is not uncommon to observe analyte-specific effects on plate height, and the nature of these effects is as varied as the chemistry the of the molecules we work with. Readers interested in learning more about this are referred to previous installments of “LC Troubleshooting” where we have focused on some of the more commonly observed phenomena, tips for assessing whether or not you are observing on of them in our work, and potential remedies (11,12).

Effects of Instrumental Factors on Observed Efficiency

Last, but not least, several instrumental factors can contribute to observed column efficiency. These factors, including injection volume, peak dispersion in tubing and connections before and after the column, and dispersion in detectors, do not actually affect the efficiency of the column itself, but they can dramatically affect the efficiency we observe based on the widths of peaks in chromatograms. The seriousness of these effects become more serious as the length and diameter of columns decrease, and with the use of smaller particles. Readers interested in learning more about these effects are referred to several prior “LC Troubleshooting” installments on this topic (13–16).

Summary

In this eighth installment on essential topics in LC troubleshooting, I have discussed the topic of column efficiency in some detail. In prior installments, I have emphasized the importance of developing expectations for our LC systems and their performance, because there is no way of knowing there is a problem with our systems unless we have clear expectations for what normal performance should look like. Expectations around peak width are among the most important, because peak width is a strong determinant of resolution, and peak width is strongly influenced by column efficiency. The major determinants of column efficiency are flow rate and particle size. When comparing different columns or conditions involving changes in particle size and column diameter, it is very important to examine the dependence of plate height on mobile phase velocity, and looking at this dependence in both reduced and non-reduced terms is instructive. Finally, it is very helpful to be aware that retention factor, analyte size, analyte chemistry, and instrumental factors can also impact the apparent column efficiency we calculate based on the widths of peaks in our chromatograms. Developing expectations for the column efficiency we should observe is an essential skill for any effective LC troubleshooter.

References

1. Theoretical plate, Wikipedia. https://en.wikipedia.org/wiki/Theoretical_plate (accessed 2025-07-06).
2. Peters, Jr., W. A. The Efficiency and Capacity of Fractionating Columns. J. Ind. Eng. Chem. 1922, 14, 476–479. DOI: 10.1021/ie50150a002
3. Stoll, D. R. Essentials of LC Troubleshooting, III: Those Peaks Don’t Look Right. LCGC N. Am.2022, 40 (6), 244–247. DOI: 10.56530/lcgc.na.az3781g7
4. Di Marco, V. B.; Bombi, G. G. Mathematical Functions for the Representation of Chromatographic Peaks. J. Chromatog. A 2001, 931, 1–30. DOI: 10.1016/S0021-9673(01)01136-0
5. Barth, H. Chromatography Fundamentals, Part VI: The Gaussian Distribution and Moment Analysis. LCGC N. Am.2019, 37 (4), 269–273.
6. van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Longitudinal Diffusion and Resistance to Mass Transfer as Causes of NonIdeality in Chromatography. Chem. Eng. Sci.1956,5, 271–289. DOI: 10.1016/0009-2509(56)80003-1
7. Snyder, L. R.; Kirkland, J. J.; Dolan, J. W. Basic Concepts and Control of Separation, in: Introduction to Modern Liquid Chromatography, 3rd ed,; Wiley, 2010: pp. 25–28.
8. Desmet, G.: Song, H.; Makey, D.; Stoll, D. R.;Cabooter, D. Experimental Investigation of the Retention Factor Dependency of Eddy Dispersion in Packed Bed Columns and Relation to Knox’s Empirical Model Parameters, J. Chromatogr. A2020,1626, 461339. DOI: 10.1016/j.chroma.2020.461339
9. Carr, P. W.; Stoll, D. R.; Wang, X. Perspectives on Recent Advances in the Speed of High-Performance Liquid Chromatography. Anal. Chem.2011,83, 1890–1900. DOI: 10.1021/ac102570t
10. Jorgenson, J. W.; Capillary Liquid Chromatography at Ultrahigh Pressures. Annu. Rev. Anal. Chem. 2010, 3, 129–150. DOI: 10.1146/annurev.anchem.1.031207.113014
11. Stoll, D. R. Strongly Adsorbing Analytes: What, Why, and How to Fix It, LCGC N. Am. 2023, 41 (7), 242–244. DOI: 10.56530/lcgc.na.vr7280f7
12. McCalley, D. V.; Stoll, D. R. But My Peaks Are Not Gaussian! Part III: Physicochemical Causes of Peak Tailing. LCGC N. Am. 2021, 39, 526-531, 539.
13. Stoll, D. R.; Broeckhoven, K. Where Has My Efficiency Gone? Impacts of Extracolumn Peak Broadening on Performance, Part I: Basic Concepts. LCGC N. Am. 2021, 39 (4), 159–166.
14. Stoll, D. R.; Broeckhoven, K. Where Has My Efficiency Gone? Impacts of Extracolumn Peak Broadening on Performance, Part II: Sample Injection, LCGC N. Am. 2021, 39 (5), 208–213.
15. Stoll, D. R.; Broeckhoven, K. Where Has My Efficiency Gone? Impacts of Extracolumn Peak Broadening on Performance, Part III: Tubing and Detectors, LCGC N. Am. 2021, 39 (6), 252–257.
16. Stoll, D. R.; Lauer, T. J.; Broeckhoven, K. Where Has My Efficiency Gone? Impacts of Extracolumn Peak Broadening on Performance, Part IV: Gradient Elution, Flow Splitting, and a Holistic View, LCGC N. Am. 2021, 39 (7), 308–314.

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