Chromatography Theory, Part III: Calculating Elution Times and Capacity Factors

In our ongoing quest to understand how to optimize methods for chromatographic separations, we will see how to calculate retention times and introduce the concept of the capacity factor. The capacity factor normalizes retention time so that these quantities can be compared to each other even if separations are performed under different conditions. The capacity factor will help us predict how well analytes will be separated from each other, and we will find that this depends upon an important concept from the last column, the fraction of time the analyte spends on the stationary phase, F_s.

Following up on our discussion of quantitating the separation of mixtures from the last column, here we will discuss further how to calculate the expected retention time of an analyte, and then define what are called capacity factors. The capacity factor is a way of normalizing retention times so we can compare chromatograms that are measured, for example, at different flow rates.

How to Calculate Elution Times

Recall that in the last column (1), we derived an equation that allows us to calculate the elution time for an analyte, as seen in Equation 1:

where t_e is the retention time, t₀ is the void time in minutes, and F_s is the fraction of time spent adsorbed on the stationary phase.

Note that the quantity (1-F_s) is in the denominator. So as F_s goes up, that is, the higher the fraction of time an analyte spends adsorbed and immobile on the stationary phase, (1-F_s) goes down and hence t_e goes up. If you imagine our different types of analyte molecules are in a horse race as in the last column (1), Equation 1 says, for example, that a horse that spends more time standing around grazing, as horses are wont to do, and not moving will get to the finish line later than a horse that spends all its time running.

Equation 1 is, in my opinion, the most important equation in chromatography because it relates the quantity we care about as measured by a chromatograph, the elution time of an analyte, to a property of the analyte­—the fraction of time it spends on the stationary phase. We will be seeing Equation 1 a lot going forward because it will provide us with the theoretical foundation for understanding how different chromatography separation methods work. As I have mentioned (1), and will probably mention again, the tricky part in achieving a separation is getting F_s to be different for all the analytes in a mixture you wish to separate.

Figure 1 shows a chromatogram from last time (1). We assumed the sample was tetrahydrocannabinol (THC) dissolved in ethanol, and that the ethanol molecules had no affinity for the stationary phase, did not adsorb, and hence had a F_s value of 0. We also established last time (1) that the t₀ for ethanol is 2 min. The elution time of ethanol then using Equation 1 is:

which shows that for unadsorbed molecules the elution time is the void time, which here is 2 min.

Now Equations 2-4 were an easy calculation because we were not dealing with an analyte. Last column we assumed (1) that our THC molecule spent ½ of its time on the stationary phase, or in other words its F_s is 0.5. How then would we calculate the retention time of this analyte?

Using Equation 1 as such:

The reason THC has an elution time of 4 min in Figure 1 is that 2 x 2 min = 4 min. Note that the elution time for THC is twice that of ethanol because it spent half its time stationary whereas the ethanol spent no time stationary.

The beauty of Equation 1 is that it allows us to calculate the retention time of an analyte as long as we know F_s. The practical problem here is that we rarely know F_s exactly because it depends upon several factors including the chemical natures of the analyte, mobile phase, and stationary phase. Thus, our lack of knowledge of the exact value of F_s means we must experiment with chromatographic conditions to optimize separations. However, Equation 1 is still true and can help guide our experiments. Given knowledge of the chemical structures of the analytes to be separated, one can deduce based on Equation 1 what types of experiments to perform to obtain different values of F_s for the analytes and hence achieve a separation. More on that in future columns.

The Capacity Factor: k

As with all things chromatography, I refer you to publications (2,3) and references therein for a more detailed discussion of what is presented here.

So far, we have only considered a single chromatogram measured under controlled conditions. A problem with chromatography is that variations in the experimental variables affect retention time. For example, when we first discussed Figure 1 (1), we stated that the flow rate was 1 mL/min which gave t0 = 2 min. Now, if we slow down the flow rate for this separation to 0.5 mL/min, an unadsorbed analyte molecule will take twice as long to elute as before and give a t₀ of 4 min. Note that in the calculation for the retention time of THC in Equations 5-7 they all depend on t₀. Specifically, Equation 6 says that the retention time for THC equals 2t₀. By slowing down our flow rate to 0.5 mL/min and increasing t₀ to 4 min the elution time for THC is now 8 min, compared to 4 min at a flow rate of 1 mL/min. These results mean that separations performed at different flow rates have different retention times, which ultimately means retention times in chromatograms run at different flow rates cannot be compared to each other.

A practical problem with chromatography, particularly with the pumps used in liquid chromatography, is that a given pump may not give the same exact flow rate over time. Worse yet, pumps on different instruments set to the same flow rate may give measurably different flow rates. This means that elution times for the same sample can vary from instrument to instrument, which is a real problem when comparing data across instruments and developing methods.

Fortunately, there is a solution to this problem. We calculate for each peak in a chromatogram what is called its capacity factor, k. The capacity factor gives us a way of normalizing retention times so we can compare them across separations on different chromatographs. For example, capacity factors allow us to compare retention times when separations are performed at differing flow rates. The equation for calculating a capacity factor for a given chromatographic peak is seen in Equation 8:

where k is the capacity factor, t_e is the elution time, and t₀ is the void time.

Let’s calculate the capacity factors for the two peaks in Figure 1. For ethanol t_e = t₀, which means (t_e – t₀) = 0 hence k = 0. This shows that for all void time peaks the capacity factor is 0. From Figure 1 t_e for THC is 4 min and t₀ is 2 min, hence its capacity factor is:

Figure 2 shows the chromatogram from Figure 1 with its peaks labeled with their capacity factors.

We said above that if we slowed our flow rate down to 0.5 mL/min t₀ for our separation becomes 4 min and the t_e for THC becomes 8 min. We have established that the capacity factor for all peaks at t₀ is 0. What is the capacity factor for THC at this new flow rate?

Note that despite the change in flow rate the capacity factor for THC is still 1 as seen in both Equations 10 and 12. The reason for this is that in equation 8 the numerator is in units of time, the denominator is in units of time, and when we calculate k the two units cancel, meaning the capacity factor is a unitless quantity that does not depend upon flow rate. This means that by using chromatograms labeled with their capacity factors we can compare retention times to each other even if they are measured at different flow rates.

Figure 3 shows a chromatogram with a void volume peak and two analyte peaks. In this chromatogram t₀ is 2 min and the analyte elution time for peak A is 4 min and the elution time for peak B is 8 min. What are the capacity factors for the peaks in this separation? Well, the first two peaks are the same as in Figure 1, so their capacity factors are still 0 and 1, respectively. The capacity factor for peak B, k_b, is calculated as follows:

kb = 3

Peaks A and B are said to be baseline separated because there is no overlap between them and there is a flat baseline between them. Note that they also have quite different capacity factors, which indicates that the relative capacity factors of two peaks might be a measure of the quality of a separation. More on that next time.

Conclusions

We presented the equation from last time for how to calculate elution time, emphasizing the importance of the fraction of time, F_s, an analyte spends adsorbed and immobile on the stationary phase. We then calculated some elution times based on assumed values of F_s. We found that elution times for the same separation measured at different flow rates are different, and that the capacity factor, k, is a flow rate independent quantity that normalizes retention time and allows us to compare retention times measured under different conditions. Finally, we saw that for baseline separated peaks their capacity factors are different.

References

1. Smith, B.C., Chromatography Theory, Part II: How Separations Occur Illustrated and Quantitated, Cannabis Science and Technology®, 2023, 6 (1), 7-12.
2. Skoog, D.A.; West, D.M.; and Holler, E.J.; Analytical Chemistry: An Introduction,Saunders College Publishing. 1994, 6.
3. Robinson, J.W.; Undergraduate Instrumental Analysis, Marcel Dekker, 1995, 5.

About the Columnist

Brian C. Smith, PhD, is Founder, CEO, and Chief Technical Officer of Big Sur Scientific. He is the inventor of the BSS series of patented mid-infrared based cannabis analyzers. Dr. Smith has done pioneering research and published numerous peer-reviewed papers on the application of mid-infrared spectroscopy to cannabis analysis, and sits on the editorial board of Cannabis Science and Technology. He has worked as a laboratory director for a cannabis extractor, as an analytical chemist for Waters Associates and PerkinElmer, and as an analytical instrument salesperson. He has more than 30 years of experience in chemical analysis and has written three books on the subject. Dr. Smith earned his PhD on physical chemistry from Dartmouth College. Direct correspondence to: brian@bigsurscientific.com.

How to Cite this Article:

Smith, B., Chromatography Theory, Part III: Calculating Elution Times and Capacity Factors, Cannabis Science and Technology, 2023, 6(2), 8-11.