Chromatographic Theory, Part IV: Measuring Separation Quality with Capacity and Separation Factors

In previous columns we established the importance of F_s, the fraction of time an analyte spends on the stationary phase, in determining elution time. We can now move on and use the elution times in a separation to establish its quality. We will use the capacity factor of chromatographic peaks as discussed last time (1) to calculate separation factors, whose value tells us how well two peaks are or aren’t separated.

Review of Capacity Factors

In my last column (1), we discussed how the capacity factor is used to normalize the peak positions in a chromatogram. For example, separations performed at different flow rates will have different elution times, but by calculating the capacity factors for the peaks in these chromatographs we can take the different flow rates into account and compare peak positions in different chromatograms to each other. To review, the capacity factor for a chromatographic peak is given by Equation 1.

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

To see the impact of the capacity factor please look at Figures 1 and 2.

Recall (1) that Figure 1 shows the retention times for two peaks we assumed were ethanol and tetrahydrocannabinol (THC). The problem here is that if the conditions of a separation are changed, such as the flow rate, the retention time will change. Figure 2 shows the chromatogram of Figure 1 with the peaks labeled with their capacity factors as calculated from Equation 1. Again, the k’s for these peaks should be the same even if variables like flow rate affected retention time.

To be more realistic, Figure 3 shows a chromatogram with three peaks labeled with their capacity factors.

We will assume the first peak with a capacity factor of zero is unretained ethanol molecules, the peak with k = 1 is THC, and for the sake of argument let’s assume the peak with k = 3 is from cannabidiol (CBD).

Now, note that k is an x-axis unit that describes peak position in a similar fashion to retention time. Our goal in chromatography is to obtain a separation for all peaks that are baseline separated; that is, there is a region of flat baseline between all peaks. All the peaks in Figure 3 are baseline separated. Since k is a measure of peak position, the capacity factors of different peaks can tell us something about how well they are separated. For example, two peaks with k’s of 2 and 2.1 may overlap and not be baseline separated, whereas peaks with capacity factors of 1 and 3 are clearly baseline separated as seen in Figure 3. This means that the relative value of the capacity factors of two peaks tells us something about how far apart they are on the x-axis and can give us a measure of separation quality. This quantity is called the separation factor.

Separation Factors

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.

The separation factor for a pair of chromatographic peaks is given by Equation 2:

where α is the separation factor, t_b is the retention time of peak b, t_a is the retention time of peak a, t₀ is the void time, and t_b > t_a.

The restriction in Equation 2 is that the retention time of peak b must be greater than that of peak a is so that the separation factor is a number >=1.

Note in Equation 2 that the numerator, (t_b – t₀), is the retention time for analyte b minus the void time t₀, the time spent in the column by an unabsorbed molecule. Thus, (t_b – t₀) measures the amount of time analyte b spends immobile and adsorbed to the stationary phase. Similarly, (t_a – t₀) is a measure of how long analyte a spends immobile and adsorbed to the stationary phase. The separation factor then depends upon the ratio of the time two analytes spend adsorbed on the stationary phase. The greater the difference in these times, the further apart these analyte peaks will be and the better their separation will be. This is consistent with what we have said in the past about the importance of F_s, the fraction of time an analyte spends on the stationary phase, in determining elution times and separation quality.

Using Equation 1 we can write the capacity factors for peaks a and b as such for Equations 3 and 4:

where k_a is the capacity factor for analyte a, k_b is the capacity factor for analyte b, and all other terms have been previously defined.

If we take Equations 3 and 4 and divide them into each other and rearrange, we find Equations 5 and 6:

Note that the right-hand sides of Equations of 2 and 5 are the same, thus:

In other words, the separation factor depends upon the capacity factors for any pair of chromatographic peaks, in this case peaks a and b. Thus, the separation factor and hence the quality of the separation of two peaks can be calculated from knowledge of their capacity factors.

The Separation Factor in Action

The purpose of this section will be to illustrate what the separation factor is for different qualities of separation. The worst possible scenario for a separation, of course, is when two analytes elute at the same time, otherwise known as co-elution. In cases of co-elution, t_b = t_a. Substituting this into Equation 2 we find that in Equation 7:

Note that the denominator and numerator are the same in Equation 7, so two co-eluted peaks have a separation factor of 1.

For a more realistic example, let’s imagine that two peaks both elute at 4 min, and that t₀ is 2 min. The separation factor in this case is seen in Equation 8:

In this case both the numerator and denominator are 2, and 2/2 equals 1. So, the worst possible separation factor a pair of peaks can have is 1, which is a lousy separation and something to be avoided.

What is the separation factor for the first two peaks in Figure 3 with capacity factors of 0 and 1?

Using equation 6 we find that this is found in Equation 9:

OOPS! We are trying to divide by zero here, which is an undefined mathematical operation. A sure way to crash most any computer is to tell it to divide by zero, and yes, I speak from personal experience here. What Equation 9 shows us is that we should not use void time peaks to calculate separation factors, only peaks with nonzero values of k will work.

What is the separation factor for peaks a and b in Figure 3? Again, using Equation 6 we find that in Equations 10 and 11:

Note that these two peaks are far apart, are baseline separated, and have a separation factor of 3. What we know so far then is that a pair of peaks with α = 1 are poorly separated and a pair of peaks with α = 3 are well separated. Thus, the separation factor is a measure of the quality of the separation between
two peaks.

Limitations of the Capacity Factor

So far in our calculations of separation quality we have only used elution time, that is the value at the top of a chromatographic peak. However, chromatographic peaks clearly have widths in addition to heights as is clearly seen in Figures 1–3, and our equations so far have ignored this fact. This can be trouble in the case, for example, of the chromatogram seen in Figure 4.

Using Equation 2 we find that the separation factor for peaks a and b, ignoring peak widths, as seen in Equation 12 is:

Note that the chromatograms in Figures 3 and 4 both have the same separation factor of 3, but in Figure 3 the two peaks are baseline separated, whereas in Figure 4 the two peaks are clearly overlapped. This means that because the separation factor ignores peak widths in some cases it has limits in terms of how well it tells us the quality of a separation. We will see in the next column how to take account of peak widths to get around this problem.

Conclusions

We reviewed capacity factors and established that they are a measure of elution time. Then, we defined the separation factor and found that it depends upon the fraction time, F_s, two analytes spend in the stationary phase. We also established that the separation factor is a function of the ratio of the capacity factors for a pair of chromatographic peaks, establishing that capacity factors can be used to determine separation quality. We found that for co-eluting peaks α = 1 whereas for a pair of well separated peaks α = 3. Lastly, we saw how that by ignoring peak width the separation factor has limitations and sometimes gives misleading information on the quality of a separation.

References

1. Smith, B., Chromatography Theory, Part III: Calculating Elution Times and Capacity Factors, Cannabis Science and Technology, 2023, 6(2), 8-11.
2. Skoog, D.A.; West, D.M.; and Holler, E.J., Analytical Chemistry: An Introduction, Saunders College Publishing, 1994, 6th Edition.
3. Robinson, J.W., Undergraduate Instrumental Analysis, Marcel Dekker, 1995, 5th Edition.

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., Chromatographic Theory, Part IV: Measuring Separation Quality with Capacity and Separation Factors, Cannabis Science and Technology, 2023, 6(3), 8-11.