Anal. Chem. 1995,67,631-640
Correlation of Quantitative Analysis Precision to Retention Time Precision and Chromatographic Resolution for Rapid, Short=ColumnAnalysis Timothy J. Bahowick and Robert E. Synovec* Center for Process Analytical Chemistry, Department of Chemistty BG- 10, University of Washington, Seattle, Washington 98 195
When chromatographic analysis involves the use of two or more chromatograms, e.g., for performing calibration or for assessing composition changes among different samples, optimization based solely on chromatographic resolution, R,, may not yield the most rapid or precise analyses. A simple model is developed that predicts quantitation precision for analysis of ill-resolvedpeaks as a function of retention time precision and R,. This model implies that use of shorter columns can provide rapid and precise quantitation. Short-column analyses have improved retention time precision and S/N ratio, which offset the detrimental quantitation effects of decreased R,. A quantitation precision study for liquid chromatography was done beginning with well-resolved peaks having R, = 1.06. Quantitation precision as percentage relative standard deviation (%RSD)was theoretically calculated and experimentally measured for two diverse experimental paths in which R, was diminished by decreasing either the selectivityratio or the column length. The quantitation method chosen was deconvolution of mixture chromatograms by performing a classical least-squares fit to chromatograms of pure standards. The quantitation precision model agreed with calculated and measured %RSDvalues to within 3 percentage points. The short-columnanalyses yielded improved quantitation precision and shorter analysis time at equal R, compared to selectivity-limited analyses. For an analyte having a peak height ratio of 1:2.5 and a peak width ratio of 1:1.2 relative to an adjacent overlapping peak, similar quantitation precision as %RSDwas obtained at equal selectivity ratio for a 50 cm column (3.1%)and a 7.5 cm column (4.3%),despite the decrease in R, from 1.06 to 0.27. The short-column advantages are also applicable to gas and supercritical fluid chromatographies and possibly to capillary electrophoresis. Chromatographic optimization is an active field, as shown in a siffniscant review by Guiochon,l recent books by Schoenmakers2 and Bemdge,3 and an entire journal volume! The recommendation5and subsequent wide use of chromatographic resolution, R,, (1) Guiochon, G. In High-Pelfonnance Liquid Chromatography-Advances and Perspectiues; Horvath, C., Ed.; Academic Press: New York, 1980; Vol. 2, pp 1-56. (2) Schoenmakers, P. J. Optimization of Chromatographic Selectivitj-A Guide to Method Development Elsevier: Amsterdam, 1986. (3) Benidge, J. C. Techniquesfor the Automated Optimization of HPLC Separations; Wiley: New York, 1986. 0003-2700/95/0367-0631$9.00/0 0 1995 American Chemical Society
as a separation criterion followed an early distinction by Obet$ between chromatoffraphic efficiency and selectivity. Chromatographic optimization based on minimum required R2 is linked to peak capacity as defined by Giddings? and it is well-suited to physically isolating desired analytes. However, approaches based on information theory8 focus directly on obtaining qualitative and quantitative information. Hayashi, Matsuda, and co-workers have reported extensively on chromatographic optimization based on the function of mutual information (FUMI).9 Compared to R,based strategies, FUMI was preferred for optimizing quantitative analysis precision (total information) and rapidity (information flow) .1OJ1 When the analytical goal is rapid, accurate, and precise quantitation, the frequently specified minimum R, values of 1.5 or even 1.0 may be limiting because of the wealth of chemometric techniques applicable to ill-resolved peaks.12 The success of available techniques for ~ingle-channel'~-'~ and multichannel data16-18 implies that partial peak overlap alone is not fatal to the analysis. Instead, other artifacts of separation bring difficulty to low-resolution analysis. Nonlinearity of the detector19 or the adsorption isotherm20can limit the concentration range of successful deconvolution. Novel calibration schemes have addressed these issue^.^^^^^ Noise is a universal problem for chromatographic data analysis, especially for nonnormal noise Techniques for chromatographic noise reduction and signal (4) Glajch, J. L.; Snyder, L. R, Eds. Computer-Assisted Method Development in Chromatography. ]. Chromaton. 1989,485. (5) Desty, D. H., Ed. Gas Chromatography; Butterworths: London, 1958; p xi. (6) Ober, S. S. In Gas Chromatography; Coates, V. J., Noebels, H. J., Fagerson, I. S., Eds.; Academic Press: New York, 1958; pp 41-50. (7) Giddmgs, J. C. Unitied Separation Science; Wiley: New York, 1991;pp 131136. (8) Eckschlager, IC; Stepanek, V.; Danzer, IC ]. Chemom. 1990,4, 195-216. (9) Hayashi, Y.; Matsuda, R Chemom. Int. Lab. Syst. 1993,18, 1-16. (10) Matsuda, R; Hayashi, Y.; Suzuki, T.; Saito, Y. Anal. Lett. 1991,24,20832091. (11) Hayashi, Y.; Matsuda, R Chromatographia 1990,30,171-175. (12) Smit, H. C.; Van den Heuvel, E. J. In Chemometn'csand Species Identitication; Topics in Current Chemistry 141;Springer-Verlag: Berlin, 1987; pp 63-89. (13) D'AUura, N. J.; Juvet, R S., Jr. 1.Chromatogr. 1982,239,439-449. (14) Goldberg, B. J. Chromatogr. Sci. 1971,9, 287-292. (15) Bahowick, T. J.; Synovec, R E. Anal. Chem. 1992,64,489-496. (16) Sanchez, E.; Ramos, L. S.; Kowalski, B. R]. Chmmatogr. 1987,385,151164. (17) Gemtsen, M. J. P.; Tanis, H.; Vandeginste, B. G. M.; Kateman, G. Anal. Chem. 1992,64, 2042-2056. (18) Cecil, T. L.; Poe, R B.; Rutan, S. C. Anal. Chim. Acta 1991,250,37-44. (19) Dose, E. V.; Guiochon, G. Anal. Chem. 1989,61,2571-2579. (20) Conder, J. R ]. High Resolut. Chromatogr. 1982,5, 341-348. (21) Dose, E. V.; Guiochon, G. Anal. Chem. 1990,62,816-820. (22) Laeven, J. M.; Smit, H. C. Anal. Chim. Acta 1985,176, 77-104.
Analytical Chemistry, Vol. 67, No. 3, February 1, 7995 631
enhancement have been d e s ~ r i b e d ,including ~ ~ , ~ ~ matched filteringH and Fourier smoothing.25 Moore and Jorgenson recently demonstrated median filtering for removal of troublesome lowfrequency background drift26Retention time precision is affected by run-to-runvariation in flow rate,n column temperature,%mobile phase liquid composition,29or other factor. Some remedies for retention time imprecision include internally3O or externally added31timing standard compounds, multivariate retention time and data axis manipulation^.^^ Retention time variation adversely affects deconvolutionand quantitation of overlapped peaks whenever the analysis involves more than one chromat0gram13-16 or when matrix effects necessitate calibration using standard Workers are increasingly addressing the effects of retention time precision on chromatographic data analysis of overlapped peaks.15J7J**35 In this paper, we demonstrate that R, alone is insufficient and ambiguous for judging quantitative analysis precision for overlapped peaks because the contributions of detected noise and retention time precision to quantitation precision depend on how R, is experimentally established. We develop a model that accurately predicts quantitation precision as a simple function of R, and retention time precision. The model is applied to independently calculated quantitation precision values as obtained from Goldberg’s deconvolution method14 for quantifying illresolved chromatographic peaks by classical least-squares (CLS) fitting to chromatograms of pure standards. The quantitation precision model predicts good precision and diminished analysis time by using shorter columns for repetitive analyses or for process monitoring applications that involve calibration or composition determination using more than one chromatogram. Hayashi and Matsuda have theoretically and experimentally established that shortcolumn analysisimproves both analysis time and quantitation precision for resolved peaks because the peaks are taller and harper.^^.^^ To extend their work to deconvolution of strongly overlapped peaks, we will show that the ratio of retention time variation to peak width improves in proportion to the decrease in R,. These two simultaneous advantages for short columns preserve good quantitation precision, despite decreases in R, and plate count. (23) Amino, R In Advances in Chromatography; Giddmgs, J. C., Grushka, E., Cazes, J., Brown, P. R, Eds.; Marcel Dekker: New York, 1977; Vol. 15, pp 33-67. (24) Van den Heuvel, E. J.; Van Malssen, K. F.; Smit, H. C. Anal. Chim. Acta 1990,235,343-353; 355-365. (25) Felinger, A; Pap, T.L.; Inczedy, J. Anal. Chim. Acta 1991,248,441-446. (26) Moore, A W., Jr.; Jorgenson, J. W. Anal. Chem. 1993, 65, 188-191. (27) Foley, J. P.; Crow, J. A; Thomas, B. A; Zamora, M. J. Chromatogr. 1989, 478, 287-309. (28) Melander, W. R; Horvath, C. In High-Peflonnance Liquid ChromatographyAdvances and Perspectives; Horvath, C., Ed.; Academic Press: New York, 1980; Vol. 2, pp 113-319. (29) Abbott, S. R;Berg, J. R;Achener, P.; Stevenson. R L.J. Chromatogr. 1976, 126, 421-437. (30) Bahowick, T.J.; Dunphy, D. R; Synovec, R E. J. Chromatop. A 1994, 663, 135-150. (31) Guillemin, C. L. J. Chromatofl. 1982,239, 363-376. (32) Smilde, A K.; Coenegracht, P. M. J.; Bruins, C. H. P.; Doombos, D. A J. Chromatop. 1989, 485, 169-181. (33) Konstantinides, F. N.; Garr, L.; Li,J. C.: Cerra, F. B. J. Chromatogr. Sci. 1987,25, 158-163. (34) Vandeginste, B. G. M.; Leyten, F.; Gerritsen, M.; Noor, J. W.; Kateman, G. J. Chemom. 1 9 8 7 , 1, 57-71. (35) Booksh, K. S.; Kowalski, B. R J Chemom. 1994, 8.45-63. (36) Hayashi, Y.; Matsuda, R Anal. Sci. 1991, 7, 329-331. (37) Hayashi, Y.; Matsuda, R In Advances in Chromatography; Brown, P. R, Grushka, E., Eds.; Marcel Dekker: New York, 1994; Vol. 34, pp 347-423.
632 Analytical Chemistry, Vol. 67,No. 3,February 1, 1995
We report an experimental study of the quantitation precision model and use of short columns. Beginning with a well-resolved separation, quantitation precision is measured for two experimental paths in which R, is diminished by decreasing the selectivity ratio or the column length, respectively. These data are compared with the model to evaluate the model accuracy and applicability for both experimental paths. Rapid shortcolumn analyses are demonstrated that maintain good quantitation precision as o b served for a substantially longer column, without having to apply a timing standardNor other retention time adjustment31-33 These quantitation precision results are understood by accounting for the improvements in S/N ratio and retention time precision that counteract the decrease in R,. Although Gourlia and BordeP previously demonstrated rapid short 1.0. The experimentally measured quantitation precision results as %RSD for the length path (IA-B-C-D) and selectivitypath (IA11-III) are shown in Figure 6 as a plot of %RSDfor the short peak, cumene, versus R, for cumene and dibutyl phthalate. Values of %RSDfor the length path were significantly better compared to the selectivity path at equivalent R,, as evidenced by the 95% confidence intervals. The cause of improved quantitationprecision for the length path was studied by preparing and quantifying a 4times more concentrated mixture using selectivity path conditions I1 and 111. This experiment evened out the S/N ratio advantage experienced by corresponding separations C and D for the length path. The cumene quantitation precision values as %RSD for the new mixture, 11.5%and 23.5%,were close to the %RSDvalues corresponding to I1 and I11 for the original mixture (Figure 6). Therefore, the improved precision for length path separations C and D was largely explained by the improved peak width-based retention time precision, 6 (eq 2). (43) Draper, N. R; Smith, H. Applied RegressionAnalysis;Wiley: New York, 1966; Chapter 2. (44) Renn, C. N.; Synovec, R E. Anal. Chem. 1988, 60, 1829-1832.
0.0
0.2 0.4 0.6 0.8 1.0 Resolution, cumene & phthalate
Figure 7. Measured quantitation precision as %RSD for cumene 0). Calculated for the length path from Figure 6 (IA-8-C-D, contributions to %RSD (eq 5) due to retentiontime precision, %RSDa (eq 9, 0),and due to detected noise, %RSDn (eqs 5-8, A), assuming YoRSDn = 2.9% for separation IA. Labeling follows Table 2.
Also shown in Figure 6 is that quantitation precision for cumene remained nearly constant along the length path for separations using a 50 cm column (IA) compared to using a 7.5 cm column (C) under identical separation conditions. We accounted for these promising length path results as follows. The measured %RSDprecision values for cumene from Figure 6 were replotted in Figure 7, along with calculated contributions due to detected noise, %RSD, (eqs 5-8), and retention time precision, %RSDd (eq 9). The values of %RSD, were based on a % E D , value of 2.9% for separation IA,as obtained from Table 3 by variance subtraction as per eq 5. The S/N ratio advantage for short columns held % E D , in check as R, decreased along the length path (Figure 7) because both S/N ratio and condition number (eq 6) increased together with 1/R, (Theory). In contrast, %RSDs slowly increased along the length path, similarly to Figure 1, becoming dominant for separation C. Regardless of experimental path, application of CLS under extreme peak overlap warrants two familiar cautions. First, qualitative analysis and interferent detection are obviously more difiicult for overlapped peaks. Second, CLS quantitation errors for adjacent overlapped peaks are negatively correlated to each other because of the fitting although the shortcolumn improvements in S/N ratio and 6 diminish these quantitation errors. Contribution of Retention Time Precision. Because the contribution to quantitation precision due to retention time precision, %RSD6, depends heavily on 6, we evaluated use of column length to control 6. The measured 6 values from Table 2 are plotted in Figure 8 versus Rs for cumene and dibutyl phthalate. The solid and dashed lines in Figure 8 are idealized relationships between 6 and R, for the selectivity path (IA-II111) and the length path (IA-B-C-D), respectively. For the selectivity path, 6 remained essentially constant. For the length path, 6 was lowered from 0.0161 for separation IA to about 0.005 by using shorter columns to diminish N (eq 4) while holding the selectivity ratio, a,constant. In general, discrete data sampling influences reported 6 values that correspond to retention time standard deviations (eq 2) of smaller duration than several data sample periods. Here, one sample period (Experimental Section) corresponded to a 6 value of 0.0031 for cumene. This sampling rate was adequate to measure 6 for the selectivity path (TA-IIIII). The sampling rate was also adequate for separation B, as implied by the agreement of the measured 6 value, about 0.005, Analytical Chemistry, Vol. 67, No. 3, February 1, 1995
637
T
a 0.021
0.0
0.2 0.4 0.6 0.8 1.0 Resolution, cumene & phthalate
Figure 8. Measured values of 6 (eq 2) for all experimental conditions. Values of 6 were nearly constant for the selectivity path (IA-11-Ill, A) and were correlated to Rs for the length path (IA-BC-D, 0). Labeling follows Table 2.
a)
20
z-
0
Lo
10
n v)
a
8 0 0
10
20
%RSD (measured) Figure 9. Comparison of calculated %RSDd values (eq 9) to measured %RSD precision values for cumene. Calculated %RSDd values were obtained three ways by specifying equal peak height and peak width for cumene and dibutyl phthalate (0);equal peak widths and correct 1:2.48 peak height ratio (+); correct peak height ratio and correct 1:1.24 peak width ratio (0).Labeling follows Table 2.
with the idealized length path value in Figure 8. However, we believe the sampling rate inflated st and 6 (eq 2) for separation D and possibly C (Table 2). The sampling rate did not hamper measurement of quantitation precision. Because the proposed model, eq 10, was developed using numerical evaluation of eq 9, we evaluated the accuracy of these calculations that use eq 9 to obtain %RSDd. This evaluation was done by using measured values of 6 and R, to calculate %RSDd for cumene (Appendix). These %ED8values are plotted in Figure 9 versus corresponding measured %RSD precision values for cumene. As a further test, the effects on %RSD6due to relative peak height and peak width for cumene and dibutyl phthalate were evaluated. This evaluation was done by calculating %RSDdfor all six separation conditions (Table 2) in each of three cases for relative peak height and peak width, as specified in Figure 9. The measured %RSDprecision values for cumene agreed closely with the most realistic calculations (Figure 9). Separation IA was an exception, as shown in Table 3 and Figure 7, because detected noise dominated. Also, %RSDd increased as the correct height ratio (0.403) and correct width ratio (0.805) for cumene were incorporated into the calculations (Appendix). This observation agreed with theory, Figure 3, that deconvolution of the short and narrow overlapped peak for cumene should be less precise. In Figure 10, quantitation precision values as obtained from numerical calculation (eq 9) and from experiment are evaluated in the context of the model, eq 10. The measured %RSDvalues 638 Analytical Chemistry, Vol. 67, No. 3, February 7, 7995
-0.01 0 0.01
0.03 0.05 6 / R s (1 - Rs/R,*)
0.07
Figure 10. Experimental evidence that quantitation precision is correlated to a function of 6 and Rs,as plotted according to the model in eq 10. Measured quantitation precision as %RSD for cumene (0) and dibutyl phthalate (A). Calculated (eq 9) values of %RSDd using correct peak height and peak width ratios (Figure 9) for a selectivity path where 6 = 0.0161 (-). Labeling follows Table 2. If omitted, the 95% confidence intervals fell within the plot symbols.
for both cumene (0) and dibutyl phthalate (A) are plotted according to eq 10, using the measured 6 and R, values from Table 2 as well as the appropriate parameters from Table 1. For comparison, %RSDd was plotted as per eq 10 for both analytes (solid lines) for an idealized selectivity path, holding 6 constant at 0.0161 as for separation IA. In agreement with theory, Figures 3 and 4, the model, eq 10, accounted for and unified quantitation precision for the length and selectivity path experiments. Consistent with Figure 4, the experimental data in Figure 10 lay on a single function for both analytes. The slight curvature in Figure 10 for cumene was consistent with Figure 3 for deconvolving a short overlapped peak. As expected from Figure 2, the taller and wider peak, dibutyl phthalate, had a smaller value of Rs*,0.845 (Table 1). Note that eq 10 should not be used for dibutyl phthalate for separation IA because R,, 1.06, exceeded R,*. The experimental deviations from the idealized selectivity paths in Figure 10 were relevant to practical application of short columns. For separation LA,the measured %RSDprecision values for both analytes were significantlylarger than the idealized values (solid lines). As discussed in Figures 4B and 7, this is explained by the dominating contribution of detected noise, %RSD,, for separation IA. In contrast, the contribution of retention time precision, %RSDS,dominated for separations I1 and 111. Si-fold analysis time savings was possible by shortening the column to 7.5 cm, beyond which point %RSDbdominated over %RSD, (Figure 7). Also, the length path appeared to fail for the 3.0 cm column (D), as shown in Figure 6. Actually, the observed increase in %RSD for separation D was largely explained by its inflated value of 6 relative to an idealized length path, as discussed in Figure 8. By using the measured R, and 6 values for separation D (Table 2), eq 10 predicted %RSD reasonably well as shown in Figure 10. Implementation of the 6/R, Concept. Using Figure 6, we can describe a separation developmentthat ultimately applies 6/Rs to save analysis time. Consider that separation I11 represents the first unsatisfactory separation attempt. From 111, the analyst improves efficiency and selectivity ratio as previous workers have taught.-4 The separation thus moves along a selectivity path from I11 to IA (Figure 6), where desired analytes are well resolved, qualitative analysis is readily accomplished, and conventional quantitation by peak height or area is usually applied. When an application involves few analyses, optimization usually ceases at
separation IA. When short analysis time is vital, optimization should continue, increasing R, further by maximizing efficiency and sele~tivity.’-~Apart from using high eluent flow rates, which worsens plate height and S/N ratio,’ time savings and slightly improved quantitation precision36arise from then shortening the column length but still maintaining baseline R,. We have demonstrated greater time savings by shortening the column further, thus sacrificing baseline R, but holding 6/R, constant. Although conventional quantitation is no longer feasible, the S/N ratio and retention time precision advantages for short columns enable quantitative analysis by deconvolution of overlapped peaks, as shown in Figures 7 and 10. What are the limits to time savings by using short columns? Even when “acceptable”quantitation precision is sought rather than compete separation, S/N ratio and selectivity ratio, a,are still limiting factors. We compare the roles of S/N ratio and a via some calculations for cumene. Using Tables 1and 2 with eq 10, the contribution to quantitation precision due to retention time precision, %RSDd,had a value of 1.75%for separation IA (50 cm column), compared to 0.51% as obtained using eq 9 (Table 3). From R,, N, and k ’, a was calculated’ as 1.081. However, %RSD, dominated for separation IA,having a value of 2.9%. For a hypothetical 7.5 cm column, holding a constant and 6/R, at 0.0152 (Table 2), the value of %RSDd using eq 10 was 3.89%, just dominating over %RSD,, and R,was calculated as 0.41. We stress that the time savings for this 7.5 cm column was possible only by first “maximizing”a. A poorer a of only 1.065 yields a value of 3.0%for %RSD6with a R, of 0.86 for the 50 cm column, with little opportunity for time savings because %RSDdwould dominate over %RSD,. For macroanalysis, where %RSD, is negligible, the limit to time savings occurs where %ED8exceeds some predetermined limit beyond injection volume variation. To relieve the dependence of %RSDdon a,6 may be diminished independently of R, by precise control of separation conditions. To illustrate, suppose that 6 were doubled unforeseen from 0.0063 to 0.0125 for the 7.5 cm column analysis mentioned in the preceding paragraph. To maintain a value of 3.89%for %ED6 would require increasing R, from 0.41 to 0.65 and a from 1.081to 1.136. Because of finite instrument precision, apparent retention time variation can also be diminished as discussed in the i n t r o d u ~ t i o n . ~Such ~ - ~ ~action was not needed in this study. Our use of 12 pm packing material provided data to support both the model, eq 10, and the use of short columns. The analyst may prefer a 9 cm column having 5 pm packing, thus yielding as many plates as the 50 cm column having 12 pm packing, but with a shorter analysis time and a larger S/N ratio.’ Unfortunately, further column shortening could be constrained by extracolumn band br~adening‘~ and limitations of column construction and data handling. Analysis time savings must sometimes be balanced with qualitative analysis and interferent detection. For a new interferent to be discerned using CLS, the regression residuals43due to the interferent must be recognizable above the underlying effects of detected noise and retention time variation. Because short column analysis improves S/N ratio and 6, it is reasonable that interferent detection capability is better for the length path compared to the selectivity path at equal R,. To illustrate, consider a selectivity path, labeled A in Figure 11, representing a chromatogram of (45) Guiochon, G.; Colin, H. In Microcolumn High-Perfonnunee Liquid Chmmatography; Kucera, P. Ed.; Elsevier: Amsterdam, 1984; pp 1-38.
0.0
0.2
0.4 0.6 Resolution, R,
0.8
1.0
Figure 11. Calculated quantitation precision as %RSD for typical analytes versus S for known interferents. Results were calculated for a 50 cm column (A) and a 5 cm column (8)under constant separation conditions. Arrows indicate mapping of curve A onto B along length path lines of constant 6/Rsfor a given analyte-interferent pair, with little increase in %RSD.
various desired analytes using a 50 cm column. One can interpret this curve as quantitation precision for desired analytes as a function of R,between desired analytes and known interferents. The detected noise contribution, %RSD,, for this selectivity path was 2% for R, = 1, and it increased as R, decreased (Figure 4A). By substituting a 5 cm column, R,was diminished by 1/(10)1’2, yet 6/R, was held constant for all analyte-interferent pairs of peaks. For the 5 cm column, a similarly calculated selectivity path, labeled B in Figure 11, yielded better quantitation precision at equal R, compared to the 50 cm column. Each point for curves A and B is linked by a length path line, having a particular 6/R, value for a particular analyte-interferent pair of peaks. Several such lines are shown in Figure 11. In the same way that quantitation precision was largely preserved for the 5 cm column (Figure l l ) , we expect that many potential interferents could still be discerned at the lower R, values for the 5 cm column. We are investigating the 6/R, concept as applied to short column partial separationsusing multichannel UV detection. The added spectral selectivity should enhance qualitative analysis well beyond the limits of singlechannel CLS and should diminish requirements for chromatographic selectivity and precise instrument control. Even so, the underlying chromatographic limitation is a,not R,. “Excess’’separation can be diminished intelligently through use of short columns, thus exploiting the improvements in S/N ratio and retention time precision. Subsequent application of second-order chemometric techniques+18 may then extract qualitative and quantitative analytical information successfully from ill-resolved peaks, as their developers intended. ACKNOWLEDGMENT
This work was supported by the Center for Process Analytical Chemistry (CPAC), a National Science Foundation, University/ Industry Cooperative Research Center at the University of Washington. APPENDIX
Calculation of %ED&. The CLS model for quantitation of an unknown binary mixture chromatogram, m(t), is
m ( 0 = (c,/~,?s,(O
+ (~b/c
