Anal. Chem. 1992, 6 4 , 489-496
9001412). A.M.P. acknowledges support from DGICYT of the Ministry of Education and Science of Spain for the grant that made possible his research in Professor Warner's laboratory.
REFERENCES (1) Szejtli, J. Cyclcdextrlns and Thek Inc/us&n Complexes; Akademiai Klado: Budapest, 1952. (2) Kwihara, M.; Hirayama, F.; Uekama, K.; Yamasaki, M. J . Inclusion phenom. 1990, 8 , 363-373. (31 . . Duchene, D.. Ed. cvclodexMns and 77W Industrial Uses; Editions de Sante: Paris, 1987. (4) uekama, K.; Otagiri, M. Crtl. Rev. Ther. Drug Carrhw Syst. 1987, 3 , I-AO.
(5) M&wn, L. 8.; Warner, I. M. Anal. Chem. 1990, 62, 255R-267R. (6) Armstrong, D. W.; DeMond, W. J . chrometogr. Sci. 1984, 22, 411-415. (7) Gazdag, M.; Szepesi, G.; Huszar. L. J . C h r m t o g r . 1988, 436, 31-38. (8) Armstrong, D. W.; He, F.-Y.; Han, S. M. J . Chrometogr. 1988, 448, 345-354. (9) Hamai. S. .I Aiys. Chem. 1989. 93, 2074-2078. (10) Maw!.M.; MOChida, K. Bull. Chem. Soc.Jpn. 1979. 52, 2808-2814. (11) Buvari. A.; Szepii. J. J . Inclusion phenom. 1983, 1 , 151-157. (12) Connors. K. A. Blndlng Consbnts: A Maesurement of Molecular Complex SbbiMy; John W h y & Sons: New Ywk, 1987. (13) Mohseni, R. M.; Hwtubise. R. J. J . Chrometogr. 1990, 499. 395-410. (14) Fujimura, K.; Ueda, T.; Masashi, K.; Takayanagi, H.; Ando. T. Anal. Chem. 1988, 58, 2868-2674. (15) Dong, D. C.; Winnik, M. W. photochem. Photoblol. 1982, 35, 17-21.
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(16) Blyshak, L. A.; Dobson, K. Y.; Patonay, 0.; Warner, I. M.; May, N. E. Anal. Chem. 1989, 61, 955-960. (17) Cam, J. W.; Harris, J. M. Anal. Chem. 1988, 58, 626-631. (18) Nelson, G.; Patonay, 0.;Warner, I. M. Anal. Chem. 1986, 60, 274-279. (19) Nelson, 0.;Warner, I. M. J . M y s . Chem. 1990, 94, 576-581. (20) Hashlmto, S.; Thomas, J. K. J . Am. Chem. Soc. 1966, 707, 4655-4662. (21) Kusumoto, Y. Chem. phys. 1987, 136, 535-538. (22) Hemal, S. J . phys. Chem. 1988, 92, 6140-6144. (23) W m a , K.; Hirayama, F.; NSSU, S.; Matsuo, N.; I r k , T. Chem. phenn. Bull. 1978, 26, 3477-3489. (24) Patonay, G.; Fowler, K.; Nelson, 0.; Warner, I . M. Anal. CMm. Acte 1988, 207, 251-258. (25) Nelson, G.;Patonay. G.; Warner, I. M. J . Inclushm phenom. 1988, 6 , 277-289. (26) Tam, M. A.; Nelson, G.; Patonay, 0.; Warner, I. M. Anal. Lett. 1988, 21, 843-856. (27) Mu'ioz de la P e k , A.; M u , T. T.; Zung. J. 8.; Warner, 1. M. J . phys, Chem. 1991, 95. 3330-3334. (28) Mu'ioz ds la P e h , A.; Ndou, T. T.; Zung. J. 8.;Greene, K. L.; Live, D. H.; Warner, I. M. J . Am. Chem. Soc. 1991, 173, 1572-1577. (29) M u b z de la P e k , A.; Anigbogu, V. C.; Ndou, T. T.; Warner, I. M. Anal. Chem. 1991, 63, 1018-1023. (30) w i i k , B.; Thuaudu, N.; Piquion, J.; Behar, N. J . Chromtcgr. 1987, 409. 61-69.
RECEIVED for review July 29, 1991. Revised manuscript received December 2,1991. Accepted December 10,1991.
Sequential Chromatogram Ratio Technique: Evaluation of the Effects of Retention Time Precision, Adsorption Isotherm Linearity, and Detector Linearity on Qualitative and Quantitative Analysis Timothy J. Bahowick and Robert E. Synovec* Center for Process Analytical Chemistry, Department of Chemistry BG-10, University of Washington, Seattle, Washington 98195
A study of the sequentlal chromatogram ratlo technique was done to characterlze and correct for retentlon tlme variation, peakshape change due to concentratlm exceeding the adsorption Isotherm Ilnear range, and mlld detector nonUnearlty over a broad concentratlon range. A rrensltlve mlnlmlzatlon routine was developed to overlay chromatographic peaks from sequentlal lnjectlons udng a slmple displacement along the tlme axls. Predse peak allgnment to wlthln 0.8% of the b a r d h peak wldth was necessary for qualltatlve Interpret a t h of the ratlo chromatogram. For qwnttlatbn of the ratlo of InJectedconcentratlons for pure-component peaks, tlme shMs as large as 2.5% of the base-llne peak wldth caused a negatlve blas of 1% or less. A peak-shape analysb method, barred on the scaled polnt-by-point difference of normallzed peaks, was adapted as a dlagnostlc test for peakshape change and as a means of correctlng the ratlo value of InJectedconcentratlons. At a concentratlon 4 tlmes the llnear Isotherm Ilmlt, the ratlo chromatogram was dktorted, and the etlmated correctlon to the ratlo value was +2.3 % of the true value. Thus, restrlctlon of the ratlo technique to wlthln the hear Isothm reglon b recommended. A nonllnear power-law detector model, whkh preserved the qualltatlve features of the ratlo chromatogram, was eadly Incorporated Into the quantnatlon method. Detector nonllnearlty corrections to the ratlo values, equal to +2.0% and +5.8%, were obtahd when the tnn, rrrtkr d I n w e d concentrath were 2.0 and 8.0, respectively. 0003-2700/92/0364-0489$03.00/0
INTRODUCTION Recently we introduced the sequential chromatogram ratio technique, based on calculating the point-by-point ratio of sequential chromatograms that have been corrected for background.' The shape of the resulting ratio chromatogram allows identification of the regions of the chromatogram where a single component elutes, thus providing qualitative information. Moreover, the ratio chromatogram provides the relative change in concentration between sequential injections. Our previous paper' explored the concepts of identification and quantitation of analytes, flagging of interferents, monitoring of concentration, and peak deconvolution without curve fitting. Ratio processing of data from sequential samples has been studied by infrared spectroscopists. Following the original concept of Hirschfeld? Koenig and co-workers3used the ratio of wquential spectra to determine scale factors for subtracting reference spectra from mixture spectra, and also to estimate constituent spectra. Later work demonstrated quantitation without external calibration for binary mixtures?.5 These scientists recognized that the high probability of overlapped bands in mixture spectra limited the usefulness of ratio techniques. Indeed, the success of many chemometric techniques employing spectral data depends upon the orthogonality of the constituent spectra? Chromatographypossessee two advantages for ratio processing of data: each component possesses only one peak, and overlap can potentially be controlled by manipulating the separation conditions. 0 1992 American Chemical Soclety
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 5, MARCH 1, 1992
Because the ratio chromatogram technique differs from the absorbance ratio technique,' in which the ratio of dual wavelength data collected simultaneously from the same chromatugram provides peak purity assessment, the sequential ratio technique has some unique concerns. For the absorbance ratio technique, retention time variation and adsorption isotherm nonlinearity are irrelevant, provided that the chromatogram contains a pure-elution region for each analyte. For the sequential ratio technique, however, the two data vectors are collected sequentially from chromatograms in which the analyte concentrations vary. Thus, retention time variation and adsorption isotherm nonlinearity can cause qualitative misinterpretation and quantitative bias. Loss of qualitative information is a result of detector nonlinearity for both ratio techniques. In addition, detector nonlinearity can bias the result for ratio of injected concentrations. Therefore, greater understanding of the dependence of the sequential ratio technique on retention time precision, adsorption isotherm linearity, and detector linearity is needed. Chromatographers have long struggled with the three issues just mentioned. Dolane has tabulated typical day-to-day variations in retention times due to changes in flow rate, temperature, and mobile-phase composition. Retention time variation is a problem with some chemometric approaches such as generalized rank annihilati~n.~ In this context, Kim'O and Sanchez" have suggested algorithms for time axis synchronization of peaks. Conder12has reviewed the sources of peak distortion in chromatography. The most important are thermodynamic in origin, relating to the behavior of the adsorption isotherm.13 Peak distortion under various adsorption isotherm models has been studied theoretically by solving the mass transport equations (ref 14 and references therein) and by employing the Craig distribution model (ref 15 and preceding series articles). Equivalence of these approaches has been shown.16 Numerical methods for peak-shape analysis include curve fitting"JS and the method of statistical moments, which is known to be sensitive to noise in the tail regions. Excoffier and co-workersl8have evaluated peak-shape comparison using the distribution function method and a procedure which sets the peak maxima in coincidence. However, the above methods do not readily allow correction of quantitative results for errors due to peak-shape change. In contrast, Balkemhas suggested a peak-shape comparison method, based on calculating the point-by-point difference of normalized peaks, which, as shown in this paper, can estimate a correction for peak-shape change. Nonlinearity of absorbance detectors for liquid chromatography has been studied in relation to detector band pass, analyte concentration, and detection wavelength.21-22 McDowell and co-workersZ3measured the effect on peak height, width, and area due to detector nonlinearity that approached saturation, and they were able to distinguish detector effects from adsorption isotherm effecta. Dorechel et alaunoted that slight detector nonlinearity often occurs in the so-called "linear" concentration range and must be addreased to avoid errors. They recommended either restricting the use of such a nonlinear detector to a narrow concentration range or using a nonlinear model such as the "almost linear" detector model of Fowlis and Scottz5or other detector models.22g26 The goal of this study was to investigate the ratio technique in situations involving retention time variation, concentrations exceeding the linear isotherm limit, and detector nonlinearity. Application to a single analyte was evaluated as a prerequisite to future multicomponent work. The approach was to use the ratio technique to measure a constant factor-of-two concentration increase over a large range in concentration. A minimization routine was developed to obtain a time-axis displacement with which to overlay sequential chromatograms.
Diagnostic tests were used to uncover the nonlinearity of the adsorption isotherm and of the detector with respect to injected concentration. The peak-shape analysis method of BalkeZ0and the detector model of Fowlis and Scottz5were adapted to the ratio technique to obtain corrections to the ratio value of injected concentrations. THEORY Review of the Ratio Technique for a Single Component. Extension to two or more components was presented previously.' If chromatography with absorbance detection is used to monitor the concentration of component A, then initial sampling and chromatographic analysis gives a detected signal, Sl(t),for the chromatogram. A Gaussian model, with retention time, t,, and peak standard deviation (width), st, is used because of its familiarity.
However, an exact peak-shape model is not required. In eq 1, CA,1 is the injected concentration, while Vi and F a r e the injected volume and eluent flow rate, respectively. For the detector, is the Beer's law absorptivity, and b is the optical path length. In practice, Sl(t) is a digitized record of the detected signal at equally spaced sampling periods beginning from the point of injection. An objective correction to center the background at zero2' is applied to all chromatograms. At some later time, a similar sample is used to obtain a subsequent chromatogram, S,(t), which is identical to eq 1 except for the subsequent injected concentration, CA,2.The ratio chromatogram,R(t),is formed from element-by-element division of the background-corrected sequential chromatograms.
In this division, all terms in eq 1 cancel except for the injeded concentrations, including the peak-shape model terms. Therefore, in the absence of noise, the ratio chromatogram maintains a flat, time-invariant ratio value throughout the duration of the peak for component A. This ratio value is equal to the ratio of component A concentrations in the two injections. A ratio value of 1.0 indicates no change in concentration. Knowledge of the initial concentration and the ratio value allows calculation of the subsequent concentration. In the case that the peak for component A overlaps with other peak(& the constant ratio value is maintained only in the pure-elution region for component A.' The ubiquitous presence of noise complicates the ratio process represented by eq 2. First, R(t)becomes numerically unstable as S,(t) approaches the background that was centered at zero. Therefore, to increase interpretability, R(t) is set to zero in regions where the detected signal for either chromatogram falls below a preset threshold value. Second, the noise introduces randomness into R(t),causing it to fluctuate about the true concentration ratio. The time-dependent, local variance in R(t),sR2(t),due to random concentration-independent noise, is derived by propagation of errors where sb,l is the noise standard deviation of SI@), and K is the ratio of the noise standard deviation of S z ( t ) to that of Sl(t). (3) In practice, K equals the ratio of the detector full-scale response for S1(t)to that for s&). Because of the randomness of R(t),a statistical estimation technique is used to calculate the ratio of injected concentrations,preferably using the entire pure-elution region so that signal averaging reduces the error of estimation. We have used a weighted-average ratio calculation over the pure-elution region, which compensates for
ANALYTICAL CHEMISTRY, VOL. 04, NO. 5, MARCH 1, 1992
the inhomogeneous variance of R ( t ) (eq 3). The procedure is to define a moving window of fixed width, M,and to move the window along the pure-elution region of R(t),thus producing a sequence of N overlapping intervals. The mean, Rp, is calculated for each interval, p , and is multiplied by an interval weight, w,,,, that is inversely proportional to sR2(t). Since is unavadable, a local estimate, sp2, is used instead, and the weights are conveniently scaled so that their sum equals N. N wp
= ( N / S P 2 ) / ( Cl/s,2)
(4)
p=l
Now, tip2should represent the randomness of R(t) associated only with noise and should not be affected by a nonzero slope of R(t) in the pure-elution region due to retention time variation, concentrations exceeding the linear isotherm limit, or detector nonlinearity. This requirement is adequately met by defining sp2 as the mean-squared residual2* of a linear regression of R(t)versus time, performed within window p ,
where ti = (1,2, ...,j , ...,W , and the mean for ti is l/'(M 1). The weight-average ratio value, R,, is calculated as
+
N
R , = (1/N) c wpp
The use of a moving window and regression-based weights differs from our previous work.' Retention Time Variation and Peak Alignment. Retention time variation between sequential injections severely distorts the shape of R(t) and may alter R, from the true concentration ratio. To understand this, we employ the Gaussian model from eq 1,substituting hl for the collection of coefficients and substituting a dimensionless time variable, z = (t = tr)/st,into the exponential term. The simplest case for a single analyte is that the retention time variation, Atr, is a displacement (shift) along the time axis. The dimensionless peak shift is defined as 6 = At,/s,. With these substitutions, the equations for the sequential chromatograms become
Sl(z) = hl exp(-z2/2)
(7)
Sz(z)= h2 exp(-f/,(z (8) Thus, the ratio chromatogram is not time invariant if the peaks are shifted.
R(z,6) = CA,~/CA,~ exp(z6 - 6'/2)
(9)
The value of R, for shifted Gaussian peaks may be derived by integration (Appendix). Since the chromatograms are digitized, the peaks may be moved into alignment by shifting the numerator peak by an integral number of sampling periods. A high data sampling rate of 300-500 points across the peak width is advantageous because it allows small shift increments. Since peak misalignment makes R(t)norhorizontal (eq 9), the number of sample periods that will best overlay the Chromatograms may be obtained by iteratively calculating the weighted standard deviation of R(t) about a horizontal line located at R,. sw
= ( -(N Y 1 pylwSp2 - NR,')
The minimum value of s,, corresponding to optimum horizontal flatness of R ( t ) for the aligned peaks, is further interpreted as the within-run standard deviation of R, for a given pair of chromatographicpeaks. Theoretical derivations and subsequent simulations involving Gaussian peaks have shown that the correct minimum is reached, even for peaks having different widths. Peak-Shape Analysis. For the remaining discussion, the sequential peaka for component A are considered to be aligned as discussed above. If the peak shape changes between sequential injections, then R(t) will not be flat because the peak-shape model terms in eq 1 will not cancel (eq 2), but instead will contribute to R(t)throughout the duration of the peak. The peak-shape analysis method of Balkemcan be used to evaluate and correct the ratio technique for concentrations that exceed the linear isotherm region, for which peak shape depends on concentration. The two peaks to be compared are truncated to the same number of points at corresponding time slices, and the peak areas, Al and A', and the peak heights, hl and hz, are obtained. The peaks are normalized to unit area, and the point-by-point difference (i.e. residuals) of the normalized peaks is calculated and plotted versus time. If the two peaks have the same shape, the residuals will be randomly scattered about zero. A change in peak shape is indicated by noticeable departure of the residual curve from zero. The residual plot is useful for qualitatively characterizing the peak-shape change in terms of broadening, asymmetry, etc. By scaling the residuals according to the height of the normalized peaks, the resulting residuals, DNP%H(t) (i.e. difference of normalized peaks, percent of height), are made quantitative.
(6)
p=l
)"'
(10)
401
DNP%H(t) = lOO(SZ(t)/A2 - Si(t)/AJ/h
(11)
h = f/z(hz/Az + hi/Ai)
(12)
where
is the average height for the normalized peaks. The value of DNP%H(t) a t the peak maximum is equal to the percentage change in peak height due to shape change alone, and it is considered to be an upper limit to the error in R, due to exceedii the linear isotherm region, since the ratio algorithm weights more strongly near the peak maximum. We have considered the effect of noise level because DNP%H(t) generally poaseases the independent sum of noise variances of both peaks. However, for this study, as well as for many chromatographic applications, the noise level is negligible for analyte concentrationsthat exceed the linear isotherm region. Detector Nonlinearity Correction. Correcting R, for detector nonlinearity is facilitated by employing the "almost linear" detector model of Fowlis and .This model represents mild calibration curvature over a broad concentration range by a power-law equation
S(t) = BC(t)X
(13)
where S(t) and C ( t ) are the time-dependent detected signal and analyte concentration at the detector, B is the apparent detector sensitivity a t unity concentration, and x is the detector response index, the value of which is reasonably close to unity. This model, eq 13,does not apply in the case of detector saturation.m Following the reasoning that led to eq 2, the ratio of injected concentrations is given by CA,2/CA,1
= R(t)'/"
(14)
Thus, the existence of flat regions in R(t) in the pure-elution regions is not affected by this type of detector nonlinearity. Further, the correction to R(t) (eq 14) depends on the concentration ratio, not the concentration level. If the nonlin-
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 5, MARCH 1, 1992 0.005
earity is slight (Le. x is close to unity), then instead of using eqs 14 and 6 to recalculate R,, a convenient detector nonlinearity correction may be made by replacing R ( t ) with R, in eq 14. Carr has discussed circumstances in which nonlinearity of both the adsorption isotherm and the detector occur^.^,^ For a Fowlis and Scott detector (eq 13))however, a constant ratio is still maintained in the pure-elution region (eq 14))implying that the detector-induced change in peak shape is independent of concentration. Moreover, it can be shown that the detector nonlinearity effect for a Gaussian concentration profile is to increase the width by a factor of llx. In contrast, adsorption isotherm nonlinearity is expected to degrade the “flatness” of R(t),as indicated by visual means, by a significant increase in the s, statistic, and by the shape analysis method (eqs 11 and 12). Although independent corrections are possible, it is probably safer to eliminate nonlinear adsorption isotherm behavior by quantitative dilution.
EXPERIMENTAL SECTION The HPLC system consisted of a reciprocating single-piston pump connected through an in-line filter and pulse dampener to a 10-port injection valve (Valco, Houston, TX) equipped with a 5-pL injection loop and an electric actuator. The column (Econosphere C-18, 5 pm, 4.6 X 250 mm, Alltech Associates, Deerfield, IL) was connected directly to the detector inlet (ISCO, model V4, Lincoln, NE). The detector time constant was 0.05 s, and a relatively insensitive wavelength (273 nm) was chosen so that analyte concentrations exceeding the adsorption isotherm linear range would fall within the dynamic range of the detector. The full-scale sensitivity was adjusted within a range of 0.14.002 absorbance units to maximize sensitivity. Reagent-gradebenzene (carcinogenic,researchers should choose other analytes, e.g. toluene), HPLC-grade methanol, and filtered deionized water were used to prepare the eluent (20%v/v water in methanol, 1.1mL/min) and a series of 10 benzene solutions in methanol ranging from 0.253 to 129 mM in constant factors of 2. The experimentally measured signal-to-noiseratios, S/N, defined as the ratio of the peak height to the background noise standard deviation, ranged from 11to 4100. By diluting in factors of 2 and 4,only five serial dilutions were required. Each solution was injected 10 times. Automated sample injection and data acquisition were achieved using an 8088ba4ed personal computer equipped with a laboratow interface. The software was written in Quick Basic 4.0 (Microsoft, Redmond, WA). The sampling rate was 48 Hz, exceeding the Nyquist minimum sampling rate of 40 Hz for the detector. The base-line peak width, determined by the width at half-height method, was about 400 points (8.3 s). The data postprocessing routines were written as functions within the Matlab software environment (The Math Works, Inc., South Natick, MA). Background correction was performed as described previously.n R ( t ) (eq 2) was set to zero whenever either chromatogram fell below a threshold of 3 times the background noise standard deviation (determined from blank injections). For the four highest-concentration solutions,a larger threshold equal to 1%of the peak maximum was used instead. An iterative peak alignment procedure was followed in which the user selected the number of sample periods and the direction by which to shift (i.e. displace)the numerator chromatogram relative to the denominator chromatogram. For each iteration, the chromatograms were truncated to the same number of points by retaining only the time-axis portions where both peaks were above the threshold. The optimum number of sample periods was identified at the minimum value of the weighted standard deviation (eq 10). The weight-average ratio value was calculated using eqs 4-6. The interval width, M, was 25 points (Le. about 11% of the width at half-height), selected as a compromise between responsiveness and accuracy of the weights. When the detector full-scale sensitivity differed for the two chromatograms, simple multiplicative corrections to the ratio values were made afterward. For the study of retention time variation, the replicate chromatograms to which the threshold had been applied were normalized to unit area using Simpson’s rule for numerical inte-
At = 10%
(A)
Of S 1
5
Time, s
if
“jl
00
235
I
I
240
I 245
250
255
Time, s F l p e 1. (A) Normaked sequential benzene c h r o m a t ~ m s (4.0 mM) showing a retentkm time change of -0.21 s (Le. 10% of s,)for S d t ) (-) r e l a t h to S , ( t )(---). The detected slgnal threshold was 3.6% of the peak height. Column: Econosphere C-18, 5 pm, 4.6 X 250 mm. Eluent: methanoVwater, 80:20 v/v, 1.1 mL/min. (6)Ratio chromatogram (eq 2) obtained a (---) before ailgnment of peaks, b (-) after alignment.
gration. For each solution, 10 pairs of replicate chromatograms (each chromatogram used twice) were first aligned, as just described. Peak shifts (i.e. trial numbers of sample periods) ranging from -25% to +25% of S, were processed.
RESULTS AND DISCUSSION Retention Time Variation and Peak Alignment. Before examining the ratio technique under conditions of nonlinearity of the adsorption isotherm or the detector, it was necessary to overlay the sequential chromatograms so that the ratio was calculated from corresponding time positions on the peaks. The alignment procedure was successfully applied throughout the concentration range studied to benzene chromatograms from solutions that were successively doubled in concentration. The output of the alignment procedure was the retention time shift, At,, of the pair of peaks. The mean value of At, was 0.08 s, and the sample standard deviation was 0.71 s for 90 pairs of sequential peaks. Although the At, values were small relative to the retention time of about 245 8 , they were significant when compared to the peak standard deviation (width), s, of 2.1 s. Such retention time shifts need to be considered when the ratio technique is applied. To illustrate, we consider normalized replicate chromatograms of 4.0 mM benzene, shown in Figure lA, in which Atr for the numerator peak was -0.21 s (Le. 10% of st), and the signal-to-noise ratio, S/N, was 165. The detected signal threshold, shown in effect by the abrupt truncation of the peaks, was 3.6% of the peak height. From the ratio chromatogram, R(t),Figure 1B curve a, calculated (eq 2) from the data for Figure lA, little analytical information can be gained. Instead of horizontal flatness, which would confirm peak identification,’ exponential behavior (eq 9) is seen in Figure 1B curve a. Furthermore, the weight-average ratio value, 8, (eq 6)) was biased from the expected value of unity to 0.994 (mean for 10 replicates). For Atr equal to 20% of st, the 8, result was 0.965. After the peaks were aligned, R(t)was flat, Figure 1B curve b, and fi, was 1.OOO. The run-to-run sample standard deviation of R, was 0.009 for 10 replicates. The within-run precision of R,, as given by s, (eq lo), the weighted
ANALYTICAL CHEMISTRY, VOL. 64, NO. 5, MARCH 1, 1992 LU
5 ._
I
498
I
2.5
d .
E c
F
0.0
s
._ -2.5 L
P
lij -5.0 .~ -20 a;
2
-10
0
10
20
Peak Shift, % of s , 0.101 (B)
10
100
1000
10000
Signal-to-Noise Ratio
1
Peak Shift, % of s t
Flgw 2. (A) Percentage wor in ratb value of injected concentrations
(eq 6) due to peak mlsallgnment: Mean results (0) f twice the sample for 10 replicates for the 4.0 mM benzene standard devlatbns (0) sokrtkn; theoretical curve for Gaussian peaks (-). Minimum peak shlft to observe a slgnlflcant bias was 13% of st. (8) Response of the weighted standard devietbn (Le. “flatness”) statlstic (eq 10) to peak shlft for the 4.0 mM solutlon. Minimum peak shift to observe a signlflcant increase In the statlstlc (eq 10) was 3% of s,. Labels a and b correspond to Figure 18.
standard deviation, was 0.011 f 0.006 (mean f twice the sample standard deviation). The equivalence of the within-run and run-brunprecisions demonstrates an ideal performance level for chromatography, obtained by successive replicate injections. A computational study of the effects of retention time variation, At,, was done to establish tolerances for At, in order to avoid the qualitative and quantitative errors illustrated in Figure 1B. To eliminate concentration effects and injected volume variation from the At, study, replicate chromatograms, normalized to unit area, were used. The implication of At, for quantitative analysis of pure peaks, shown in Figure 2A, was that perfect alignment of sequential chromatograms was not necessary for accurate quantitation. The (solid) theoretical curve for Gaussian peaks having randomly distributed noise (Appendix) showed negative errors in R, for all nonzero At,. The magnitudes of the errors were less than 1% for values of At, within f10% of st, increasing to 3% when At, was f20% of st. The mean experimental curves for the 4.0 m M solution (0, S/N = 165) and the other solutions (not shown) closely followed the theoretical curve. The fi, values from Figure 1B for Atr values equal to 10% of st (a) and zero (b) are labeled in Figure 2A. Precision was assessed by plotting run-to-run variability bands (0) in Figure 2A, equal to &2 times the sample standard deviation of R, for 10 replicates. As shown in Figure 2A, the bias in R, exceeded the run-brunvariability when At, was at least 13% of sI. As expected, the width of the variability bands decreased with increasing S/N, and the bias was significant for smaller values of At,. The implications of At, for qualitative analysis are shown in Figure 2B, in which the mean value (0) of the “flatness statistic”, s, (eq lo), used as an objective function to align sequential peaks,is plotted versus At, for the 4.0 mM benzene solution. Intervals (0) equal to f2 times the sample standard deviation of s, for 10 replicates are also shown. The sharp minimum in s, facilitated precise peak alignment, nearly
Flgure 3. Run-twn uncertainty in sequential chromatogram alignment versus S/N. Results obtained as in Figure 28: ratlo of chromatograms ratio of chromatograms of solutions doubled of the same solution (0); in concentration (0). Results from peak centrokl calculations (A).
always to within 1 sample period (i.e. 0.021 s, 1% of s t ) for a given pair of sequential peaks. On the other hand, the sensitive response of s, to At, indicates that perfect horizontal flatness of the ratio chromatogram,R(t),is difficult to obtain in practice because achieving it necessitates high data acquisition rates of at least 50 tQ 100 per st, according to Figure 2B, in order to “zero in” on the minimum for s,. The actual sampling rate was 100 per st. In addition, small changes in peak shape adversely affect the flatness of R(t). Thus, use of a visual assessment for the flatness of R ( t ) in the pureelution region as a method for analyte identification’ can be of limited utility. In combination with a visual inspection of R(t)for aligned peaks, it is useful to measure the departure of R(t) from horizontal flatness. This is done by calculating the value of s,/R, and testing for a significant increase compared to an established value of s,/R, obtained from ratios of aligned, replicate chromatograms of standard solutions. This statistical test is useful for recognizing small changes in sequential chromatograms due to internal time shifts, peak-shape changes, or the appearance of interferents. As an illustration, we determined the minimum retention time shift for replicate injections, At,(minimum), such that a significant increase in s, was observed. Division by R, (- 1.0) was omitted, since the peak shift had little effect on R, (Figure 2A). As shown in Figure 2B, At,(minimum) was specified as the value of At, for which the run-brunvariability band for s, did not overlap with the variability band where At, = 0. The value obtained was 3% of st. The calculations shown in Figure 2B were repeated for the other benzene solutions, and the results for At,(minimum) versus S/N are shown in Figure 3 for chromatograms of the same solution (0) and for chromatograms of solutions doubled in concentration (0) versus the S/N of the more dilute solution. The value of At,(minimum) was 3-5% of st (i.e. 0.06-0.10s) except for a large increase at low SIN. These valuea were equivalent to the run-brunprecision in sequential chromatogram alignment using superposition of the peak centroids (A). Such high repeatability of retention times is not normally posaible. However, by using an isolated constituent peak as an internal “timing standard”, the sensitivity of s, for revealing small changes in sequential chromatograms can be exploited. For ratio analysis of pure peaks, our findings were that the tolerance for At, may be substantially more lenient for quantitative analysis than for qualitative interpretation. The larger tolerance for quantitation occurs because the ratio chromatogram for misaligned peaks, as illustrated in Figure 1B curve a, possesses two offsetting regions in which R(t) is greater than the true injected concentration ratio in one region and is less than the true ratio in the other. This offsetting behavior in R(t) can be lost for quantitation of overlapped peaks. With two analytes, for example, the effect of a decrease in resolution is that part of one offsetting region in R(t)for
494
ANALYTICAL CHEMISTRY, VOL. 64, NO. 5, MARCH 1, 1992
01
10
1
100
1000
Concentration, mM
FIpw 5. Detector linearity plot (seetext) indicatlng concevsdownward cwvature of the caiibration c w e . 95% CI of the mean (10 replicates) are shown.
0
3 a
I
Time, s
184
0 1
'
1
Concentration,
10
I
100
mM
Flgurs 6. Ratio value of injected concentrations (eq 6) for solutions doubled in concentration, versus concentration of the more dilute solution. 95% C I of the mean (10 replicates) are shown. No corrections are shown except that the peaks were aligned.
1
I
a -I O
._t
235
240
245
250
255
260
Time s
Flgurs 4. (A) A significant increase in peak width occurs beyond the linear isotherm iimlt of about 20 mM for benzene. 95% confidence intervals (CI) of the mean (10 replicates) are shown, which coincide with the plot symbols at higher concentrations. (B) Well-behaved ratio chromatogram ( x l 6 , ---) obtained within the llnear isotherm region. Distorted ratio chromatogram (alp, -) obtained above the linear isotherm iimlt. (C) Shape analysis: Difference of normalized peaks, scaled as percentage of m i i z e d peak height (DNP%H(t),eq 11). concentratbns are shown within the linear isotherm region (x - 6,-4 and above the linear isotherm region (a- @, -). The retention time was 245 8 . Greek letters identify solution concentrations from A.
each analyte is "lost" to the overlap region.' Therefore, as the resolution decreases, the At, tolerance for quantitation decreases to the more stringent value required for qualitative analysis. Adsorption Isotherm Nonlinearity and Peak Shape Analysis. In Figure 4A the upper limit of the linear isotherm region for benzene, as easily discerned by the significant increase in peak width at half-height, was about 20 mM. Working within the linear isotherm region ensured constancy of peak shape over a useful concentration range of nearly 2 orders of magnitude. This range could easily be extended toward lower concentrations by using a more sensitive detection wavelength (ExperimentalSection). Constancy of peak shape in sequential chromatograms leads to a well-behaved ratio chromatogram, as shown in Figure 4B ( x / 6 , - -) for the 16.2 mM (x)and 8.1 mM (6) solutions that were within the linear isotherm region. For the 129 mM (a)and 64.5 mM (0) solutions that exceeded the linear isotherm region, peak broadening distorted the ratio chromatogram, as shown in Figure 4B (a/@,-). Therefore, restriction of the ratio technique to within the linear isotherm region is recommended. The nonflatness of R ( t ) was further indicated by a significant increase in (s,/R,) (100%) from (0.6 f 0.2)% (mean f 95% confidence interval (CI) for 10 replicates) for x / 6 to (1.7 f
-
0.2)% for a/@. The method of Balkem allowed quantitation of the injected concentration ratio for the 129 and 64.5 mM solutions despite the peak-shape change. An estimated upper limit correction factor to R,, equal to 1.023, was obtained as follows. The difference of the normalized peaks, scaled according to peak height, i.e. DNP%H(t), was calculated using eqs 11 and 12, and is shown in Figure 4C (a- @, -). The pattern of the residuals, negative near the peak maximum and positive in the head and tail regions, showed that the peaks were shortened and broadened for concentrations exceeding the linear isotherm region. The value of DNP%H(t) at the peak &um, -(2.3 f 0.3)% corresponded to an upper limit correction factor of 1.023, raising R, from 1.91 f 0.01 to 1.95 f 0.01. The a-(3 case just discussed yielded the largest correction for adsorption isotherm nonlinearity. The corrections diminished rapidly as concentration decreased, since the DNP%H(t) plot indicated no differences in peak shape for concentrations within the linear isotherm region, as shown in Figure 4C (x - 6, - - -). Detector Nonlinearity Correction. The detedor linearity Figure 5 (i.e. peak height divided by concentration versus logarithm of concentration), revealed slight concavedownward detector nonlinearity over a broad concentration range, not including detector saturation. Despite the detector nonlinearity, flat ratio chromatograms were observed within the linear isotherm range, as in Figure 4B. This observation supports the choice of the Fowlis and Scott detector model.% To apply the model, the detector response index (eq 13) was determined using least-squares regression of the logarithm of peak height versus the logarithm of Concentration. The detector response index (i.e. slope) was 0.9714, with a standard error of 3 x and the coefficient of determination was 0.9998. For a true injected concentration ratio of 2.0, the correction (eq 14) for R, was 0.039, i.e. 2.0% of the true ratio. The effect of detector nonlinearity was also apparent from Figure 6, in which the mean R, values (10 replicates each) for solutions successively doubled in concentration were plotted versus the concentration of the more dilute solution. NO correction for adsorption isotherm nonlinearity is shown in
ANALYTICAL CHEMISTRY, VOL. 04, NO. 5, MARCH 1, 1992
Table I. Overall Accuracy for Chromatographic h t i o Measurements of Concentration Increases
true ratio
a
Ped height ratio
ratio technique
peak area ratio
2 8
uncorrected 1.97 f 0.Ma 1.95 f 0.06 7.5 f 0.3 7.6 f 0.3
1.98 f 0.08 7.7 f 0.4
2 8
corrected for detector nonlinearity 2.01 f 0.06 1.99 f 0.06 8.1 & 0.3 8.0 f 0.4
2.02 f 0.09 8.2 f 0.5
95% confidence intervals for the mean.
Figure 6 for concentrations exceeding 20 mM. Despite the scatter in Figure 6, most of the R, values were below the expected value of 2.0, consistent with a detector response index below unity. As an overall assessment of the bias due to detector nonlinearity, the overall mean for nine R, values in Figure 6 was 1.97 with a 95% CI of f0.06 (3% relative). The detector nonlinearity correction, which was conveniently applied (eq 14), was not statistically significant, since the corrected mean was 2.01 f 0.06. The nonlinearity was further explored by rearranging the data and by repeating the calculation procedures to estimate a true injected concentration ratio of 8.0. In this case (Table I), the detector nonlinearity correction (eq 14) was signifcant, improving the overall mean for R, from 7.6 f 0.3 to 8.1 0.3. Parallel calculations for ratios of peak heights and peak areas, shown in Table I, gave equivalent results as the ratio technique, although the ratio technique may be applied to merely a portion of a peak in the case of overlap with another peak.' The scatter of the mean R, values in Figure 6 was consistent with the observed fluctuations in peak height shown in Figure 5. From a least-squares regression of peak height divided by concentration versus logarithm of concentration (Figure 51, the regression standard deviation for percentage error in peak height (or concentration) was 2.5%, which is reasonable, considering the possibility of serial dilution error, evaporation, or drift in separation conditions. From this uncertainty result, the 95% CI for R,, adding the relative errors, was *3.8% relative concentration change. In general, the increase in relative error as a result of error propagation is a disadvantage of the ratio technique, especially until long term run-to-run precision in chromatography can be improved. Application Note. It is instructive to describe the implementation of the algorithm. The detector should be calibrated using at least 3-5 concentration levels, spanning the desired range, to determine the response index, r , in eq 13, and to validate the detector model. A component whose peak is base-line separated from the desired analyte(s) is added to both sample and analyte standard solutions to serve as an internal timing standard. The internal timing standard provides unambiguous peak alignment of all chromatograms as discussed earlier. At least 3-5 sequential ratios should be performed, using chromatograms of the analyte standard solution(s),both with sample chromatograms of the analyta(s) of intereat, and with replicate chromatograms of the standards. The variation in I?, reflects run-to-run precision of the instrument. The 'flatness" of the ratio chromatogram, R(t),as indicated visually and by the value of s,/R,, is useful for analyta identification. Analyta identification is accomplished by comparing the 'flatness" results, just mentioned, for sequential ratios of the analyte standard with the unknown component peak,with the corresponding results for sequential ratioe of replicate chromatograms of the analyte standard. The comparison is a sensitive test for analyte identification, since a mismatch in either retention time (relative to the timing
*
485
standard peak) or in peak shape will upset the flatness of the ratio chromatogram, and will raise the value of s,/R, significantly. Therefore, false positive identification is considered unlikely. For concentrated samples, a quantitative dilution may be necessary to eliminate possible nonlinear adsorption isotherm effects. Following successful analyte identification, the injected concentration ratio is immediately available from the value of R,, to which the detector nonlinearity correction is applied (eq 14). If desired, the bias due to detector nonlinearity may be made negligible, as shown in Table I, by performing additional quantitative dilution. Run-to-run variation in the peak spacing for multicomponent samples, due to flow rate variation, drift in separation conditions, etc., may limit application of the sequential chromatqram ratio technique. We are investigating this issue for sequential chromatogram analysis of multicomponent samples, particularly as the distance in time between the analyte peak and the internal timing standard peak increasea.
ACKNOWLEDGMENT We thank the Center for Process Analytical Chemistry for support of this work. APPENDIX Effed of Retention Time Variation on the Ratio Value of Injected Concentrations. The weighbaverage ratio value, R,, for the case of two Gaussian peaks shifted by an amount 6 = AtJs,, may be calculated by substituting continuous functions into the defining equation for a weighted average. l_R(z,6)&,6) dz
R,@)
=
(AI)
l,&6)
dz
where o is the weight function and i = (t - tr)/sp First, into eq 3 substitute R(z,6) from eq 9 for R(t) and substitute Sl(z) from eq 7 for Sl(t). Use the result from eq 3 to substitute for o(z,6) = l/sR2(z,6)in eq Al. After substituting eq 9 into eq Al, the result is R w ( 6 ) = (cA,2/cA,l)(Il(6)/~2(6))
(A2a)
where
0(2,6)
-k ( ( C A , ~ / C AeXP(226 ,~)~
- b2))
(A2d)
Equations A2a-d were evaluated numerically using built-in Matlab functions.
REFERENCES (1) Synovec, R. E.; Johnson, E. L.; Bahowick, T. J. Sulya, A. W. A M I . them. 1990, 62, 1597-1803. (2) Hkschfeld, T. Anel. C h m . 1978, 48, 721-723. (3) Koenig, J. L.; DEsposb, L.; Antoon, M. K. App/. Spwtmsc. 1977, 31, 292-295. (4) Koenlg, J. L.; Kormos, D. Appl. Specbpsc. 1979, 33, 349-350. (5) Koenlg, J. L. Pvo A@. C h m . 1982, 54. 439-448. (8) Kallvas, J. H. Appl. Specbpsc. Rev. 1989, 25, 229-259. 1077, 134, (7) Yost, R.; Stoveken, J.; Medean. W. J . Chroma-. 73-82. ( 8 ) Dden, J. W. LC-QC 1990, 1 1 , 842-844. (9) Sanchez. E.; Remos, L. S.; Kowabki, B. R. J . Chfun?ato(r. 1987, 385, 151-184. (10) Kim, R. R. Ph.D. Disseftatlon, Messachusetto InstlMe of Techndogy, Cambridge, MA. 1885. (1 1) Sanchez, E. Ph.D. Mssertatkn.University of Washington, Seattle, WA, 1957. (12) COndeC. J. R. CC, J . R o s ~ ~C I ~h .fun?am C .h V m a m . Commun. 1982, 5 , 341-348. 397-403.
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RECEIVED for review July 30,1991. Accepted December 10, 1991.
Determination of Chlorite and Chlorate in Chlorinated and Chloraminated Drinking Water by Flow Injection Analysis and Ion Chromatography Andrea M. Dietrich,* Tracey D. Ledder, Daniel L. Gallagher, Margaret N. Grabeel, and Robert C. Hoehn Environmental Engineering and Environmental Sciences Division, Department of Civil Engineering, Virginia Tech, Blacksburg, Virginia 24061-0105
Thls research Investigated the determlnatlon of chlor'e and chlorate concentratlons In drlnklng water by flow InJectlon analyrls (PIA) wlth lodometrlc detection and Ion chromatog raphy (IC)wlth conductlvlty detectlon. The FIA and IC methods were accurate and effectlve for reagent water. The IC method was accurate for measurement of chlorite and chbrate concentrath h drkrkkrg water even In the presence of other oxldants Includlng chloramlnes. However, F I A was affected by chbramlnea and other oxldants In drlnklng water, rewltlng In Inaccurate determlnatlons. Whlle chlorlte concentratlono were unstable In chlorlnated drlnklng water, addltkn d sodlun oxalate Increased the stabUtly to up to 3 days and addltlon of ethylenedlamlne Increased stablllty to up to 18 days. Chlorate concentratbn were stable In drlnklng water for up to 18 days with or without a preservative.
INTRODUCTION Chlorine dioxide (C102)is a widely used disinfectant and bleaching agent that is currently being used by many drinking water treatment utilities in the United States, Canada, and Europe for oxidation and di~infection.'-~It is frequently applied as an initial oxidant during treatment, followed by chlorination (gaseous Clz or HOC11 as a final disinfectant. When used as an oxidant, chlorine dioxide reacts to form chlorite (CIOz-)and chlorate (C103-),which have been shown to cause hemolytic anemia in laboratory animals." Both C10, and C103- concentrations are currently under consideration for regulation by EPA and because of possible adverse health effects will likely be regulated with a maximum contaminant limit of
