Digitization Errors in the Measurement of Statistical Moments of Chromatographic Peaks Stephen N. Cheder and Stuart P. Cram' Department of Chemistry, University of Florida, Gainesvilie, Fia. 32601 THEEFFECT OF LOCATING the limits of integration and random noise on chromatographic peaks has been discussed in a previous publication ( I ) where the errors in measuring statistical moments from digital data were treated in detail, These errors were found to be a function of the peak symmetry, the rate of analog-to-digital conversion, the location of the limits of the peak start and end, the order of the moment, and the signal-to-noise ratio for the chromatographic system. Additional considerations in treating the precision and accuracy of digital data are the instrumental effects of the analogto-digital converter (ADC) itself. Because the ADC has a finite conversion time, it will act as a low pass filter and thereby smooth the data to a certain extent. The "band pass" of the ADC will depend on the type of converter used, the presence or absence of a sample and hold amplifier, and the clock rate. For fast successive approximation converters, for example, this error will be small for chromatographic peaks. On the other hand, slow data acquisition rates and voltage-to-frequency converters (as used in digital integrators) will show an appreciable smoothing function and heavily filter the noise. Thus, the type of converter (2) and data acquisition rate (1) are of paramount importance in retaining the integrity of the analytical data. The second type of error introduced by the ADC which has a significant effect on the measurement of the statistical moments is the amplitude error. The magnitude of this error is determined by the resolution of the digital converter. Equation 1 shows that the error in the nth moment is a direct function of the error in the amplitude, y z ,
c mno =
YZ(X
c
-
IO0
$
a 0
K K
IO"
w
\
w
L I-
a A
w
IO'*
K
lo-a
0
0 I
I 0' I
I
l
l
1
IO
I
I
l
l
1
1
IOZ
PlY
2 - , n 2 2 Yr
where x is the time of measurement of y,, p1 is the normalized first moment of the peak profile, and n is the order of the total moment, mo. Further it is seen that this amplitude digitization error is not cancelled by the normalization of the moments and that it will affect the absolute value of the moment. It is also important to point out the dependence of the moments, particularly the higher order moments, on the accuracy of measuring the time of each data point with the highest precision possible. To minimize this error, we have used a quartz crystal programmable clock, a sample and hold amplifier (S/H), and a successive approximations ADC in our experimental system (3). The latter has a data conversion Present address, Analytical Chemistry Division, National Bureau of Standards, Washington, D.C. 20234. (1) S. N. Cheder andS. P. Cram, ANAL.CHEM.,43, 1922 (1971). (2) S. P. Cram, ibid., submitted for publication. (3) S. P. Cram and J. E. Leitner, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy,Cleveland, Ohio, March 6-10, 1972, Paper No. 337.
I
l
l
I os
NO. OF DATA POINTS
(1)
z
2240
IO'
Figure 1. The total error in calculating the ( A ) fourth moment with a 10-bit ADC (@I,( B ) fourth moment with a 13-bit ADC 0 and without an ADC (limit error = o), (C) zeroth moment with a 10-bit ADC (a), and ( D ) the zeroth moment with a 13bit ADC Cot and without an ADC (limit error = 0 ) for a Gaussian curve at i = O . O l % integration limits as a function of the number of data points taken per peak time which is independent of the amplitude of the input signal and, when operated under the control of the programmable clock, has a precision and accuracy which is better than the errors discussed previously. With the S/H-ADC configuration, the point in time in which the conversion is made can be accurately assigned to the end of the 12-psec tracking time of the S/H amplifier. In this manner, the time of each data point in Equation 1 is known and is independent of the amplitude of the datum. The timing error becomes particularly significant for the second and higher order moments and weights the data points on the leading and tailing parts of the elution curve most heavily where the relative error in the amplitude measurement also becomes very large and means that a programmable gain amplifier is essential to the data acquisition system (3).
ANALYTICAL CHEMISTRY, VOL. 44, NO. 13, NOVEMBER 1972
.
..a
t
m
NO. OF DATA POINTS
Figure 2. The total error in calculating the ( A ) fourth moment with a 10-bit ADC (e),( B ) fourth moment with a 13-bit ADC (0, and without ADC (limit error = e), (C) zeroth moment with a 10-bit ADC (e),and (D)the zeroth moment with a 13bit ADC (0,and without an ADC (limit error = 0 ) for a Type I1 curve at *O.Olz integration limits as a function of the number of data points taken per peak CALCULATIONS
All of the results presented here were simulated on a PDP8/L laboratory computer with 8K of core (Digital Equipment Corp.). This system includes the following peripheral devices which were used in this work: ASR-33 Teletype (Teletype Corp.), a four-tape magnetic tape cassette system (Model 4096, Tri-Data Corp.), a high speed paper tape reader (Mark V, Datascan), 15-inch display scope (Model 1735D, International Telephone and Telegraph), Model 547 laboratory oscilloscope (Tektronix, Inc.), and a 10-bit digital-to-analog converter built from a Model COO2 ADC (Digital Equipment Corp.). The amplitude error in the ADC conversion for a complete moment analysis is discussed here. The peak shapes and limits of integration used in this work are the same as those previously defined ( I ) . The Gaussian and Type I1 peaks were normalized to a maximum amplitude of 10.23 V and 8.191 V in order to simulate the digitization error in a 10-bit and 13-bit successive approximations converter, respectively. The 10-bit converter was chosen for this study because it is used extensively for analog-to-digital conversion in most laboratories. The resolution of such a converter is limited to 10 mV for the least significant bit (LSB) on a 10-volt
N
d
4
i
-2
-
t
t
v
2 Y
v)
v1
V Y
u
I
ANALYTICAL CHEMISTRY, VOL. 44, NO. 13, NOVEMBER 1972
I
2241
Table 11. Errors in the Statistical Moments Due to ADC and Integration Limit Effects for a Type I1 Curve N ~ of. -Errors at =tl.Oz Errors at f 0 . 1 Errors at 10.01 Moment“ data pts 10 Bits 13 Bits Limit 10 Bits 13 Bits Limit 10 Bits 13 Bits Limit 1.10 10 0.565 0.585 0.586 0.225 22.6 -26.1 -29.5 -29.5 -24.5 100 1.62 1.62 1.62 0.108 0.141 0.136 1.70 X lo-’ 2.78 X 10-2 -2.46 X 10-2 500 1.59 1.59 1.60 0.167 0.158 0.157 5.30 X 2.08 X 10-2 2.21 X 10-2 2.09 1.92 2.10 2.09 1.93 1.93 1.11 10 2.37 2.38 2.37 2.22 2.22 2.53 100 2‘22 2.53 2.53 2.57 2.58 2.58 2.23 500 2.23 2.23 2.53 2.53 2.53 2.57 2.58 2.58 1.12 10 41.7 42.0 42.0 25.9 25.2 19.0 29.1 28.3 28.4 35.5 8.78 100 35.4 35.4 8.79 8.75 3.67 1.76 1.70 500 35.0 35.0 35.0 8.74 8.51 8.50 4.02 1.63 1.59 1.13 10 71.0 71.2 71.2 36.2 35.2 35.1 27.8 25.2 25.4 64.3 64.2 24.0 100 64.2 23.8 23.8 12.1 6.32 6.15 500 63.8 63.7 63.8 23.8 23.3 23.3 13.0 5.99 5.92 54.5 53.3 57.3 10 85.4 85.5 85.5 40.5 35.9 36.1 80.5 40.3 100 80.5 80.5 40.1 40.0 23.8 13.8 13.6 40.0 39.4 39.4 500 80.1 80.1 80.1 25.2 13.3 13.2 Skew (1.1~3/1.12~”) 10 34.8 34.7 34.7 0.022 -0.27 18.2 -20.9 -23.3 -23.2 100 31.1 31.1 12.8 12.6 12.6 31.1 7.04 3.79 3.71 500 30.9 30.8 30.8 12.6 12.4 12.3 7.53 3.71 3.63 Excess(fi4/pa2) 10 57.1 57.0 57.0 17.1 16.5 34.5 -18.3 -24.8 -24.5 100 53.2 53.2 53.1 28.3 28.0 28.0 17.9 10.7 10.6 500 52.9 52.9 52.9 28.0 27.6 27.6 18.8 10.5 10.3 a Errors are calculated as relative errors from the true value of the moments and expressed as a per cent True values are: p o = 1.58; fil = 2.59; 1.12 = 25.4; p g = 352; fi4 = 8.93 X lo3; skew = 2.75; excess = 13.8.
z
z
(z).
input. Therefore, the 13-bit converter (LSB = 1 0 . 6 mV) was also studied in order to examine the effect of the higher resolution and precision of these converters. The 13-bit converter would also be equivalent to using an autoranging amplifier (with a maximum gain of eight) and a 10-bit ADC. The simulation calculations and programs are the same as those previously reported ( I ) except that the resolution error of the ADC is included here. Also, it was assumed that the peak-to-peak noise was less than the least significant bit of each converter which is a reasonable assumption for most clean, properly operated, well shielded and grounded ionization detectors. The errors introduced into the moment calculations by a successive approximations type of ADC with a negligible timing error (as previously discussed) are tabulated in Tables I and 11. These tables give a complete error analysis for each of the moments for Gaussian and Type I1 curves at three different limits of integration for the case of a 10-bit ADC, 13-bit ADC, and no ADC at all (limit error) as the number of data points per peak are increased. The errors are tabulated as either relative errors or difference errors (because the odd numbered moments of a Gaussian curve equal zero). It is very important to note that these “errors” are really a total error, and as such represent the sum of the limit error and the ADC error. Thus the error due to the ADC alone may be calculated from the difference between the 10-bit and 13-bit errors given in Tables I and I1 and the limit error for any of the limits of integration or number of data points. The total error is given here as it will be the most useful figure and represents the real limiting error in statistical moment calculations. It should be noted for both the Gaussian and Type I1 error tables that: a. For any reasonable clock rate or fixed number of data points, the error for the 10-bit ADC is greater than the 13-bit ADC. b. The error for the 13-bit ADC ,approaches the limit error itself when enough data points are taken, regardless of the order of the moment. 2242
c. As the order of the moment increases, the total error for a converter of any precision increases. d. The errors are significantly larger for the asymmetrical peak. e. The total error for any converter decreases as the integration limit is increased (e.g., from = t l . O O ~ to +0.01%). The first four of these effects can be more clearly seen in Figures 1 and 2. Figure 1 represents the boundaries of a family of curves for the zeroth through the fourth moments for a Gaussian curve, and Figure 2 indicates the limiting error curves for a Type I1 curve. Because the skew and the excess are ratios of the other moments calculated, their error curves will also fall within the limits shown here. It can be seen that the errors in the zeroth moment for a Type I1 curve are a factor of ten larger than the corresponding values for Gaussian peaks, and that the errors are two orders of magnitude larger for the fourth moment calculations. The large errors associated with the 10-bit ADC in Figures 1 and 2 are a direct result of the relatively large value of the least significant bit for a converter with a resolution of one part in 1024. This difference in converters introduces an additional error of only 0.01-0.03% in the peak area calculation and no appreciable error in the peak mean for 1 0 . 1 limits on a Type I1 curve. In both cases, the digitization error introduced by the 13-bit converter was less than one part in ten thousand, However, errors as large as 10-12z in the fourth moment may be attributed to the differences in the ADC resolution. As the integration limits are increased, the error for any converter decreases as expected because we are getting a better approximation to the true peak area and including more of those points which are most heavily weighted in the moment calculations (see Equation 1). That also explains why the errors increase as the order of the moment increases for any given integration limit. The oscillation of the error curves as they approach a fixed error level is real, but should not be taken as absolute values. Rather they are included to show that they do occur and that
ANALYTICAL CHEMISTRY, VOL. 44, NO. 13, NOVEMBER 1972
because the errors are so close to zero, a large scatter is to be expected. Further, it does not make sense to choose the number of data points for characterization in this region of the curve. Absolute values of the errors were plotted and thus they can be expected to oscillate around a minimum measurement error. These oscillations will decrease with the order of the moment because the weighting of the moments in-
creases for points which are further removed from the center ofgravityofthepeak.
RECEIVED for review May 1, 1972. Accepted July 19, 1972. Financial support of the National Science Foundation under Grant No. GP,.14754is gratefully acknowledged.
Fluorometric Assay of Methyl Ketones David N. Kramer, Lucio U. Tolentino, and Ethel B. Hackley Physical Research Laboratory, Research Laboratories, Edgewood Arsenal, Md. 21010
THEREIS A PAUCITY of simple spectrophotometric methods for the microdetermination of methyl ketones. Recent studies of the nitroprusside method by Yampol'skii and Geller (1) showed that the sensitivity achievable is of the order of 250 /*g/ml. Levin and Taterka (2)proposed the use of vanillin in alkaline medium for the detection of aliphatic ketones, especially methyl ketones. In addition to its insensitivity, the method is nonspecific and requires a prolonged heating step. The nonspecific reaction of methyl ketones with hypohalite solutions to form haloform has been suggested as a photometric method for methyl ketones (3). Also, the insensitive Meisenheimer reaction ( 4 ) employing polynitrobenzenes in alkaline media for the detection of active methylene compounds has been proposed. Adamiak (5) reported a colorimetric determination of methyl ketone in air using furfural in alkaline media followed by a treatment with concentrated sulfuric acid. The method required a two-hour reaction time and had a sensitivity of 3 /*g/ml. Bayer and Drewsen (6) and Tanasescu and Georgescu (7) found that the condensation of o-nitrobenzaldehyde 1 with a-methyl ketones in the presence of alkali yielded ketols of the type 2. Further reaction in the presence of alkali results in the formation of indigo. Compounds 1, 2, and 3 have been postulated (6). The reaction may be depicted as shown in Scheme I. Feigl (8) has suggested the use of 1 as a reagent for the qualitative detection of compounds containing the CH3C0 grouping. According to Feigl, the formation of indigo proceeds by the intermediate formation of y-nitrostyrene. We wish to report results of our reexamination of this reaction in an attempt to quantitatively analyze for methyl ketones and in addition elucidate the mechanism by which indigo is ._
(1) M. Z. Yampol'skii and B. E. Geller, Tr. Kom. Anal. Khim., Akad. Nauk SSSR. h i . Geochim. Anal. Khim., 13, 78 (1963); Chem. Abstr., 59, 6993d (1963). (2) V. E. Levin and M. Taterka, Anal. Chim. Acta., 15, 237-245 ( 1956). (3) V. Sedivec, Chrm. Listy., 51, 63 (1957); Chem. Abstr., 51, 11933f (1957). (4) T. Momose, Y.Ohkura, and K. Kohashi, Chcm. Phnrm. Bid/., 11, 301 (1963); Chem. Abstr., 59, 5469e (1963). ( 5 ) J. Adamiak, Chem. A m / . (Warsaw). 9, 1051 (1964); Chem. Absrr., 62, 16871c (1965). (6) A. V. Bayer and V. Drewsen, Ber., 15, 2859 (1882). (7) I. Tanasescu and A. Georgescu, Buff.SOC.Chim. Fr., 5 l P 234 (1932). (8) F. Feigl, "Spot Tests in Organic Analysis," Elsevier, New York, N.Y., 1956, p 224.
2
1 -
4 -
3
moH __C
indigo
H indoxyl
-5
6 -
Scheme I formed. Our preliminary studies had indicated that 1 and acetone in the presence of dilute alkali undergo a series of reactions yielding initially a fluorescent substance, then a nonfluorescent yellow dye, and finally precipitation of indigo. We now report a simple direct method for the assay of methyl ketones based on the intermediate formation of the fluorescent indoxyl, using acetone and acetophenone as model compounds in the development of the assay. EXPERIMENTAL
Reagents. All experiments were performed with reagent grade chemicals and pure solvents. Ortho-nitrobenzaldehyde was employed as received from Matheson, Coleman and Bell, East Rutherford, N.J., mp 42-44 "C. The ortho-phenylacetyl ketol 2 was prepared in the following manner: 0.001 mole of 1 was reacted in acetone solvent in the presence of sufficient 10% NaOH to produce traces of indigo. The solution was stirred for a half-hour and filtered to remove the indigo. An equal volume of water was added and the solution was extracted with ligroin. The ligroin was charcoaled and concentrated in cacuo. The ketol crystallized on cooling. Recrystallization of material from ligroin yielded a product melting sharply at 62-62.4 "C uncorrected [reported (64-66 "C)] (7). In a similar fashion, the ketol is prepared from acetophenone (mp 62.5-64). A 10-*M stock acetone solution was prepared and diluted with distilled water to obtain desired concentrations.
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