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Width Based Quantitation of Chromatographic Peaks. Principles and Principal Characteristics. Akinde Florence Kadjo, Purnendu K. Dasgupta, Jianzhong Su, SuYu Liu, and Karsten G. Kraiczek Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.6b04857 • Publication Date (Web): 28 Feb 2017 Downloaded from http://pubs.acs.org on March 3, 2017

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Analytical Chemistry

Width Based Quantitation of Chromatographic Peaks. Principles and Principal Characteristics. Akinde F. Kadjo, Purnendu K. Dasgupta∗ Department of Chemistry and Biochemistry, University of Texas at Arlington, Arlington, TX 76019-0065, USA

Jianzhong Su, SuYu Liu Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, USA

Karsten G. Kraiczek Agilent Technologies, Hewlett-Packard Strasse 8, D 76337 Waldbronn, Germany

* E‐mail: [email protected] Fax: 817‐272‐3808.

1

ACS Paragon Plus Environment


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ABSTRACT Height‐ and area‐based quantitation reduce two‐dimensional data to a single value. For a calibration set, there is a single height‐ or area‐based quantitation equation. High‐ speed high‐resolution data acquisition now permits rapid measurement of the width of a peak (Wh), at any height h, (a fixed height, not a fixed fraction of the peak maximum), leading to any number of calibration curves. We propose a width‐based quantitation (WBQ) paradigm complementing height or area based approaches. When the analyte response across the measurement range is not strictly linear, WBQ can offer superior overall performance (lower root mean square relative error over the entire range) compared to area‐ or height‐based linear regression methods, rivaling weighted linear regression, provided that response is uniform near the height used for width measurement. To express concentration as an explicit function of width, chromatographic peaks are modeled as two different independent generalized Gaussian distribution functions, representing respectively the leading/trailing halves of the peak. The simple generalized equation can be expressed as Wh = p(ln ))q where is hmax/h, hmax being the peak amplitude and p and q being constants. This fits actual chromatographic peaks well, allowing explicit expressions for Wh. We consider the optimum height for quantitation. The width‐concentration relationship is given as

where a, b, and n are

constants. WBQ ultimately performs quantitation by projecting hmax from the width, provided that width is measured at a fixed height in the linear response domain. A companion paper discusses several other utilitarian attributes of width measurement.

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Since the inception of quantitative chromatography, the height and/or area of a peak have been used for quantitation. Area is a true representation of the solute quantity while the peak height is an oft‐used substitute.1 Chromatograms were once recorded manually; analog chart recording appeared in early 60’s.2 Area measurements were made with a planimeter, or approximated by triangulation, or paper cut‐outs of peaks weighed.3 The first digital integrators appeared in the mid‐70’s.4,5 Today one or more high performance liquid chromatography (HPLC) manufacturers allow data collection at 200 Hz.6 High speed data acquisition is vital to preserve the fidelity of resolution in fast HPLC.7 Data rates up to 500 Hz are used in commercial gas chromatography (GC) systems;8 and up to 20 kHz in research GC‐GC systems.9 Available software generates area‐ or height‐based calibration plots and quantitates chromatographic peaks with little or no user input. Height is better than area especially if peaks are poorly resolved;1,10 it is less affected by asymmetry and overlap (high asymmetry increases overlap probability).11,12 Over a large concentration span, linearity of area is better and area is preferred for better accuracy and precision.1,13 Noise filtration may affect peak height but not area.14 But both are affected by detector non‐linearity. In the case of detector saturation, changing parameters, e.g., a different wavelength15 or isotopologue16 is advised. Typical practice of area‐ and height‐based quantitation involve a single standard linear regression equation for quantitation. Standard linear regression minimizes absolute errors; the relative error (RE), often more important, may become large at low concentrations. Herein we describe the principles of width‐based quantitation (WBQ). We show that WBQ often provides less overall RE where the height or area data does not quite 3



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fit a linear relationship, especially due to divergence from linearity at higher concentrations, whether because of detector characteristics or chromatographic conditions. WBQ has previously been used in Flow‐Injection Analysis (FIA), pioneered by Ramsing et al.17 and Stewart;18 and also pursued by the present senior author.19,20 Due to space limitations, the use of WBQ in FIA is relegated to the Supporting Information (SI). However, these experiments have generally used an exponential dilution chamber or other dispersing elements to create conditions in which at least one side of the peak has an exponential profile so a (pseudo)‐linear relationship can be established between the peak width Wh at some fixed height h and the logarithm of the concentration (ln C). Such deliberate dispersion would be an anathema to chromatographers. Also, except when an exponential dilutor is used, the calibration equations have no fundamental mathematical basis. Applying WBQ to chromatography also requires a mathematical basis of relating width to concentration. It is an adage that chromatographic peaks cannot be described by a single mathematical model,1 but efforts to do so through various modified Gaussian models (see [21] and citations therein) have long abounded. However, an explicit expression of width at any height from such models is not straightforward. We propose a general model that provides good fits to both Gaussian and non‐Gaussian peaks and allows explicit expression of the width at any height, thus providing a sound basis for WBQ. We examine here the advantages/disadvantages of WBQ. Note that Wh refers to measurements at some fixed height h and not at some fixed fraction of the peak maximum (hmax), e.g., when asymmetry is measured at 5% or 10% of hmax.

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PRINCIPLES Gaussian Peaks. Chromatographic peaks are ideally Gaussian, the expected norm for a partition process. Many real peaks are highly symmetric, but rarely truly Gaussian. Still this is an appropriate beginning point. We assume for simplicity that our Gaussian peak is centered at zero time (t = 0); thence:

h =

*

∗

...(1)

where s is the standard deviation of the Gaussian peak. In order to calculate the width Wh at any particular height h, the corresponding t values on the ascending and descending sides of the peak (th,a, and th,d, respectively) are first needed. These are:

,

where =

 ...(2)

,

/ .

The width is then the difference between these two t values: Wh =

,

,

= 2s

...(3)

Hence, ∗

...(4)

If h is small enough to be in the linear response domain of the detector/analyte/column system, the ascending peak at h has no foreknowledge of whether the peak maximum will remain within the linear response domain, or in the extreme case, become completely clipped. Similarly, when descending through h on the trailing edge, it has no memory of the 5



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actual maximum value registered. Consequently, hmax computed from eq 4 is the height that would have been registered if the analyte peak remained within the linear domain regardless of whether it was actually so. Therefore, hmax computed from eq 4 is linearly related to the concentration C, giving more general forms of eq 4: ...(5) where a and b are constants. Non‐Gaussian Peaks. Non‐Gaussian peaks (tailing, fronting, or both) have been modeled as exponential or polynomial modified Gaussian peaks.22‐25 The width at a fixed height for a specific function is easily numerically computed; obtaining generally applicable analytical expressions is another matter. For most non‐Gaussian peaks, the peak is also asymmetric: the trailing edge of the peak is not a mirror image of the leading edge. Thus far peak modeling has focused on using a single function. We propose here to model the peak as two separate generalized Gaussian distribution (GGD) functions. The most general situation is where the two GGD functions may not share a common apex or have the same amplitude, as illustrated in Figure S1 in the Supporting Information (SI):

h =

∗ ∗

| | | |

,

...(6)

,

However, essentially all real peaks we have looked at fit very well with a shared apex with the two GGD functions thus having the same amplitude. Any departure observed

6


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occurs very close to the peak apex or the base, neither extreme being of great value to WBQ. Again, assuming peak apex location at = 0:

h =

∗

| |

∗

,

...(7)

,

where the top/bottom equations pertain respectively to leading/trailing halves of the peak. In proceeding similar to that for eq 3, Wh =

+

...(8)

where the first and second term on the right pertain to the leading and trailing halves, respectively and may be independently pursued for shape considerations,26 and also possibly for quantitation. Parameter limitations in eqs 6‐8 are easily imposed. Peak shape considerations (Figures S2, S3 in the SI) will indicate that for real chromatographic peaks the values of m and n in eq 8 will generally lie between 1 and 2, 1/m and 1/n (eq 8) therefore falling between 1 and 0.5). If Wh is measured between 5‐95% of hmax (1/ = 0.05‐0.95, = 1.05‐20). Within these constraints, it is readily shown that Wh can be expressed by a single term with 1% Root Mean Square (RMS) error (see Figure S4 in the SI and attendant discussion): Wh = (p(ln ))q ...(9) As an illustration, we chose some simple random values for the variables in eq. 8, that also results in a far from Gaussian peak:

7



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h =

∗

| | .

,

∗

.

,

...(10)

The peak resulting from these two functions is illustrated in Figure S5. Fits to similar equations for a number of illustrative real peaks are shown in Figures S6‐S9 in the SI. Following eq 8, Wh for this peak can be explicitly given as: Wh = .

∗

+ . ∗

...(11)

This is approximated with high accuracy following eq 9 to: Wh .

.

∗

...(12)

ln hmax can in this case be then expressed as: . WBQ is fundamentally based on

∗

.

...(13)

being linearly related to C. For quantitation, the

general relation of C as a function of Wh (and vice‐versa) are best expressed by: ...(14) Eq 5, representing a purely Gaussian peak, is simply a special case of eq 14 with n=2. It is worthwhile noting that values of n >2 produces a flat‐top (increasingly with increasing n, see Figure S3), not common in chromatographic peaks.

EXPERIMENTAL SECTION Ion chromatography (IC) data were generated with an IC‐25 system with an isocratic pump or a ICS‐5000 system with a gradient pump. Other components included G40 electrodialytic KOH eluent generator, injection volume of 10 L (unless otherwise stated), 2 8


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mm bore AG20/AS20 guard and separation column, LC30 temperature controlled oven (30 °C), ASRS‐Ultra II anion suppressor in external water mode, and a conductivity detector intgrated in the system (all foregoing: www.thermofiser.com). For all real chromatograms, the data array picked for height/area measurement contained at least 5 sampled points on each side of the putative peak and a best fit baseline was drawn through the extreme 5 points on each side. The sum of all the ordinate values in the baseline‐corrected data array was taken as the area and the maximum ordinate value as the height. Caffeine chromatographic data were generated in using a 1290 HPLC system on an Eclipse XDB‐C18 column (4.6 x 150 mm, dp =5 µm), using an isocratic 85:15 water:acetonitrile eluent at 1 mL/min. The diode array detector response time and optical slit width was set at 0.5 s and 4 nm, respectively (all foregoing; www.agilent.com). The absorbance measured at 272 nm averaged over 4 nm was referenced against measurements at 380 nm, averaged over 40 nm. Simulations. For all simulations (all peaks Gaussian, standard deviation ), the baseline was set at zero and the peak apex assigned a location of t = 0. The area was calculated from ‐5 to +5 . The height was then taken as the value of the highest datum in that domain.

RESULTS AND DISCUSSION Theoretical Limits of Uncertainty and Accuracy of Height, Area, and Width Measurement of an Ideal Gaussian Peak. In a typical quantitation situation, we refer to a plot of the measurand (height or area) against standard concentrations (the independent

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variable) as the calibration curve. For an unknown sample, the concentration is then interpreted from an appropriate equation that best fits the curve. In many practical situations, the major contributor to the errors and uncertainties in the ascertained concentrations is the imperfect fit of the calibration equation to the data. In a perfect world, an UV absorbance based quantitation system detection system should never, for example, have a finite intercept in a height‐ or area‐based linear relationship with concentration. However, even in this ideal world, there are finite limits to the attainable accuracy and precision for the measurands. We first explore these limits below. UV absorbance‐based quantitation, arguably the most common practice in HPLC, is assumed. With an ideal zero intercept linear calibration equation, uncertainties in height/area are proportionately translated into quantitation uncertainty. We assume a perfectly Gaussian band with  = 1 s under both no noise condition and a realistic amount of noise and stray light (0.05%). The base case has a true peak amplitude of 1 mAU. Because of stray light there will be a minute (‐0.05%) error in the measured absorbance. The peak to peak baseline noise is assumed to be 20 AU at a sampling frequency of 10 Hz (this is the best case for a present‐day diode array detector, isocratic conditions, no refractive index‐related noise). The true absorbance amplitude is not observed until the sampling frequency (f) is sufficiently high (see Table S1 and discussion in SI regarding adequacy of f to achieve a specified accuracy); however, the computed area is not affected.7 Height Measurement. Quantitation begins with ascertaining the beginning and the end of a peak, generally through a threshold slope or a minimum area specification. Within the domain of the peak so‐defined, finding the height maximum is thereafter straightforward. The associated inaccuracy and uncertainty, are however, affected by noise. To simulate 10


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random noise, the results below represent 10,000 computational trials. Taking 1 mAU as the true value, the error in the average height ranges from ‐1.7% at 10 Hz to +1.6% at 50 Hz, a combined result of inadequate f (dominant factor at low f), noise, and stray light; the relative standard deviation of this perceived height (the uncertainty) is in the 0.3‐0.4% range from 10‐50 Hz (Figure 1). As will be seen below, precision improves as absorbance increases; accuracy behaves similarly until stray light induced error become dominant. Area Measurement. Here errors/uncertainties stem from locating the beginning and the end of the peak in a noisy baseline. The success of different algorithms used in commercial software differs; a comparison is beyond the scope of this paper. The accuracy is unaffected to within 0.01% with a detection span > 5. A lower span does not capture all of the area, resulting in a negative error; a larger span only increases noise‐induced uncertainty. With an integration span of 5, the error is ~0.05%, arising primarily from stray light, while the uncertainty is ~0.2%. (Figure 1). Width Measurement. To determine Wh, one must first find the temporal locations of the specified h on the ascending/descending edges (hereinafter designated th,a and th,d, respectively) and thence determine Wh as th,d ‐ th,a. No digitized ordinate value may precisely equal h, however. If one takes the nearest point, its distance from the true location of h and the associated error will decrease with increasing sampling frequency (f) due to increased data density. Rather than the nearest point, th may be interpolated from discrete values h‐h’ and h+h’’ corresponding respectively to the locations t’ and t’’ (adjacent, f = 1/(t’‐t’’)) whose corresponding ordinates bracket h. Predictably, any type of interpolation gives better results than the nearest point approach. We used the simplest, a linear interpolation method. If f is sufficiently high, more sophisticated interpolation 11



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methods (e.g., a cubic spline fit) do not further reduce error, see Table S2 in the SI for illustrative error magnitudes obtained using nearest datum and interpolations by linear, and 4‐point spline methods. The interpolation error will decrease with increasing f. The RMS error as a function of f across a span of 1/ = 0.05‐0.95 is shown in Figure S10. At a given f, the error will depend on the specific value of h where the width is measured, as well as how the sampled points line up relative to the peak position. The error is maximum when h’ and h’’ are both significant (h’h’’ 0) and a minimum when the sampled datum precisely falls on h (h’ or h’’ = 0; if h’=h’’=0, the error is zero). The error oscillation frequency increases and the amplitude decreases with increasing f (Figure S11). But regardless of f, as a function of h the error decreases with decreasing sensitivity of Wh to changes in h, a minimum is reached at 1/ of ~ 0.6 (see following section on optimum height); the error sign changes thereafter. Although in Figure S11 the minimum errors seem to reach zero in each oscillation cycle, a magnified view (Figure S12) will indicate that is not actually the case. Also, the absolute error magnitudes are acutely dependent on the relative alignment of the sampled data and the peak. For the error to be exactly zero, not only must one sampled point fall exactly on th,a, Wh must be an exact multiple of 1/f so that th,d coincides with another sampled point. In any case, these ideal no‐noise case error magnitudes are overall too small to be of concern. Width Measurement in the Presence of Noise. Additional errors arise in locating th in the presence of noise. If th is sought ascending from the baseline, it is likely to be prematurely located because of noise, resulting in a Wh greater than the true value and thus a positive error in concentration. Conversely, if th is sought descending from the peak, premature identification will result in a negative error in concentration (see Figures 12


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S13a,b). Averaging the locations suggested from bottom‐up and top‐down searches will minimize the error, but not eliminate it. Note that if noise is truly random, each time an illustrative peak with noise is generated, the exact error will change. For this reason, we average the results of 10,000 computational trials in the presented data, as in Figure 1, which shows the relative error in concentration (assumed to be the same as the relative error in hmax), as predicted by WBQ using eq 5, plotted as a function of f. The error ranges from ‐1.4% at 10 Hz to 150 AU, even for the 1 mAU peak. We can deduce the optimum 1/ for measuring the width of a Gaussian peak. First principle considerations (see SI) suggests that the minimum sensitivity of

to occurs at

√

, i.e., at ~60% of the peak maximum

(Figure S15). However, the sensitivity remains relatively low and flat over a large span of 1/ from ~0.3 to 0.9, (and virtually constant between 0.4 and 0.8, Figure S15a). When constructing a width‐based calibration curve, one is obligated to choose a height that accommodates the lowest concentration calibrant. However, advances in memory storage and computing speed makes it practical to store not just width‐based calibration curves at several heights, but the entire profiles of the calibrant peaks. For an unknown, it is thus 13



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possible to refer to a stored width‐based calibration at a height nearest to the optimum (1/  0.6) of the unknown peak height, or to generate such a calibration equation from the stored data on the fly. If calibration concentrations are not close enough to permit choosing 1/  0.6 to accommodate concentrations below that of the unknown, an appropriate h is 90% of the hmax of the lowest concentration standard to be included in the calibration. Confining the calibration to the most relevant region improves accuracy.27 Errors and Uncertainties as a Function of Sampling Frequency and Peak Amplitude. Figure S13b makes it obvious that increased data density (increasing f) will reduce the interpolation error as the error in locating h decreases. Accordingly, regardless of the precise height at which width is measured, the error steeply increases as f is lowered below 10 Hz (Figure S16). The curves for h = 150 AU are less monotonic than those at 600 or 850 AU because of the greater effect of noise. Otherwise, for all three values of h, at f20Hz, the errors are all below 0.6%; the curves for h = 600 and 850 AU can barely be distinguished; the relative errors are ~0.1% for both at 50 Hz. The major difference between measurement at h = 150 vs. 600 or 850 AU is in the relative precision. Whereas in the entire f = 10‐50 Hz span, the magnitude of the uncertainty for the h = 150 AU measurement is always above 2% (frequently near 3%), those for 850 and 600 AU do not exceed ~0.6 and 1%, respectively. Similarly, at a fixed value for 1/ =0.15 for width measurement, as the peak amplitude is increased in steps from 1 to 10,000 mAU, the value of h proportionally increases, accordingly reducing the relative noise and thus the relative uncertainty (Figure S14). In contrast, there is hardly any change in the accuracy with increasing peak absorbance until at very high absorbance when stray light‐induced error becomes significant (the latter can obviously be avoided by measuring width at a lower 14


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height), the accuracy for 10‐1000 mAU are all generally better than 0.5% at f30Hz) and are all superior to that at 1 mAU (see Figure S17 for a magnified view). Measurement at 1.5 AU for a peak with 10 AU amplitude deteriorates the accuracy to ~1% but this is still far superior to what will be possible with height or area based quantitation at such an absorbance. Utility of WBQ under conditions of detector saturation are discussed in the companion paper.26 Quantitation based on Real Width‐ Height‐ and Area‐based Calibration Curves. The foregoing section on the limits of accuracy and precision on the quantitation of a single ideal Gaussian peak indicates that even under the stringent conditions of our base test case, the performance parameters are quite similar for the three different quantitation approaches. Based on purely theoretical considerations of a single ideal peak, height vs width determinations involve locating 1 vs. 2 points and thus the uncertainty cannot in principle be less for the latter in a well‐behaved system. However, as previously stated, the major uncertainty in quantitation from a linear calibration curve often lies in the failure of the data to perfectly fit the linear model. Below we focus on quantitation errors of real data sets by the three different approaches. As an indication of conformity to linearity, the r2 value obtained by linear regression is often used. This algorithm minimizes absolute errors. Relative errors, of greater interest to an analytical chemist, become large at the low end of the measurement range. Weighted linear regression addresses this and is often recommended over unweighted regression28,29 but not as often used. When the relative uncertainty is approximately constant across the calibration span, 1/x2‐weighting provides the least overall relative error;30 this is typical in most bioanalytical situations.31 Here we judge performance by the RMS relative error (RMSRE, see SI for illustrative calculation). 15



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We use IC data below, admittedly due to the predisposition of the senior author, but also because this represents a demanding test: responses of different analytes can be intrinsically linear or nonlinear, fronting and tailing or both are not uncommon, and while a detector response may become nonlinear, it is never completely saturated and thus does not provide obvious cues to abnormalcy. WBQ with Illustrative Ion Chromatographic Data. Capillary scale IC data for isocratic separation of 7 anions over a 1000‐fold concentration range was supplied by the manufacturer (Figure S18, S19). At the highest concentrations, every peak overlaps some with an adjacent one but with appropriate choices of the studied range and h, overlap effects can be avoided. We quantitated fluoride, chloride and phosphate, as these are least affected by overlap. Area was measured by the “vertical cut from the valley” method that is least affected by overlap (see SI in Ref. 26). As none of the peaks are really Gaussian (see [26] for shape analysis), eq 14 rather than eq 5 was used throughout. The best fit parameters were obtained using a nonlinear least squares sum minimization routine (Microsoft Excel SolverTM). The performance metrics appear in Table 1 and a sensitivity plot32 is in Figure S20. The latter clearly indicates that overall response is not linear. WBQ is self‐constrained to a height 20,000 ng, the peak is truncated, accounting for the precipitous fall in height or area response. In each case, the test concentration was quantitated using a calibration curve generated using two points above and two points below (but not including) the test concentration. Width was measured at 0.90 hmax of the lowest concentration in the curve. WBQ does better at the lowest concentration as well as at amounts >4000 ng. Note than the right y-axis has a break to accommodate the very high errors generated in area or height-based quantitation at concentrations >10,000 ng.

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Table 1. Weighted and Unweighted %RMS Errors for. Area, Height, Width based Quantitation.

Analyte

Injected Concentration Range, mM (n )

Unweighted, % RMSRE (r 2) Area

Width Eq 14

Height

1/x 2-weighted, % RMSRE (r2) Area

Height

Width Eq 14

5 (0.9834)

24.8 (0.9819)

Width Measurement Height (fraction of Hmax of Cmin)

A. Peaks Without major Fronting/Tailing Chloride Figures S17, S18 Phosphate Figures S17, S18 Fluoride Figures S17, S18 Chloride Figures S17, S18 Chloride Figure 3 Bromide Figure 3 Nitrate Figure 3

5.6E-3 - 2.8 (9)

11 (1.0000)

192.4 (0.9968) 26.7 (0.9838) 4.8 (0.9980)

61.4 (0.9967) 9.0 (0.9966) 1.0 (1.0000) 5.3 (0.9798)

2.8E-2 - 2.8 (7)

1.8 (1.0000)

2.8e-3 - 0.28 (7)

34.6 (0.9996) 21.8 (0.9999) 5.9 (0.9985) 6.8 (0.9888) 5.5 (0.9950) 5.9 (0.9985)

0.1 (0.29)

4.2E-3 - 2.1 (9)

57.1 (1.0000) 749.4 (0.9741) 31.2 (0.9782) 8.7 (0.9891) 17.2 (0.6395) 28 (0.9761)

0.1 (0.63)

2.1E-2 - 2.1 (7)

9.6 (1.0000) 232.9 (0.9751) 11.1 (0.9951) 2.7 (0.9990) 19.4 (0.6507) 11.1 (0.9952)

0.5 (0.48)

2 (0.45)

2.1E-3 - 0.21 (7)

56.2 (0.9988) 24.1 (0.9996) 9.8 (0.9960)

8.0 (0.9911) 9.8 (0.9944)

0.02 (0.29)

2.1E-3 - 1.05 (9)

163.4 (0.9980) 650.3 (0.9781) 26.2 (0.9836) 5.6 (0.9921) 15.1 (0.7360) 22.4 (0.9824)

0.14 (0.56)

1.05E-2 - 1.05 (7)

54.3 (0.9980) 201.1 (0.9786) 16.6 (0.9891) 5.2 (0.9837) 16.7 (0.7492) 16.6 (0.9883)

0.9 (0.63)

1.05E-3 - 0.105 (7)

33.2 (0.9997)

5.6E-3 - 2.8 (9)

11 (1.0000)

192.4 (0.9968) 26.7 (0.9838) 4.8 (0.9980)

2.8E-2 - 2.8 (7)

1.8 (1.0000)

61.4 (0.9967) 9.0 (0.9966) 1.0 (1.0000) 5.3 (0.9798)

2.8e-3 - 0.28 (7)

34.6 (0.9996) 21.8 (0.9999) 5.9 (0.9985) 6.8 (0.9888) 5.5 (0.9950) 5.9 (0.9985)

0.10 (0.29)

0.05-5.0 (6)

53.5 (0.9990) 31.3 (0.9997) 0.9 (1.0000) 9.5 (0.9697) 8.1 (0.9857) 0.9 (1.0000)

0.40(0.65)

0.05-5.0 (6)

61.2 (0.9986) 80.4 (0.9977) 5.4 (0.9988) 10.8 (0.9623) 15.6 (0.9315) 5.3 (0.9990)

0.17 (0.76)

0.05-5.0 (6)

67.1 (0.9985) 67.4 (0.9985) 5.9 (0.9985) 13.2 (0.9497) 13.4 (0.9492) 5.9 (0.9986)

0.17 (0.65)

15.6 (0.997)

9.7 (0.9740

9 (0.9965)

0.66 (0.90)

9.0 (0.9967) 6.5 (0.9875) 6.2 (0.9953) 5 (0.9834)

9 (0.9948)

0.04 (0.33)

24.8 (0.9819)

0.66 (0.90)

9 (0.9965)

2.00 (0.45)

B. Fronting/Tailing Peaks Formate Figure 4 Nitrate Figure 4 Trifluoro acetate Figure 4

0.1-10.0 (8)

110 (0.9939)

390 (0.9000)

3.1 (0.9996) 13.7 (0.9159) 36.5 (-0.8452) 3.1 (0.9996)

3.0 (0.51)

0.1-10.0 (8)

7.2 (1.0000)

190 (0.9783)

5.4 (0.9983) 3.7 (0.9993) 18.9 (0.7448) 5.4 (0.9983)

2.0 (0.51)

0.1-10.0 (8)

11.4 (0.9999) 54.5 (0.9985) 3.3 (0.9996) 3.1 (0.9990) 6.4 (0.9830) 3.3 (0.9996)

1.5 (0.37)

.

25



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Page 26 of 29

Table 2. Errors as a Function of the Height Chosen for Width Measurement Chloride

Bromide

%RMSRE

Height for Width Msmt, µS/cm

Eq 14 Best fit r2

0.9997

2.6

0.08

0.20

0.9995

3.4

0.30

0.9998

0.40 0.50

Nitrate

%RMSRE


Eq 14 Best fit r2

%RMSRE

0.9979

7.1

0.09

0.9977

7.1

0.11

0.9980

7.2

0.13

0.9970

8.3

2.2

0.14

0.9984

6.4

0.17

0.9985

5.9

1.0000

0.9

0.17

0.9988

5.4

0.21

0.9985

6.0

0.9997

2.7

0.50

0.9984

4.7

0.50a

0.9987

4.9


Eq 14 Best fit r2

0.10

an =6 in all cases except for nitrate at h = 0.5 S/cm, the lowest (50 M) datum cannot be

included due to insufficient height; in this case n=5.

26


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Width Based Quantitation of Chromatographic ... - ACS Publications

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