(6) CF 3
(a)
/ CF3-CH2-0-C
I\
CF3
O H CF3 A
CF3 H
\I
/
/
c-o-h
CFs
OH CF3
B
In A , the CF3triplet (in dilute acetone solution) is found at -2.44 ppm from TFA(ext), whereas the corresponding triplet of unreacted CF3CH20Hoccurs at -0.12 ppm. In B, C F s groups (a) give a band at -4.03 ppm compared with a doublet (J = 6.6 Hz) at - 1.42 ppm for CF3 groups in (CF3)2CFOH. The band observed for (a) in the adduct (9 lines) results from coupling of (a) with the methine proton( J = -6 Hz) as well as coupling between fluorines (a) and (6) with J = 2.5 Hz. The latter coupling is also apparent in the septet due to fluorines (b)at $3.31 ppm (the usual HFA adduct band for this alcohol). In dilute solutions containing CF3CHz0H and HFA, the
triplet due to CF3 in the former appears only when its concentration exceeds that of HFA indicating that the reaction to form A goes nearly to completion. In contrast, solutions containing (CFJ2CHOH and HFA, with either component in excess, show bands due to both reactants and to the adduct. Bands which measure the concentration of each component are in this case conveniently spaced and separated in the spectrum, so that the relative amounts of each can be determined by integration. By using known initial concentrations of (CF,),CHOH and HFA in acetone and by measuring the concentration of each component at equilibrium in this way, the equilibrium constant for the reaction (CF3)zCHOH f (CF3)s C=O
i?
(CF,)?CHOC(OH)(CF,)L
was found to be 6.2 liters per mole at 32OC. A mixture containing reactants at initial 1M concentrations is thus approximately two thirds reacted. RECEIVED for review September 9, 1969. Accepted October 24,1969.
Moments Analysis for the Discernment of Overlapping Chromatographic Peaks Eli Grushka,’ Marcus N. Myers, and J. Calvin Giddings Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 The higher central moments in the form of the derived quantities, skew and excess, were examined for the information they contain on peak contamination and peak overlap. In particular, we have investigated the possibility of utilizing skew and excess in the discernment of low resolution peaks. Single and double peaks of various forms, especially bi-Gaussian and Poisson distribution, were generated mathematically. The skew and excess of double as compared with sin le peaks of the same asymmetry type confirmed the esired discrimination. Experimental results, on the other hand, showed that while the skew and the excess of double peaks are consistently different from those of single peaks, the actual recognition of a double peak i s not as clear as in the theoretical cases and may require an internal standard. The results also point out the qualitative information that may be obtained from skew and excess analysis.
1
IN A PRECEDING PAPER ( l ) , it was suggested that peak-shape factors may contain information about peak overlap and contamination, about column conditions, and about peak identity. Here we examine the first of these areas: the possible use of computer-analyzed peak-shape data to distinguish single peaks from composite peaks made up of strongly overlapping single peaks. The study is equally applicable to spectroscopy and to such methods as sedimentation and electrophoretic analysis, both the latter used to establish criteria of purity in biochemical systems. 1 Present address, Department of Chemistry, State University of New York at Buffalo, Buffalo, N. Y. 14214.
(1) E. Grushka, M. N. Myers, P. D. Schettler, and J. C. Giddings, ANAL.CHEM., 41, 889 (1969).
As in the previous work, we take the highcr central moments, and especially the skew and excess which derive from these moments, as the most fundamental expression of peak shape. I n particular we propose to investigate the use of skew and excess as parameters bearing information on peak overlap. These parameters are a) dimensionless, and b) zero for Gaussian peaks. Because of characteristic a) they are insensitive to any changes in scale and therefore reflect only intrinsic peak characteristics. Characteristic b) means that these parameters are a direct measure of non-Gaussian elements of a peak, elements that certainly increase in importance with the disengagement of single peaks under a composite envelope. At first it was believed that excess alone would best indicate composite peaks because the disengagement of equal Gaussians generates excess but not skew. This is a result of the flattening of the composite peak (measured by excess) but the failure to introduce any asymmetry (measured by skew). However, for real peaks it may be advantageous to use both parameters. Recent work has appeared on the problem of unresolved peaks (2-6). The approaches and goals, while different from our own, are of considerable importance. The problem of distinguishing between single peaks and composite peaks has many ramifications. If a reference peak ( 2 ) P. D. Klein, Scpar. Sei., 1,511 (1966). (3) R. 0. Butterfield, E. B. Lancaster, and H. J. Dutton, ibid., 1, 329 (1966). (4) H. J. Jones in “The Practice of Gas Chromatography,” L. S. Ettre and A. Zlatkis, Eds., Interscience Publishers, New York, N. Y., 1967. ( 5 ) L. Enchenyi, Abstracts 155th American Chemical Society National Meeting, San Francisco, Calif., April 1968, No, B-48. (6) V. Cejka, M. H. Dipert, S. A. Tyler, and P. D. Klein, ANAL. CHEM., 40, 1614 (1968).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 1, JANUARY 1970
21
”
Rs =O
7
7
-
Excess = 0
A
Rs =0.250 Excess =-0.080
SINGLE
0.2 I
0.I 0
Figure 1. Excess of two equal and overlapping Gaussian peaks
is available and can be identified with one of the constituents, then a difference in width between the reference and composite peaks signals the contamination. This approach has been developed by Klein (2). If the constituents of the composite peak are unknown, peak width may still provide a n indication of overlap (by assuming a constant plate number obtained from known single peaks, etc.). However peak width or plate number, like most other parameters, is quite variable from one component to another. Another approach, which we investigate here, seeks to determine if the composite peak has a shape discernibly different from that of its constituent single peaks. The discernibility of shape differences is itself a complex subject. If all constituent peaks were Gaussian, contamination would be indicated by finite non-Gaussian shape factors such as skew and excess. Discernibility would be exceedingly high, limited only by the statistical noise and accuracy of the data collection system. The actual case is much more complex for four reasons: 1) the constituent peaks are not Gaussians, 2) the (non-Gaussian) peaks cannot yet be obtained by theory in exact mathematical terms, 3) peak shape depends on column parameters and operating conditions, and most importantly, 4) the peak shapes are a function of the chemical identity of the constituents. Category 4) introduces a n effective noise or uncertainty into any procedure that may be devised for the discernment of composite peaks of uncertain constituency. This is a particularly crucial matter because composite peaks in general d o not differ much in shape from their constituent peaks when resolution, Rs,is low (j 1,j>i
(7) J. C. Giddings, “Dynamics of Chromatography, Part I: Principles and Theory,” Marcel Dekker, Inc., New York, N. Y . , 1965. (8) H. Cramer, “Mathematical Methods of Statistics,” Princeton University Press, Princeton, N. J., 1946, p 212.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 1, JANUARY 1970
I2m
Time, tm, is the coordinate of the peak’s maximum and (TO and a1 are related to the width of the front and back of the peak, respectively. The first moment of a bi-Guassian peak can be determined as ml
=
t,
‘.O,
*
3
4 I
+ 2(Ul --
uo)
Consequently the retention time, being the first moment of the chromatogram, should not be taken from the peak’s maximum. The second, third, and fourth central moments are m2 = -
2(u1 -
~
+ +---u1 + f f o
0
)
n
uI3 ~
uO3
(3a)
0
0.2
0.4
0.6 Skew
0.8
1.0
1.2
Figure 4. Excess us. skew for composite bi-Gaussian peaks with BZ = BI 1 at various resolution levels
+
27dUl
+ ad
(34
When u1 = uo the peak is Gaussian and Equations 3a-3c reduce to m2 = uZ,m3 = 0, and m4 = 3u4. Also for the bi-Gaussian peak the skew and the excess are Skew
+
(0.218)B3 (0.144)BZ - (0.144)B - 0.218 m3 __ = [(0.363)B2 (0.273)B 0.363]3’2 m23’2 (4)
=
Excess =
m4
m2
(0.511)B4
+
+
-3=
+ (0.590)B3 + (0.798)B2 -I- (0.590)B 4-0.511 [(0.363)BZ+ (0.273)B + 0.36312 (5)
Where B is al/uo. Equation 4 and 5 show the following limits lim skew B+l
-
0
lim excess + 0 B-.1
lim skew + B-
-
d2(4
-
(a - 213’2
=
0.995 lirn excessB- m
3n2
- 4.n - 12
( n - 2)Z
=
0.869
(6) Skew and excess measures can also be employed for overlapping peaks. In many instances two such peaks appear as a single one (see, for example, Figure 16) and the skew and excess characteristics of that composite peak are of interest for our objectives. Overlapping pairs of bi-Guassian peaks were simulated and analyzed, with the help of a Univac 1108 computer. Because the expression for the moments as well as for the skew and the excess for such double peaks are complicated, they are not given here. Each of the two peaks in the double peak model
The resolutions of each set of points are: Rs(B1 = 1, Bz = 2): 0.000,0.125, 0.250,0.375; Rs(Bi 2,Bz = 3): 0.000, 0.083,0.167,0.250,0.333,0.417; Rs(B1 3, Bz = 4): 0.000, 0.050, 0.100,0.150, 0~200,0~250,0.300, 0.350; Rs(B1 = 4, Bz = 5): 0.000,0.042, 0.083, 0.125, 0.167, 0.208, 0.250, 0.292, 0.333. In the last two composite peaks the excess first in. creases and then, with further increase in the resolution, begins to decrease
had its own B value which varied from 1 to 5 in increments of unity. The heights of both peaks were the same. Starting with complete overlap-Le., zero resolution-the peaks were separated by certain increments and each of the resulting composite peaks was analyzed. Figure 3 shows a plot of excess us. skew for composite peaks for which B1 = B2. This case is likely to be approached frequently in chromatography where two closely eluted compounds have similar profiles. The solid line in Figure 3 is the theoretical line on which all single bi-Gaussian peaks fall. Of course all the B1 = Bz peaks with 0 resolution lie on that line. The rest of the points are for resolution >O. (The points for the case B1 = B2 = 1 are not shown; their skew is always 0 and the excess O.) Except for a distinct crossing region whose dimensions are roughly 0.5 to 0.6 in the skew axis and 0.2 to 0.275 in the excess axis, the solid line is uniquely defined as the domain of single peaks. All double peaks with B1 = Bz 5 5 with resolution greater than zero will lie on either side of the line and will intersect it only in the crossing region. Because the theoretical single-peak line is noiseless, double peaks with resolution greater than zero but less than 0.5 can be recognized, except in the small region of ambiguity mentioned above, Real systems, of course, have inherent noises in them which make the single-peak line much less defined. We shall elaborate this point later on. Figure 4 is a similar plot with B1 # Bz. We took the case where B2 = B1 1 so that the second peak is not wildly different from the first one. The most striking observation about Figure 4 is that double bi-Gaussian peaks with B1 # BZ and zero resolution can be easily differentiated from a true single bi-Gaussian. Another point of interest is that the same crossing region on the single-peak line appears in this plot. It should be mentioned here that for the case B1 = 3, B2 = 4 and B1 = 4, Bz = 5, the skew and the excess of the double bi-
+
ANALYTICAL CHEMISTRY, VOL. 42, NO. 1, JANUARY 1970
23
4.u
12 i 0 R=P,= 2 0 0 0 p = PE = 3.00 A F=P,= 400
10 -
3.0-
8 -
*
a , = 4.0 a, = I O 0
u)
b
VI
2
2.0
6 -
/
w"
-
/
//
4 -
W
f
t4
0 0
SINGLE "KINETIC TAILIN( PEAKS
d
2 1.0
/ /
/ 01
7
I
I
I
1
0.5
1.0
1.5
2.0
I
2.5
3
Figure 6. Excess us. skew for composite kinetic tailing peaks at various levels of resolution The resolutions of each set of points in order of decreasing skew and excess are: Rdal = 1/2): 0.000, 0.223, 0.354; Rs (ai = 1): 0.000, 0.167, 0.289; Rs(ai = 1.5): 0.000, 0.139 0.250; RS(UI= 2): O.OO0, 0.121, 0.224; RS(UI= 3): 0.000, 0.100, 0.189; = 4): 0.000,0.087,0.167; R.S(UI= 10): 0.000
Skew
Figure 5. Excess us. skew for composite Poisson Distribution peaks with PI = Pz at various resolution levels The resolution of each set of points in order of decreasing ex. cess and skew are: Rs(Pl = Pz = 1.5): 0,000, 0.320, 0.640; Rs(P1 = P2 = 2): 0.000, 0.267, 0.400, 0.533; Rs(P1 = Pz = 3): 0.000, 0.100, 0.200, 0.300, 0.400, 0.500; R s ( A = Pz = 4): 0.000, 0.080, 0.160, 0.240, 0.320, 0.480; Rs(Pi = P2 = 5): 0.000, 0.067, 0.137, 0.200, 0.267,0.333,0.400
and the excess is
Gaussian peaks increase until the resolution is about 0.1 and then begin to decrease as the resolution continues to increase. Poisson Distribution Peaks. These peaks have a shape more closely resembling that of a chromatographic peak. The normalized function has the form 1
n>O
(7)
For a particular n value, the parameter b determines the width of the peak. The moments of such a peak are easily found. The first four moments, with all moments above the first taken around the center of gravity of the peak, are
+ 1) + xo = xm + b m2 = b2(n + 1) m3 = 2b3(n + 1) m4 = 3b4(n + l)(n + 3)
m l = b(n
@a> (8b) (8c) (84
The first moment m l is taken around an arbitrary origin. The distance from this origin to the point where the peak begins to rise is defined as X O . In chromatography xo is the distance (or equivalently the time) from the injection to the beginning of the peak emergence. xm is the coordinate of the peak's maximum. The skew is
(9) 24
It should be noted that both the skew and excess are independent of b. As n increases then lim Skew = lim Excess + 0 n + 03 n+ a
(11)
In the other extreme, as n approaches zero, the skew goes to the value of 2 while the excess approaches the value of 6. As with the bi-Gaussian peaks, composite (double ) Poisson distribution peaks were simulated. Again the heights of the two peaks were equal as were their front widths. The ratio P of the back of the peak to its front (at an arbitrary height 0.05 relative to maximum peak height) was varied to achieve various degrees of tailing. Quantity P is similar to B for the bi-Gaussian case. In Figure 5 each of the two composite peaks in a composite possess equal P values. The resolution of the peaks in each set varies from zero to about 0.5. The zero resolution point is again equivalent to a single peak and consequently falls on the solid line, the theoretical line for all single Poisson distribution peaks. Figure 5 shows that double peaks with resolution > 0 lie above the solid line. In this case no region of ambiguity exists, at least for the P and Rs values considered by us, and double peaks with resolution less than 0.5 can be differentiated from single peaks. This differentiation is less marked for low P values than for high, however. In the case where the P values of the two peaks are unequal, P2 = PI 1, in the resolution range 0-0.5, it can be demonstrated again that none of the points (including the zero resolution point) fall on or cross the single line. Thus a double peak can be immediately recognized as such. Peaks Generated by a Kinetic Mechanism. Peak asymmetry, particularly tailing, may have its origin in the underlying kinetics of sorption and desorption. Giddings et al.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 1, JANUARY 1970
+
II I
9
"1 I I
h HEPTANE 85OC,Ret. time 105.I7 sec. A
8
PENTANE 8 5 T ,Ret time 48.47sec.
10
ACETONE, 85OC.
0 BUTANONE,
85°C.
A PENTANONE, 85OC.
I
1
Rs=O
9-
D
1
0.187
7 0.281
0
8-
0.375
6 7-
5 Iv) n
v, v,
a,
0 240 h
$ 4
u X
W
0180 b
W
0137
3
6l
i
6-
%=O
A
POISSON
5 5-
4 4 -t
/
0.129 0 0.065 0
2
0.206
0
O
A
o 0.442
I 0
0
0/0.2?2
I
0.5
1.0
0.147
0.295
I
I
1.5
2.0
2
Skew
0
1.0
1.5
2.0
2.5
3.1
Skew
Figure 7. Excess us. skew for pentane and heptane. The resolution is shown for each point
Figure 8. Excess us. skew for several ketones. The resolution is shown for each point
(9, 10) have dealt theoretically with the profiles produced by kinetic tailing. It is found that peak shape can be described in terms of Bessel functions and exponentials. Although complicated, and thus not reproduced here, the skew and the excess for that peak were computed and plotted in Figure 6. Each set of points correspond to a particular value of al, the average number of sorptions in a run. The two peaks in each composite peak were again identical. The resolution, indicated in Figure 6, increases with decreasing skew and excess. Only the set of points with al = '/z crosses the solid line in this case. With this exception, the distinction between double and single peaks can again be made by skew and excess analysis. EXPERIMENTAL
The instrumentation and computer program for analyzing chromatographic moments were described previously (1). No attempt was made to optimize the chromatographic system. Double Peak Injection, The Carle gas valve described before ( I ) was used for double injection as follows. An initial injection was made by switching loops in the valve. After a predetermined time interval, the loops were exchanged again for the second injection. Resolution was computed in terms of the time interval and the observed peak width. For simplicity the two peaks injected were of the same compound and of equal amounts. Materials. The chromatographic runs were made with pentane, heptane, pentene, hexene, heptene, acetone, 2butanone, and 2-pentanone. The column was 2 meters in length, as described before ( I ) . On the average, six runs were made for each data point. (9) J. C. Giddings and H. Eyring, J. Phys. Chem., 59,418 (1955). (10) J. C. Giddings, ANAL.CHEM., 35,1999 (1963).
0.5
RESULTS AND DISCUSSION
Our results confirm that the skew and excess of double peaks are consistently different from those of a single peak. However there is a good deal of variation in these values for single peaks of different compounds. This suggests either the use of a reference peak or that skew-excess or similar relationships, rather than either quantity singly, must be used for overlap identification. An estimation of the general level of precision was gained .from single peaks. For our group of 11 different compounds, the standard deviation was typically about 5 % of the mean with a range from 2-10 %. The precision in the measurement of excess for the same single peaks was less satisfactory, usually about 10 % but varying from 6-30 %. Figure 7 shows an excess US. skew plot for heptane and pentane at 85 "C. The flow velocity was varied along the heptane runs. The resolution corresponding to each point is indicated on the graph. For single peaks (shown in boldface) Rs = 0. The results shown in Figure 7 suggest that experimental double peaks with low resolution (in the range 0.08-0.41) may be shifted left of the line (the solid line in the figure) joining single peaks, as suggested by theory. When temperature is changed, single-peak points are obtained which no longer lie on the line. Further investigation is needed to ascertain the effect of other variables. Figure 8 shows a similar plot for some ketones. The zero resolution point of acetone, for reasons which are not clear, is beneath the point for which resolution > 0. Here again the single peaks (Rs = 0) have different skew and excess values than those of the double peaks. The same observation can be
ANALYTICAL CHEMISTRY, VOL. 42, NO. 1, JANUARY 1970
0
25
12
1.25
. 0
ACETONE
Rs = O
1.00
0
1 V
0.75
0.197
0
-1
0187
0281
0 375
\
'I
0.50
2
2
0.25
0
2
W
200
400
0462
600
1
I
I
800
1,000
1,200
1,400
1,6C
Number of Theoretical Plates
0
- 0.25 .
Figure 10. Excess us. plate number for several ketones. The resolution is shown for each point
0.290
0.11700 0
0.234
0.351
'Rs=O 0.436
-0.50 .
- 0.75
0.468
I
0.25
I
0.50
I
0.75
1-01
Skew Figure 9. Excess us. skew for several alkenes. The resolution is shown for each point made from Figure 9 which shows the skew and the excess values for a few alkenes. In this case the skew and excess of hexene and heptene first increase and then decrease with increased resolution. Neither Figure 8 nor 9 shows the clear distinction between double and single peaks evident in Figure 7. Either the single peak line (or band) is too broad as in Figure 9, or the double peak points do not adequately shift out of that band as in Figure 8. In Figures 8 and 9 the bi-Gaussian and Poisson distribution lines are shown for comparison. The two theoretical curves are seen to be in qualitative accord with the trend in the data. A visual inspection of Figures 7-9 shows that the single peak
26
A
I
0
points for each homologous series fall into different regions of the coordination system. This phenomenon shows that qualitative analysis information may be forthcoming from skew and excess parameters, as suggested in the introduction. We have examined here mainly the excess us. skew characteristics of unresolved peaks. Other possibilities may also hold promise. For instance, an excess us. plate number plot should have some differentiating characteristics. Plate height will decrease both with double peak resolution and with column nonidealities if the peak is single. However, excess will respond differently to these two factors; it is expected to decrease with resolution and increase with nonidealities. Figure 10 verifies this expectation for several ketones. The double peaks tend to lie below and to the left of the single peaks. While the above results show that the higher statistical moments reflect the emergence of resolution, the patterns are not always entirely clear, undoubtedly because of detailed variations in peak distribution. Considerably more experimental work is needed to delineate the proper conditions for applying this approach to diverse analytical systems where adequate resolution is not readily obtained. RECEIVED for review June 25, 1996. Accepted October 27, 1969. This investigation was supported by Public Health Service Research Grant GM 10851-12 from the National Institutes of Health.
ANALYTICAL CHEMISTRY, VOLi 42, NOi 1, JANUARY 1970
