Computer Characterization of Chromatographic Peaks by Plate Height and Higher Central Moments Eli Grushka, Marcus N. Myers, Paul D. Schettler,’ and J. Calvin Giddings Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Computer methods make possible the detailed analysis of chromatographic peak shape, and can be used to obtain the various moments of experimental peaks. The importance of the statistical moments and related properties i s discussed here. The moments characterize a peak completely and are related to parameters such as theoretical plate height and forms of asymmetry such as peak tailing. They may also reflect peak contamination. A rapid and accurate method for measuring moments is described. This involves a digitizer coupled with a gas chromatograph. The data obtained are analyzed by a Univac 1108 computer.
THEDETAILED FORM of chromatographic peaks has been the subject of considerable theoretical inquiry, but not of experimental investigation. The latter often ends with a specification of peak location and area, sometimes width, and infrequently with a statement about asymmetry. Too often the Gaussian approximation by itself is assumed to contain all relevant characteristics of the peak. The advent of computerized data reduction systems has so far done little to alter this outlook. Indeed resolution is ordinarily measured to a very good approximation in terms only of peak location and width. However, the more subtle characteristics of a peak may contain valuable information of a diagnostic type. First of all, the non-Gaussian terms may contain information about strongly overlapping peaks and forms of contamination generally. Second, they may reflect the condition of the column and, if understood, could possess sufficient information to specify improved, perhaps even optimized, conditions. Third, they may be solute-selective and thus offer an adjunct tool for identification. A necessary step toward implementing these approaches is, of course, the development of a methodology for detailing peak characteristics. This is the subject of the present paper. Several options exist for the mathematical expression of the peak profile. We choose here to characterize a peak by its statistical moments (and related parameters). The importance and theoretical evaluation of these moments in chromatography has been discussed in a number of papers (1-6). The choice of moments for experimental characterization hinges on the fact that they completely specify any peak (7), and that each one is easily interpreted. Present address: Department of Chemistry, Juniata College, Huntingdon, Pa. 16652. 1
(1) J. C. Giddings and H. Eyring, J. Phys. Chem., 59,416 (1955). (2) E. Kucera, J. Chromatogr., 19, 273 (1965). (3) H. Vink, ibid., 20, 305 (1965). (4) H. Yamazaki, ibid., 27, 14 (1967). ( 5 ) J. C. Sternberg, “Advances in Chromatography,” J. C. Giddings and R. A. Keller. Eds.. Marcel Dekker, New York, Vol. 2, 1966, pp 205-270. (6) 0. Gruber, “Advances in Chromatography,” J. C. Giddings and R. A. Keller, Eds., Marcel Dekker, New York, Vol. 6, 1968, pp. 173-246. (7) . I G. A. Korn and T. M. Korn. “Mathematical Handbook for Scientists and Engineers,” McGraw-Hill Book Company, New York, 1968, p 599.
The individual moments characterize a peak in the following fashion. The zeroeth moment of the peak (if the peak is not normalized) is its area. The first moment is the coordinate of the center of gravity of the peak and thus equals the retention time (or migration distance). The second moment, when taken around the center of gravity, is a measure of the peak’s width. The third moment (also taken around the first) shows the magnitude as well as the direction of the peak’s asymmetry. The fourth moment (again around the first) is a measure of peak flattening as well as peak width. All other odd moments are related to asymmetry while the higher even moments measure characteristics similar to the fourth moment. The skew and excess are two moment-related quantities which are useful because they measure directly the deviations from a Gaussian profile. The skew is a measure of peak asymmetry and the excess reflects the flattening of a peak relative to the Gaussian norm. While many chromatographic peaks approach Gaussian shape to a fair approximation, the precise mathematical form of the peak profiles for a general retention mechanism has never been deduced. Giddings has pointed out that the simplest conceivable retention mechanism (single step sorption-desorption) does not lead to a Gaussian profile, but to a considerably more complicated form containing a Bessel function and an exponential term (8, 9). The plate model itself leads to binomial and Poisson distributions. Only in the limit of infinitely long times do all these mathematical expressions approach a Gaussian form (8). The departure from this limit within a single peak reflects the finite rate of various column processes. These rates are a function of column conditions and of the solute itself. In contaminated peaks the non-Gaussian terms reflect, in addition, a distortion due to the contamination. THEORY
Mathematical Characterization of the Peak. ized nth moment of a peak is defined by
The normal-
wheref(t) is the peak profile and t is the time (or distance) coordinate. For convenience it is better to take all moments higher than the first around the center of gravity of the peakLe., around the t coordinate of the first moment
It was mentioned before that the moments completely define a peak. Consistent with this, the function f ( t ) can (8) J. C . Giddings, “Dynamics of Chromatography, Part I: Principles and Theory,” Marcel Dekker, Inc., New York, 1965, pp
2Ck26. (9) J. C. Giddings in “Gas Chromatography, 1964,” A. Goldup, Ed., Elsevier Publishing Company, Amsterdam, 1964. VOL. 41, NO. 7, JUNE 1969
889
be expanded in terms of its moments. One such expression is (IO)
FmsSula Regulata
12
where t is now the time relative to the first moment, Hi is the ith Hermite Polynomial (111, u 2 is the variance of the peak (= m2) and the ti's are the expansion coefficients which, as Equation 3 shows, measure directly the departure from Gaussian. These coefficients are:
etc.
(4)
The skew and the excess are defined as c3 and c4,respectively; these quantities are thus seen to express the simplest departures from Gaussian. All the odd c’s in Equation 4 will vanish for a symmetrical peak and all c’s will vanish if the peak is Gaussian. In theory all the moments of a peak can be measured, but we have found that in practice one is limited to roughly the first 5 moments, mo through m4. By the nature of their definition in Equation 2 the higher moments magnify the importance of those points which are far from the center of gravity of the peak. Thus base line noise and drift will be amplified and will contribute increasingly to the magnitude of the higher moments. In fact the importance of the base line is even evident in some of the first five moments, as will be shown in the results section. Measurements of the Moments, Skew, and Excess. If the detector output is divided into finite intervals and digitized, the moments become (5) where y , is the peak height “count” in the ith interval. again, ti is relative to mlfor n > 1.
Here,
EXPERIMJ3NTA.L Apparatus. Figure 1 shows the experimental system. The gas chromatograph consisted of a constant temperature oven and a Beckman flame ionization detector and electrometer (Model GC-4). The electrometer was connected to an Infotronic CRS-30D Digitizer which in turn operated a high-speed tape perforator (Teletype BRPE 11) and an Esterline Angus Speedservo Recorder. The latter allowed the visual monitoring of peak shape and elution times. The paper-tape data were converted to card form with an IBM 46 tape-to-card punch for input to a Univac 1108 Computer. Additional information such as perforator start time, peak identification, etc., were added after the conversion of the data to card form. Peak Generation. Two peak generators were made. The first was a square waue (SW) generator which consisted of an Eveready No. 735 NEDA 900 1.5-V battery, a voltage divider, and a microswitch. The circuitry was such that when the switch was in the open position the digitizer input was shorted. A signal of about 10 mV was introduced to the digitizer by closing the microswitch for a certain predetermined time interval.
(10) F. Zernike, “Handbuch Der Physik,” Vol. 111, J. Springer, Berlin, 1928, p 448. (11) H. Margenau and G. M. Murphy, “The Mathematics of Physics and Chemistry,” D. Van Nostrand Company, Princeton, N. J., 2nd ed., 1964,p 121. 890
0
ANALYTICAL CHEMISTRY
1-1
Flame Ionization
Paper Tape
Angus - Esterline Speedservo Recorder
The second signal generator employed a cam system. The cam was connected to a 1-rpm Hurst motor. An arm, which was attached at its pivot to a Bourns 35105-1-102 1 Kohm variable resistor, followed the cam periphery. This produced a change of about 40 ohms in the resistor. A HewlettPackard Model 6215A Power Supply was used as a voltage source. Its output was varied until a signal of desirable magnitude was obtained. The signal shape resembled a chromatographic peak. Materials Used. The two columns employed were an open stainless steel tube 1520-cm long with inner radius of 0.263 cm, and a 200-cm long copper tube with an inside radius of 0.241 cm, packed with Chromosorb W coated with 5% by weight Apiezon L. Methane and pentane were used as the chromatographic solutes, while helium was employed as the carrier gas. Procedure. Sample injection was accomplished with a Carle valve (No. 2014) which switched a small amount of sample vapor into the carrier stream. Two timers were automatically started as the injection was made. One was stopped when the tape perforator was started, immediately before the emergence of the peak from the column. The second timer was used to obtain either the retention time (the peak maximum) or perforator stop time. The perforator was stopped manually when the trace on the recorder indicated the signal had reached the base line. Spacing punches were used to separate data from different peaks. Studies of the effect of the base line setting of the digitizer were carried out by first adjusting the base line to zero according to the manufacter’s instructions, then moving the base about 900 counts above and below that true base line. Computer Program. A computer program was written to calculate and print the peak’s area, retention time, skew, excess, and plate height. The latter was calculated by three methods: a) from the second moment, b) from the width at half-height, assuming a Gaussian peak shape, and c) by nonlinear least-squares fitting a Gaussian peak (12) to the observed peak and using the resultant u value. (12) D. W. Marquardt, “Least Squares Estimation of Non-Linear Parameters,” IBM Share General Program Library, No. 3094, March 1964.
Table I. Generated Peaks Study Dig. time Averages
Peak shape Square wave
...
(SK)
rnl
(sed=
0.1
...
...
... ...
Error Averages
Square wave
Error Averages
Square wave
0.1
% Error Averages
Cam generated
...
...
0.05
Cam generated
0.1
21.16 f 0.049 0.23
Error Averages
...
% Error ... a First moment. b Second moment. c Area in arbitrary units.
a
F U L L GAUSSIAN: Skew =O HALF GAUSSIAN (Shaded): Skew=0.995
0.1
mz (sec)b
8.31 f 0.042 0.50
8.44 3~0.052
... ...
0.6
...
...
1 .o
22.05 f 0.012
0.7 64.16 f 0.015 0.023 64.199 f 0.021 0.03
...
...
8.65 3ZO. 058
Areac X 9.103 f 0.036 0.40 9.180 f0.031 0.3 9.299 f0.030 0.3 12.588 & 0.0006 0.047 12.577 f 0.0014 0.11
Width (sec) height 9.9
Skew Negligible
...
... 10
Negligible
...
... 10.1
... 20.36 f 0.003 0.017 21.599 f 0.027 0.13
Excess 1.1993 f 0.0001 0.0050
1.1993 fO.0001 0.005
Negligible
1.1991 f0.0001
... -0.0779
0.005
f 0.0003
0.39 -0.07820 f 0.00017 0.21
-0.537 f 0.0003 0.044
-0.5895 f 0.0009 0.15
b
F U L L GAUSSIAN: Excess = 0 H A L F GAUSSIAN: Excess = 1.377
Figure 2. Skew and excew for normal Gaussian peaks and bisected Gaussian peaks
RESULTS AND DISCUSSION Magnitude of Skew and Excess. Because skew and excess are unfamiliar quantities to many, it is useful for interpreting the following chromatographic data to “bracket” our numerical values through easily visualized cases. We have mentioned that the Gaussian, at one end of the spectrum, has zero skew and excess. If we bisect the Gaussian (Figure 2a), the highly asymmetrical half has a skew of 0.995. By comparison a right angle triangle has the value 0.566. If we divide the Gaussian into two equal areas by a “horizontal” cut (Figure 2b) the “flattened” lower half is another good reference. Its excess is 1.377. We may also compare with a square profile, which has the negative value of excess, - 1.200. Accuracy and Precision Based on Artificial Peaks. The accuracy of the digitizer system was checked by feeding square wave and cam generated peaks directly to the digitizer. Table I shows the accuracy of the system. Ball, Harris, and Habgood (13) discussed the accuracy and sources of error for manual peak-area measurements. They found that the relative error in such measurements varies from about 0.5% up.
(13) D. L. Ball, W. E. Harris, and H. W. Habgood, ANAL.CHEM., 40, 129 (1968).
Table 11. Effect of Base Line Variation for 10 Methane Peaks Data Accumulated Roughly 3a on Either Side of the Peak Casee C Case*A Case*B Skew 0.0858 0.2039 0.3378 f0.0088 f0.0488 +0.0408 Excess -0.2184 0,5576 2.0372 f0.0419 f0.0958 f0.0416 Plate ht (cm). 0.1923 0.2182 0.2987 f0.0174 f0.0083 f0.0080 Plate ht (cm)b 0.1939 0.2091 0,2261 f0.0036 fO.Oo1P f 0,0022 Plate ht (cm)C 0.2041 0.2110 0.2159 f0.0082 f0.0099 f0.0064 Plate ht (cm)d 0.2122 0.2117 0.2131 & 0.0058 f0.0045 f0.0044 Computer-second moment. b Computer-width at half-height. Computer-least-squares Gaussian. Manual-using width-at-half-heightmethod. * Cases A, B, and C are described in Figure 3.
Table I shows that the relative errors in peak areas, as determined by the digitizer, are 0.3-0.4970 in the square wave case and about 0.1% for the cam generated peaks. It should be noted that the square wave analysis is a rather challenging test for the digitizer due to the sharp rise and drop of the signal. Moreover for peaks of that shape much of the error undoubtedly comes from the lack of exact coincidence between the peaks and the digitizer intervals. It is estimated that the latter effect itself would yield about 0.25% error, leaving perhaps 0.20y0 error in the operation of the system. Effect of Digitizer Base Line Position. This and the remaining parts of the paper will deal with the properties of “real” chromatographic peaks. First we examine base line effects. The digitizer base line, with respect to the true electrometer base line, has been variously positioned as shown in Figure 3. The base line position will affect the measured values of plate height, skew, and excess as follows: VOL. 41, NO. 7, J U N E 1969
891
~~
Table 111. Comparison of Different Methods for Computing Plate Heights Retention time (min) 33.69 33.99 33.18 33 * 35 32.73 32.77
H (ern). 0,2923 0.2925 0.3293 0.3250 *O. 3702 0.3266
33.29 f0.5
0.3131 hO.019
H (cm)b * O . 2332 0.2602 0.2679 0.2691 0.2698 0.2597
H (cm). 0.2605 0.2601 0.2681 0.2677 *O. 2970 0.2663
H
H (ern).
0.2270 0.2190 0.2170 0.2252 0.2245
0.2688 0.3034 0.2793 0.2482 0.3110
0.2225 10.0043
f0.0254
Average of the six runs f standard deviation 0.2653 =k0.0050
0.2645 10.0039
0.2822
* These values were eliminated when averages and standard deviations were computed. Computer-second moment. Computer-width at half-height. c Computer-least-squares Gaussian. d Manual-using width-at-half-height method. e Manual-base line intercept method. a b
tR
Table IV. Computer Results on Plate Height, Skew, and Excess, for Successive Runs of Pentane H (cm). H H (cm). Skew Excess
(min)
48.71 47.71 48.70 48.79 48.43
0.3910 0.4218 0.3750 0.3697 0.4013
0.2535 0.2435 0.2106 0.2321
0.2647 0.2782 0.2456 0.2566 0.2725
1.3350 1.4649 1.3341 1.4786 1.5356
4.2771 4.4867 4.2068 4.9769 4.8853
Computer-second moment. at half-height. c Computer-least-squares Gaussian. a
* Computer-width
CASE A
I
CASE B
--.----DIGITIZER BASELINE AND
I
CASE C
LINE
ACTUAL BASE
PEAK
Figure 3. Illustration of base line shift
Plate Height: Case A
< Case B < Case C
Skew: No general relationship Excess: Case A
< Case B < Case C
Table I1 shows experimental results for these cases obtained from running ten methane peaks through the open column. The relative height change was estimated to be 1.5%. In Case C the detector base line, as seen by the digitizer, is a constant background signal which manifests itself as a large value of the excess. In Case A , on the other hand, the digitizer sees only parts of the peak and thus the excess is less than its true value. The relative magnitude of the error caused by these base line shifts is larger in Case C than in Case A because in the former the background signal contributes indefinitely to the error. Thus the base line output of the digitizer must be calibrated to coincide with, or be “placed” very slightly above the true detector base line. This is of major importance since any base line fluctuation or “noises” 892
ANALYTICAL CHEMISTRY
will contribute greatly to the moments as well as to the skew and excess. Normal Chromatographic Peaks. Table 111 shows additional plate height values from runs of methane on the 2-m packed column without the base line shift. In Tables I1 and I11 it is seen that the plate height values obtained from the second moment (Method a) are larger than all other values. The reason probably relates to the fact that this method receives a more substantial contribution from peak tailing than the other methods. Table I11 also shows that the plate height values measured manually from the base line intercept method have the least precision, but that generally the manual methods yield a comparable precision. This conclusion probably reflects variations in the peaks themselves. The better precision of the earlier “artificial” peaks confirms this. The lack of peak reproducibility may stem from nonreproducible injection and similar factors which could be improved in a high precision system. Thus we believe that, ultimately, machine methods will yield much better precision and accuracy. Table IV illustrates the calculation of skew, excess, and plate height values for successive pentane peaks. In this particular experiment the perforator mispunched some of the points in several peaks, thus simulating noise. Nonetheless, all the values, except those for the plate height via the moments, are consistent in their agreement, and illustrate that meaningful values can be achieved by this method. RECEIVED for review January 31, 1969. Accepted March 17, 1969. This investigation was supported by Public Health Service Research Grant G M 10851-12 from the National Institutes of Health.
