Letter pubs.acs.org/ac
Further Considerations of Exact Equations for Peak Capacity in Isocratic Liquid Chromatography Thomas L. Chester* Department of Chemistry, University of Cincinnati, P.O. Box 210172, Cincinnati, Ohio 45221-0172, United States ABSTRACT: Further consideration is made of the exact equation for peak capacity by Potts, L. W.; Carr, P. W. Anal. Chem. 2011, 83, 7614−7615 for liquid chromatographs with significant extra-column broadening. Equivalent exact equations are easily derived that provide comprehension and insight of how extra-column broadening degrades chromatographic performance as measured by peak capacity.
P
contributions to observed peak widths are often significant today, even in low-dispersion ultrahigh-performance liquid chromatography (UHPLC) instruments, especially when low-plate-height stationary phases are used in columns of small volume. It remains a reasonable assumption that Ncol is the same for all peaks, but when σex is significant Nobs is not constant, its degradation is worst for the earliest peak and Nobs approaches Ncol as retention increases. If the extra-column contributions to broadening are independent of the column contributions, then their variances are additive, and the observed peak width can be expressed as
eak capacity in chromatography is generally regarded as the maximum number of peaks that can be separated at a stated resolution within a given time interval.1 Giddings introduced the concept of peak capacity for size exclusion chromatography and derived a formula for its estimation, which is also applicable to isocratic high-performance liquid chromatography (HPLC) separations.2 Grushka later showed that Giddings’ formula for estimation was actually exact.3 Both Giddings and Grushka assumed that peaks are separated when the resolution, Rs, equals unity. Rs in terms of time is
R s = Δt/4σ
(1)
where Δt is the time between the maxima of two adjacent peaks and σ is their average standard deviation. Analysts may be required to seek resolution higher than unity, particularly for quantitative analyses of regulated samples, while still using peak capacity as a performance measure. Assuming constant flow rates, Giddings’ formula for peak capacity, n, as a function of time and allowing specification of the desired resolution between adjacent peaks is n=1+
Ncol 4R s
⎛ t R, n ⎞ ⎟⎟ ln⎜⎜ ⎝ t R,1 ⎠
σi =
Ncol
+ σex 2
(3)
where σi is the standard deviation observed for peak i, tR,i2/Ncol is the variance arising from the column for a peak eluting at time tR,i, and σex2 is the extra-column variance, which is the square of the standard deviation due to extra-column sources (σex). The value of σex can be measured directly by replacing the column with a zero-dead-volume union and making a small-volume injection. Vivo-Truyols et al. derived an approximate equation to estimate peak capacity in isocratic systems with significant extracolumn broadening.5 Potts and Carr later showed that the VivoTruyols et al. equation is accurate to within about 2.5% but that it is easier to calculate an exact value for peak capacity.6 The exact equation given by Potts and Carr is
(2)
where Ncol is the column plate number, tR,1 is the retention time of the first peak of interest, and tR,n is the retention time of the last peak of interest. Giddings and Grushka made no distinction between the observed plate number, Nobs, and Ncol. Both of theses authors also assumed that the plate number is constant for all peaks in the chromatogram. These were reasonable assumptions for small injection volumes at the time of these works because nearly all the width of a peak was contributed by the relatively inefficient columns then available. However, these assumptions are no longer reasonable with the history of improvements in stationary phase materials, the continuing decrease of stationary-phase particle size, and the introduction of superficially porous particles (SPP, also known as core-shell, solid-core, etc.).4 Extra-column © 2014 American Chemical Society
t R, i 2
n=1+
⎞⎤ ⎛ t Ncol ⎡ −1⎛ t R, n ⎞ ⎢sinh ⎜⎜ ⎟⎟ − sinh−1⎜⎜ R,1 ⎟⎟⎥ 4R s ⎢⎣ ⎝ σex Ncol ⎠⎥⎦ ⎝ σex Ncol ⎠
(4)
where sinh−1 is the inverse hyperbolic sine function, sinh−1(x) = ln(x +
x2 + 1 )
(5)
Received: May 31, 2014 Accepted: July 6, 2014 Published: July 6, 2014 7239
dx.doi.org/10.1021/ac502005a | Anal. Chem. 2014, 86, 7239−7241
Analytical Chemistry
Letter
Equation 4, while exact, does not provide any immediate insight regarding how extra-column broadening affects peak capacity, particularly for users unfamiliar with the behavior of the inverse hyperbolic sine function. Substituting 5 into 4 and rearranging leads to numerous, equivalent exact expressions that can provide this insight. First, consider n=1+
Ncol 4R s
⎛t + R, n ln⎜ ⎜ ⎝ t R,1 +
t R, n 2 + σex 2Ncol ⎞⎟ ⎟ t R,12 + σex 2Ncol ⎠
In this equation, the 1 on the right-hand side is the peak capacity provided by the system. This results because it is always possible to elute the void peak even when Ncol is zero (meaning there is no column). (N/col)1/2/4Rs ln(tR,n/tR,1) is the additional peak capacity contributed by the column. {(N col ) 1/2 / (4Rs)} ln[(tR,n/tR,1) × {(tR,1 + (tR,12 + σex2Ncol)1/2)/(tR,n + (tR,n2 + σex2Ncol)1/2)}] is the peak capacity lost from the column contribution due to extra-column broadening in an imperfect system. Note that the first two terms in eq 7 correspond to Giddings’ original formula for a system where σex is zero, eq 2, and that the third term in eq 7 is zero when σex is zero. With eqs 6 and 7, the dependencies and limits of n with respect to the variables are clear by inspection. In particular, n approaches eq 1 as σex approaches zero, and n steadily decreases and approaches unity as σex increases to large values (more specifically, when σex2Ncol ≫ tR,n2). Dependencies of peak capacity on interrelated parametric variables such as particle size, column dimensions, and flow rate can be deduced by making appropriate substitutions. Because we have used the time domain to express σex, it will have some dependence on flow rate. Note that σex cannot be entirely predicted from theory because the specific geometry and flow properties of the system cannot be known in advance: both bends in the tubing and turbulence or eddys in improperly assembled fittings will affect σex and its behavior with respect to flow rate. Figure 1 illustrates the extent of n loss vs σex for several combinations of system and column. The upper trace in each chart is the peak capacity using Giddings’ original equation,
(6)
This equation uses only familiar functions. Note that evaluating eq 4, as written, using an inverse hyperbolic sine function within a program or spreadsheet will result in a divide-by-zero error if σex = 0. This is a result of evaluating the argument before applying the inverse hyperbolic sine function. The exactly equivalent eq 6 will evaluate without error even for σex = 0. For educational purposes, the following rearrangement, while slightly more complicated, is even more insightful: n=1+
×
Ncol 4R s
t R,1 + t R, n +
⎛ ⎛ t R, n ⎞ Ncol ⎜ t R, n ⎟⎟ − ln⎜⎜ ln ⎜t 4R s ⎝ t R,1 ⎠ ⎝ R,1
t R,12 + σex 2Ncol ⎞⎟ ⎟ t R, n 2 + σex 2Ncol ⎠
(7)
Figure 1. Isocratic peak capacity variation as a function of extra-column broadening. In all cases the upper trace (long-dashed) is calculated using eq 2, which ignores extra-column broadening, and the lower trace (solid) is the exact solution calculated using eq 7. The difference between the two traces is the peak capacity lost due to extra-column broadening, which is the third term in eq 7. The arrows along the abscissas indicate the typical range of the standard deviation due to extra-column broadening, σex, in typical instruments of the type indicated. 7240
dx.doi.org/10.1021/ac502005a | Anal. Chem. 2014, 86, 7239−7241
Analytical Chemistry
Letter
and the lower trace is the exact peak capacity using eq 7. The difference between the two traces is the peak capacity lost due to extra-column broadening, and this also equals the third term in eq 7. In all cases the time range of n corresponds to retention factors from 0 to 20. Flow rates were chosen to provide the same reduced velocity for charts a, b, and c, all of which are for totally porous particles. The flow rate in chart d, which is for 1.3-μm superficially porous particles, is the same as for chart b because such small SPP particles are often compared with larger totally porous particles under otherwise identical conditions. The arrows along the abscissas indicate the typical ranges of σex for the properly configured commercial instruments represented in each chart. Ncol was estimated assuming the plate height is three particle diameters for the totally porous packings and two particle diameters for the SPP case. It is clear that significant peak capacity will be lost in many chromatographic systems, even for low-dispersion UHPLC systems when used with small-volume columns packed with very efficient particles as in Figure 1c. The losses are even worse when high-efficiency and/or small-volume columns are used with older HPLC systems having large extra-column broadening. In addition, serious performance degradation can easily occur in modified instruments, particularly in open-access or shared instruments where users may not be aware of all the changes that have been made in tubing and fittings.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +1-513-310-7375. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Poole, C. F. The Essence of Chromatography; Elsevier: Amsterdam, The Netherlands, 2003; pp 56−58. (2) Giddings, J. C. Anal. Chem. 1967, 39, 1027−1028. (3) Grushka, E. Anal. Chem. 1970, 42, 1142−1147. (4) Chester, T. L. Anal. Chem. 2013, 85, 579−589. (5) Vivo-Truyols, G.; van der Wal, Sj.; Schoenmakers, P. J. Anal. Chem. 2010, 82, 8525−8536. (6) Potts, L. W.; Carr, P. W. Anal. Chem. 2011, 83, 7614−7615.
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dx.doi.org/10.1021/ac502005a | Anal. Chem. 2014, 86, 7239−7241
