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Brian A. Bidlingmeyer F. Vincent Warren, Jr. Waters Chromalogaphy Division of MlillporeCorporation
34 Mapie st Milford. Mass 01757
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YUI The rapid development of the liquid chromatography (LC) marketplace has led to a proliferation of column manufacturers. This, in turn, has contributed to confusion over performance specifications of high-efficiency columns since each manufacturer chooses ita own procedure for testing, measuring, and reporting column efficiency. Although the efficiency, or plate count, of a packed chromatographic column is of considerable interest to chromatographers, it is important to remember that the efficiency value is an indication only of how well a column has been packed. The efficiency value alone cannot adequately predict column performance under all conditions in that it is designed to be primarily a measure of the kinetic contributions to band broadening. Other contributions to peak width, such as extracolumn effects and thermodynamic factors (frequently manifested as peak tailing), should ideally play an insignificant role in determining the column efficiency value, although in practice this may not he the case. Because any definition of column performance must he tied to the separation for which the column will he used, it is impractical to evaluate column performance on the basis of a single number. However, the distinction between column performance and column efficiency becomes very clear in the laboratory. For most chroma0003-2700/84/A351-1583$01 SO10 @ 1984 American Chemical Society
tographers, performance means the capability to do a particular separation, and high efficiency alone will not guarantee this. An efficiently packed column containing a poorly prepared or inappropriate packing material will be of little value in achieving desired chromatographic separations. A good example of this involves the separation of lipophilic amine compounds. For this application, certain bonded-phase column packings lead to poor peak s y n metry regardless of efficiency, whereas other packings give improved chromatography of this particular sample type ( I ) . Regardless of the specific testing procedure used, several parameters influence the determination of column efficiency (i.e., kinetic performance) including the following: the eluent's composition and viscosity (2) as well as its linear velocity; the solute used in the measurement of plate count; temperature; column length; packing type; particle size; and the method chosen for measurement and calculation. For example, some manufacturers choose a flow rate that corresponds to the optimal linear velocity for the column being tested, while others use a higher than optimal velocity. I t is therefore important that all efficiency values he accompanied by a statement of all the conditions under which the plate count value was obtained. In the ahsence of such information, it is mean-
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ANALYTICAL CHEMISTRY, \IUL 3
I I
inglesa to compare efficiency values. Unfortunately, there is at present no standard approach for measuring column efficiency. Suggestions for standardization of test conditions have been made by some researchers (3)as well as hy organizations such as the American Society for Testing and Materials (E-19 Committee) and the Fwd and Drug Administration. Because of the lack of industry standardization regarding the measurement of column efficiency, plate count values reported by column manufacturers can be a cause of considerable confusion. First, it should he recognized that in most measurements of column efficiency, no effort is made to remove the contributions of the LC system components to the ohserved peak width. Thus, the usual measurement of column efficiency is actually a reflection of total efficiency for the system and the column together. Fortunately, for most well-maintained chromatographs, the system's contribution to peak width will be minor. Guidelines for recognizing unsatisfactory system contributions and for mathematically eliminating the effect of these on the measured column efficiency have been adequately discussed elsewhere ( 4 , 5 ) . Among the variables that can influence the column efficiency value, the choice of measurement/caleulation method is one of the most frequently overlooked. This is unfortunate because this choice can have a dramatic impact on any efficiency value based on peaks deviating from the ideal Gaussian profile (6,7), as is commonly the case in real chromatograms. In this article, we will focus on alternatives for the measurement and calculation of an efficiency value (plate count) once a test chromatogram has been run. By understanding the vfuious calculation methods, users can choose the one that best fulfills their interests. We make no specific recommendations regarding the selection of test conditions for the evaluation of column efficiency. We recommend that practicing chromatographers rely on other sources, such as column manufacturers, the literature, standardization organizations, and individual experience, for this information. Another source of confusion is the variety of units in which column efficiency is reported. Published values have been expressed in terms of plates ( N )per column, plates per foot, plates per meter, and occasionally the height equivalent to a theoretical plate (HETPor H).All of these measures attempt to adjust for the linear dependence of efficiency ( N ) on column length (L)according to the equation
N = L/H 1584A
(1)
Flauro 1. Illustration of the use of various widths of a Gaussian peak profile fw the CaicUlatim of column efficiency ( N )
H is a good way to expreas column efficiency in units of length without specifying the length of the column. Many theoretical treatments of column efficiency focus on H because of its ready conversion to a reduced plate height, which can be expressed as a multiple of the particle diameter of the column packing material. The value of H reflects the ratio of column length to efficiency, unlike the other measures (e.g., plates per meter), which use the inverse ratio. Thus, H will be a small number (e.g., 0.2 mm) for an efficient column, while the alternatives will yield a high number (e.g., 5ooo plates/ft). Unfortunately, chromatographers have been conditioned to prefer a "horsepower" focus on a high number of plates per column rather than a low value of H. It is particularly important to remember that plate-per-column values fail to clarify how well a column has been packed, since even a poorly packed column can have a high plate count if it is long enough. Although we prefer the use of H to specify column efficiency, due to the prevailing preference for the inverse measures we recommend that users of packed chromatographic columns use a consistent unit, such as plates per meter, to facilitate the comparison of columns of differing length. Measurement and calculation methods Because a chromatographic peak is aasumed to result from the sample hand spreading with a Gaussian distribution of sample concentrations in the mobile and stationary phases, the calculation of theoretical plates is often based on a Gaussian model for peak
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14. DECEMBER 1984
shape. Because of this, the general formula for calculating column efficiency (in units of theoretical plates, N)is conventionally defined by the equation
where t, is the retention time for the peak and o2is the variance for the peak measured in time units. Various peak width measurements can be related to the variance according to the relationship W2=a02
(3)
where a is a constant that depends on the height from the baseline at which the peak width is measured. Using Equations 2 and 3, the general relationship for calculating theoretical plates from a Gaussian peak profile becomes
Equation 4 is the basis for a number of commonly used methods for calculating column efficiency. Figure 1shows the relationship between the Gaussian peak profile, the height at which the width is measured, and the value of constant a in Equation 4 for six possible determinations of N. Several alternatives for the calculation of column efficiency assess the peak variance differently. One approach uses the ratio of peak area to peak height. This heighthea method assumes a Gaussian profile, for which the relationship among height, area, and width is well-known. Another method avoids d u m i n g a specific peak shape model by using the second statistical moment to characterize
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I
Flgure 2. Two peaks that give the same plate count value according to the tangent method (a) rlu = 0.0, (b) 710 = 2.0, whae 7 is
time constant fw lhe exponential peak tailing hlnction
peak variance. Both this and the heighthea method require the use of a computing integrator. Two recently reported manual alternatives (7-9) use peak asymmetry measurements to approach the determination of column efficiency. For both methods, an exponentially modified Gaussian model is used, and the properties of that model are exploited to relate manually measured peak asymmetry to column efficiency. These methods, as well as those suggested in Figure 1,are discussed in greater detail helow. If a chromatographic peak were truly of Gaussian shape, then each of the calculation methods would give the 881118 result. However, in realworld situations, non-Gaussian peaks occur, and the different methods of calculation, therefore, result in quite different values for N. Often the peak shape deviates from the ideal Gaussian shape by the appearance of a “front” or a “tail.” For fronted or tailing peaks, the higher up the peak the measurement is made, the higher the calculated theoretical value for plates will be. To clarify the alternatives for calculating N, the various methods are discussed below. Inflection method. This method is the least sensitive to asymmetrical peaks because the calculation is based on the fact that at the inflection point (which occurs at 60.7% of the peak height for a Gaussian peak) the width is equal to two standard deviation units. Slight tailing or fronting in the peak w i l l not be reflected by this method since the width is measured far above any influence of tailing. Width at half peak height. This method is also insensitive to peak asymmetry since tailing often shows up helow the measurement location. A 1580A
benefit of this method is that it is generally very reproducible from person to person. There is less user decision to be made in this measurement since it is made at a convenient fraction (50%)of the peak height. Tangent method. This method is often believed to have a slight sensitivity to asymmetrical peaks as tangents are drawn to the sides of the peaks. However, the upper portion of the peak dictates the tangent line, which minimizes any contribution of a tailing segment of the peak. Even when peaks are reasonably symmetrical, the tangent method often will have variation from person to person as there is a decision to be made concerning the tangent lines. When the peak is asymmetrical, the tangent inethod will not result in a significantly lower plate count. Figure 2 provides a visual comparison of a symmetrical peak and a tailed peak having the same plate count according to the tangent method. The three, four, and five sigma methods. The sensitivity of these measures to peak asymmetry increases with their order of listing. The closer the measurement is made to the bottom portion of the peak, the more it will he influenced by tailing or fronting. Thus, the five sigma method is extremely sensitive to peak asymmetry because the width is measured at 4.4% of the total peak height. Any degree of tailing or fronting will have a direct impact on the calculated column efficiency. This measurement method is also sensitive to the ability to define the baseline (although, if the baseline is noisy, a plate count measurement may not be justified as other chromatographic parameters probably need to be corrected). The application of
ANALYTICAL CHEMISTRY, VOL. 56. NO. 14. DECEMBER 1984
the five sigma method depends on the ability to accurately measure the specified fraction of the peak height and to construct a horizontal line at this height. Small errors in the construction of this horizontal line, especially at the tailing side of the peak, can make the method imprecise. If the baseline is easily established, a little practice can lead to good reproducibility among workers measuring the same peak. Height/area method. The area of a Gaussian peak is a function of its standard deviation and peak height, according to the equation
A =a u h (5) where A is area, u is the standard deviation, and h is the peak height. Rearranging Equation 5 gives
By combining Equations 2 and 6,another relationship for plate count is obtained
The form of this equation indicates that N can he calculated once a peak’s height and area have been measured. To use this method it is necessary to have an integrator or computer that can calculate the area of a peak and measure its height. Moment method. Another approach is to make no assumptions about the peak shape and to express several parameters of the peak in terms of the statistical moments. If an appropriate data system is available, these moments can be determined from the raw data of the chromato-
matographic peaks (11-17) adequately. This function consists of a Gaussian profiie convolved with an exponential decay function described by a time constant. The totalvariance of the EMG is described by the equation
Flgun 8. Measuement of peek asynnneby by determination of the AIB ratio at 10% of peak hei@
gram and then used to calculate efficiency values. Peak area is the zeroth moment, p ~and , is calculated by a fairly stmightforward summation. The mean is the fmt moment, PI, of the elution curve and occurs at the center of maen of the peak. For a Gaussian peak,the mean and the peak maximum (mode) are coincident. The seeond central moment, pz, is the peak variance. The third and fourth momenta are measures of skewness (excess) and flattening( M i ) , respectively. Moments higher than 8 2 are not needed for determination of plate count. However, they are useful in treating the peak shape in a sophiticated fashion. For those who desire further information, a good disnrasion is given by Grubner (10). As shown in Equation 2,the number of theoretical plates can be calculated using the second moment as the value of the peak variance. In applying Equation 2,Grubner (10) and Rogers (11)have chosen to replace the retention time of the peak maximum, t,, with the retention time of the mean, pl. For a Gaussian peak the mean and . the peak maximum are identical 80 there is no difference. The specific reasoning for preferring the mew is found in the respective references. For the present work, the mode bas been
used. The use of the moment method requires the use of micrmmputers and data aepuisition techniques. Variations may arise due to the speeific choice of where the peak begins and ends and due to noise on the baseline. If an appropriate, correctly programmed calculation device is avail1688A
able, the moment method most accurately represents the number of theoretical plates. Asymmetry-based methods. T w o reports (7-9) have recently presented other approaches for determining column efficiency in a manner that explicitly uses information regardig peak asymmetry. Both methods require the graphical determination of just three parameters from the chromatogram: the retention time at the peak apex (&),the peak width at 10% of its height, and the empirid asymmetry ratio AIB. The AIB ratio can be found very simply from measurements on a c h matographic peak, as indicated in Figure 3.A perpendicular is dropped from the peak apex to the baseline, and at 10%of the peak's height the distances B and A (see figure) are measured and their ratio calculated. The A/B ratio is a convenient measure that has the advantage of agreeing with the chromatographer's intuitive perception of peak asymmetry. Unfortunately, the AIB ratio lacks any direct connection to the fundamental parameters that characterize t a i l i i peaks. The parameters accounting for peak asymmetry can he clarified only in the context of a mathematical model for chromatographic peaks. Since the Gaussian model is dearly inappropriate for the skewed or tailing chromatographic peak shapes frequently ohserved in practice, a variety of alternative models have been proposed. The exponentially modified Gaussian (EMG) (12-15) bas been shown to represent the shape of skewed chro-
ANALYTICAL CHEMISTRY. VOL. 56. NO. 14. DECEMBER 1984
where T is the time constant for the exponential peak tailing function and UG is the variance of the (underlying) pure Gaussian. (For convenience of notation the ratio of T to UG will he denoted as T / U rather than TIOG.) The value of the tau/sigma ratio is a fundamental measure of peak asymmetry. Unlike AIB, however, the T / U ratio is not easy to measure directly without use of a computer. The goal of both methods to be discussed in this section is to relate the easily measured AIB ratio to the more fundamental T / U ratio. Once T and a are known, Equations 2 and 8 allow the column efficiency to be determined immediately. Barber and Carr (9)use a series of calibration curves from which T and a can be graphically determined. In applying this approach to synthetic chromatograms of known plate count, an accuracy of i2.4W and a precision of i5% were obtained (9). The dsadvantage of this approach is the need to have on hand a set of calibration curves and to perform additional manual measurements on those curves. Foley and Dorsey (7,8)take a slightly different approach to determining the relationship between AIB and TIU. Rather than relying on calibration curves directly, a variety of chromatographic figures of merit are derived on the basis of a small set of least-squares-fittad relationships (7). One of the resulting equations gives an expression for the column efficiency NSy,in terms of the graphically measurable parameters t,, W0.1, and AIB
The designation NIY.indicates that in determining N no effort bas been made to f a h r out the influence of the chromatographic system components on peak width (extracolumn band broadening). The accuracy and precision values determined in applying this equation were roughly a fador of two better than those found by Barber and Cam.The improved precision is not surprising considering the avoidance of manual measurementson calibration curves. Comparison ol various calculation ItWthOdS
Elecause the murrence of peak tailing is a reality for chromatographers, even in some of the relatively ideal
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Figure 4. Exponentially modified Gaussian peak profiles for T I U= 0.0, 0.5, 1.0, 2.0, and 5.0; constant mean of 700 sand uGof
1O.OD s, with T increasing systems, it is critical that a measurement method give an accurate value for the plate count even in the presence of tailing. There are a numher of situations in which the chromatcgrapher ne& an efficiency value that refleds the complete peak profile (including tailing). For example, even for solutes that tend to give symmetric peak profiles, tailing may occur due to a column channel or a void. In another case, good peak symmetry may he needed to ensure reproducible quantitation. Similarly, in comparing commercially availahle columns prior to purchase, the relative accuracy of the calculation methods used hy the various manufacturers would he of great interest. If some calculation methods lead to significantly inflated plate count values, this could be an important consideration in making a fair comparison. In each of these cases, those calculation methods most sensitive to peak asymmetry will best serve the chromatographer’s needs. On the other hand, if the goal is only to monitor the efficiency of a column from its first use until the end of its useful lifetime, it is certainly true that any of the methods presented so far might be appropriate in tracking changes in the column’s relative efficiency. For the monitoring 1590A
of efficiency changes, ahsolute accuracy is not of great importance, and the most convenient method should pmhably he selected. At this point, 10 alternatives have been introduced for the calculation of e6lumn efficiency (plate count) based on various measurements of the chromatographic peak of interest. To compare all of the methods objectively, we chose to conduct a computer evaluation based on a series of synthetic exponentially modified Gaussian peaks. By varying the ?/a ratio over a range that is representative of the pmihle peak tailing likely to he ohserved by chromatographers, it WEIS possible to compare the ability of each algorithm, in the presence of tailing, to accurately determine the efficiency asaociated with each synthetic chromatogram. The method of Barber and Carr (9), which requires manual measurements involving calibration curves, was not included in the study due to the difficulty of fully automating this procedure. All of the remaining nine methods were compared. The primary objective of the comparisons to he discussed is the determination of the accuracy of each method relative to the “correct” efficiency value, in the presence of increasing degrees of peak asymmetry
ANALYTICAL CHEMISTRY, VOL. 56. NO. 14. DECEMBER 1984
(tailing). For practical purposes, this is equivalent to comparing the results of each method to the results of the moment method, since we found the moment approach to generate the correct answer (withiin roundoff error) in all cases. It has been observed in previous reports (6)that some of the commonly used calculation methods (e.g., width at half peak height) tend to give inflated estimates of the actual column efficiency. The present study should serve to clarify these ohservations in that no previous effort that we know of has made as extensive a comparison of the available alternatives for the calculation of efficiency. Comparison using a constant value of ag. One way to alter the ?/a ratio involves holding uo constant while increasing 7, to yield peak profiles such as those shown in Figure 4. This results in an overall increase in the peak variance according to Equation 8 and serves to mimic the chromatographic situation in which various sources of hand broadening such as extracolumn effects, bed collapse, or channeling cause an increase in the observed peak width. Frequently, the result is a tailed peak profde, as in Figure 4. Because the overall variance increases in this approach, the column efficiency is expected to drop off as
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results of the moment method. Overall, it can he observed that those methods that use width measurements from the upper portion of the peak profiie yield more inflated (and less accurate) plate count values than those based on measurements made lower on the peak profile. Comparison using a constant total peak variance. A somewhat clearer view of the differences among the nine methods for efficiency calculation can be obtained by increasing r/u in another manner. If the total peak variance is kept constant according to Equation 12, values of T and u can be selected to give the same values of r / a used in the previous exgmple and the peak shapes shown in Figure 6. For comparative purposes, this approach has the advantage of holding the “correct” (Le., moment) plate count constant. In addition, results of the various methods are more easily distinguished when the results are presented as in Figure 7. As in the previous comparison, the efficiency values were computer calculated for all methods. The trends in the relative accuracy of the various methods are also as before. Table I compares the nine calculation methods in a clearer manner by focusing on the efficiency values for the synthetic chromatogram having a r/u ratio of 2.0, corresponding to a
Flgure 5. Trend in column efficienciescomputer calculated by nine methods, as a function of increasing peak asymmetry for profiles of the type shown in Figure 4 Key b meho6s: 2s = two sigma. 3.9 = thee sigma. 45 = low wma, 5s = live sigma. T = tangan. H = hail pSer hem,AH = arealwght.M = moment. and A = ssymmby based
increases. In this study, r / a is varied from 0 to 2, as in previous studies (6).A r / a of 0.0 is assigned to a pure Gaussian profile, while a ria of 2.0 is associated with a moderately tailing peak. This range is adequate to illustrate the differences among the nine calculation methods.
Efficiency values were calculated entirely by computer, thus eliminating the subjectivity inherent in handdrawn tangents, etc. The results for all nine methods are presented in Figure 5. Among the manual alternatives, the five sigma method performs well, while the asymmetry-based method (Equation 9) most closely follows the
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ANALYTICAL CHEMISTRY, VOL. 56. NO. 14. DECEMBER 1984
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Flgure 7. Trend in column efficienciescomputer calculated by nine methods, as a function of increasing peak asymmetty for profiles of the type shown in Figure 6 Abbreviations88 In Figvs 5
moderately tailed peak. It is particularly interesting to note that the popular half-height and tangent methods are equally poor in their accuracy relative to the moment method. Of these two, the half-height method is much easier to execute in practice. In addition, the subjectivity of drawing tangent lines would probably introduce a greater variation from person to person when the tangent method is performed manually rather than by computer as in this study. This problem is apparent in a previous comparison of four alternatives for column efficiency measurement @-the moment, areaheight, halfheight, and tangent methods. In comparing our results to those obtained in Reference 6, we found excellent agreement with the values presented in Figures 5 and 7 for all hut the tangent method, the tangent method was executed manually in the previous study, thus explaining the discreDancies noted.
The calculation methods used hy various suppliers of LC columns are shown in Table 11. Clearly, plate count measurements can be confusing when the values obtained from different commercial sources are compared if it is not recognized that different calculation methods are used. Note that some manufacturers select a demanding measure of plate count (Table I), while other suppliers choose a less stringent measure. When judging the expected performance of a column to be purchased, the calculation method used must be considered. If peak tailing is present in the test chromatogram, some calculation methods will reflect this more accurately than others. If the test peak is truly Gaussian, each of the calculation methods will give the same result.
Conclwion It is important to remember that while the measurement/calculation
Table 1. Comparison of efflclency values for a synthetic chromatogram by nine calculation methods C.lsulMlOn m t h W
Inflection (twosigma) Half peak height Tmnt i-!alght/area ratio
Four sigma Five slgma Asymmetry-lmsed
Moments Actual
1594,.
N (plal..l&m)
11.265 10,694 10,536 8,803 7,725 6,277 5,078 4,967 4.967
Table II. Calculation methods for column efficiency used by LC column suppliers capy
lauacy
Beckman Du Pont IBM Mer& Perkin-Elmer
Least Least Least Medlum
Medium
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method plays a very significant role in the determination of column efficiency, a number of other variables related to the chromatographic test conditions (e.g., test solute, flow rate, and eluent) contribute to plate height. The specific discussion of these variables is beyond the scope of this paper. Any evaluation of column efficiency must be based on an understanding of the capabilities and limitations of the various calculation methods used to determine efficiency. In addition, it is important to have a realistic view of the factors that contribute to the calculated efficiency value. Poor peak shape does not guarantee a problem with the column's efficiency. Peak tailing may be seen in a chromatogram even when highly efficient columns are used. When acidic or hasic solutes are analyzed along with less polar solutes, the resulting chromatogram may contain a mixture of sharp, symmetrical peaks for the less polar solutes and badly tailed peaks, for example, for the bases. This problem can generally he solved hy the addition of suitable modifiers to the eluent, and it bears little relation to the efficiency of the column in use. Chromatographers must select the calculation method most appropriate to their needs. In some situations, the relative accuracy of the method may matter very little, as in the monitoring of a column throughout ita useful lifetime. Here, it is the changes in efficiency, such as those due to the appearance of voids or channels in the bed structure, that are of interest. In this case, any convenient calculation method can be used to determine whether the column's kinetic performance is significantly degraded since its first use. Another approach is to be less concerned about the accuracy of any given calculation method and more concerned about peak symmetry. From this perspective, one indicator of how well a column is packed might he the degree of agreement between one
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ANALYTICAL CHEMISTRY, VOL. 56. NO. 14, DECEMBER 1984
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ANALYTICAL CHEMISTRY, VOL. 50. NO. 14, DECEMBER 1984
1595 A
h uniqueapproach to the growing Iidd DI slate.selecled dynamics. Puts foRh
photoreactiveand collisionai resonances as the unifying concept lor intwrpreting a variely 01 dynamical phenomena, Looks at recent developmentsin electron-molecule scanering. photoionization, van der Waals complex photodissociationand predissociation. mimolecular dynamics. and state-toslate bimolecular dynamics. CONTENTS
If the most accurate methods (e.% ive sigma) and one of the least accu‘atemethods (e.g., inflection). For a ,mly Gauesian peak profile, the ratio ,f these two numbers should he close o unity. Use of such a ratio has the idvantage of expressing the effect of wak asymmetry ( A B ) BS a function If plate count values, which are familar to chromatographers. The accuracy of each calculation nethod relative to the moment methd is important in other situations, !specially when comparing columns b a t have been assigned efficiency valies by a variety of methods. This xoblem is commonly encountered by :hromatographers whenever a new :olumn is to be purchased and high eficiency is desired. Due to the lack of itandardization in the testing, calcula;ion, and reporting of column efficien:y information, a clear comparison of ;he efficiency offered by commercially ivailable columns is very difficult to 3htain. The information presented in this report should allow a qualitative issessment of reported efficiency values in terms of their accuracy. An unsmbiguous quantitative comparison would be possible only if more detailed information about the test C ~ O matogram were available for each column.
Ackowledgmnt The authors wish to acknowledge the assistance of J. Newman, R. Day, T.Tarvin, J. Ekmanis, and R. King for help in the preparation of this manuscript. We also wish to thank M. Delaney of the Boston University Department of Chemistry for allowing us to use his computer programs to generate the peak shapes and calculate various asymmetry values.
(13) GrushLa, E.Anal. Chem. 1912.44. 1133. (14) Sternberg,J. C. In ‘‘Advance8 in aph ’’. Giddings, J. C.; KelChroma ’ Dekker: New ler, R. A s s . ; arcel York, N.Y., 1966,Vol. 2, p. 205. (15) Delaney, M.F. AMlyst 1982,107,
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Brian A. Bidlingmeyer is uice-president and technical director of Waters Chromatography Diuision of Millipore Corporation. He received his AB in chemistry from Kenyon College and his PhD in analytical chemistry from Purdue University. He is presently a n adjunct associate professor a t Boston University. His research interests include the investigation of chromatographic retention mechanisms and the development of specialized applications with the longterm goal of commercializing “tailored” separation systems.
References
F. Vincent Warren is a research nanceh & &gular Distributions
Based on a symposium sponsorad by the Dwsnn 01 Ph sical Chemrstw
olthe Amencan 8hemrcalSoomh/ ACS SymposiumSeries NO 263 514 pages(1984)ClolhWund LCW16934 US 8 Canada$89.95
1590A
ISBN0841208654 Expon $107.95
togr. Sci. 1984.22, do. (9) Barber, W. E.; Cam, P. W.Anal. Chem. 1982,53,1939. (IO) Grubner, 0. In “Advances in Chromato ra hy”; Giddings. J. C.; Keller, R. A., E&%arcel Dekker: New York, N.Y.. 1958; Vol. 6. pp. 172-209. (11) Oberholtzer, J. E.; Rogers, L.B. And. Chem. 1969.41,1235. (12) Yau. W. W. Anal. Chem. 1917.49, 395.
ANALYTICAL CHEMISTRY. VOL. 56. NO. 14, DECEMBER 1984
chemist in the Waters Chromatography Diuision of Millipore Corporation. He received a BS in chemistry from Stanford University in 1975,an MAT degree from the University of New Hampshire in 1976,and a n MA degree in chemistry from Boston University in 1982.His research interests include retention mechanisms in LC and the use of computers in chromatography and spectroscopy. He is presently completing work for his PhD as part of a collaborative program with Boston Uniuersity.
