19139
Anal. Chhem. 1981, 53, 1939-1942
934 other spectra of the same compounds.
RESULTS AND DISCUSSION Matching the 431 unknown spectra of pure compounds against only the reference spectra selected as described (“test a”) retrieved all the 934 other spectra of t,he same compounds that were found searching the entire data base. The number of reference spectra searched was reduced from 41 429 to an average of 2925 (7.5%),representing spectra filed under an average of 20 MSP mass indices. The total computer time required was reduced by a comparable amount (from 2 to 0.2 9).
In additional tests (b and c), the reference spectra searched were limited to those filed under MSP mass indices corresponding to peaks in the unknown spectrum whose U + A values differ from the highest value by (b) no more than two units and (c) no more than one unit, with no requirennent for the minimum number of peaks. These tests retrieved (b) 99.2% and (c) 97.5% of the reference spectra by searching only (b) 3.8% and (c) 1.6% of the reference file. As a “reverse search” system, PBM has improved capabilities for idenlification of minor components in mixtures. The loss of performance for such components under test a conditions can be estimated from the test b and c results; the latter should represent components present in -50% and -25% concentration, respectively, of those tested under a conditions, based on the log base 2 scab of “A value” abundances. Thus a loss of only a few percent in reliability is expected for identification of a 25% component. The pioneering proposal of Dromey (6) for searching an ordered mass spectral file showed impressive results; 68% of spectra were retrieved in searching 1%of the file, 82% in 2%, 92% in 3%, and 100% over 3%. However, these represented the system’s ability to retrieve the identical spectrum from the file; in contrast, our tests utilized other spectra of the unknown compound, as would be the case for a real unknown. Further, it was reported that the presence of an impurity peak of m / z 40-99 could produce a false file-order value, severely limiting the system’s applicability to mixture spectra (6). For our system the classification of each reference spectrum according to the mass of its most “important” peak (as determined by U + A) divides the data base into 732 categories. The majority of these contain approximatelythe same number of entries, as the U + A values used are based on the statistical occurrence of peaks in the reference fie. (Classifying reference
spectra by the mass of their most abundant peak without mass-uniqueness weighting gave 608 categories with a much wider range of entries, requiring 80% more spectra to be searched.) Thus the search time for spectra filed under 20 of these categories will be small compared to that for the whlole file, while it is not surprising that the entries corresponding to the 20 (on average) most important peaks of an unknown spectrum should contain the correct reference spectra, if present. The real-time performance possible using PBM in tlhis manner on computers of GC/MS/COM systems depends on several obvious factlors, such as the capability for PBM processing in competition with higher priority tasks such as data collection. Implementation on a GC/MS/COM system is in progress; at present, search results can be obtained from up to six spectra per iminute during the GC/MS run, and a substantial increase in this performance appears feasible.
ACKNOWLEDGMENT The authors are indebted to R. G. Dromey and D. W. Peterson for helpful discussions. LITERATURE CITED (1) Burllngame, A. L.; Baillie, T. A.; Derrick, P. J.; Chizhov, 0. S. Atrial. Chem. 1980, 52, 214R. (2) Hertz, H. S.; Hites, R. A.; Biemann, K. Anal. Chem. 1971, 43, Ei81. (3) Mclafferty, F. W.; Hertel, R. H.; Villwock, R. D. Org. Mass Spectnm. 1974, 9, 690. (4) Pesyna, G. M.; Venkataraghavan, R.; Dayringer, H. E.; McLafferty, F. W. Anal. Chem. 11976, 48, 1362-1368. (5) Pesyna, 0.M.; Mclafferty, F. W. “Determination of Organic Structures by Physical Methods”; Nachod, Zuckerman, Randall, Eds.; Academic Press: New York, 1976; pp 91-155. (6) Dromey, R. G. Anal. Chem. 1979, 51, 229. (7) Atwater, B. L. Ph.1). Thesis, Corneil University, 1980. (8) Stenhagen, E.; Abrahamsson, S.; Mclafferty, F. W. “Registry of Mlass Spectral Data”, extended version of magnetic tape; Wiley: New Yiwk, 1978.
In Ki Mun Daniel R. Bartholomiew Douglas €3. Stauffer Fred W. McLafferliy* Department of Chemistry Cornel1 University Ithaca, New York 14853 RECEIVED for review December 22, 1980. Resubmitted April 20,1981. Accepted ,July 17,1981. Research sponsored by .the National Science Foundation, Grant No. CHE-7910400.
AIDS FOR ANALYTICAL CHEMISTS Graphical Method for Obtaining Retention Time and Number of Theoretical Plates from Tailed Chromatographic Peaks Wllliam E. Barber and Peter W. Carr” Deparfment of Chemistry, University of Minnesota, Minneapolis, Mlnnesota 55455
Recently there has been a great deal of interest in methods of characterizing chromatographic peaks. Ideal Gaussian profiles (1)are rarely if ever observed. Any number of internal and extracolumn processes (2-8) may lead to peak asymmetry; thus in practice peaks are usually tailed. Such asymmetry can greatly complicate the measurement of the number of theoretical plates (N) on a column. Conventionally N is defined as
N = tP2/mz
(1)
where t, and rilz are the retention time (Le,,time of appearance of the peak maximum) and the variance (i.e., second cent.ral moment) of the peak, respectively (2). The equivalent expression for a pure Gaussian is
N =
0003-2700/81/0353-1939$01.25/00 1981 American Chemical Society
tR2/O2
(2)
1940
ANALYTICAL CHEMISTRY, VOL. 53, NO. 12, OCTOBER 1981
where tR not only is the time at the peak apex but also is the center of gravity (first statistical moment, ml)and u is the standard deviation of the Gaussian profile. The most common method for estimating plate counts (N) is given by the following equation: 2
N = 5.54(
i)
(3)
WI 12
where W l p is the peak width measured at half-height. The accuracy of the above method is contingent upon how closely the actual peak conforms to the equation of a pure Gaussian profile. Since pure Gaussian profiles are seldom observed experimentally, the above technique can lead to poor estimates of N (9). By far the most commonly used model of a tailed peak is the so-called exponentially modified Gaussian equation. Kissinger (10) most recently wrote a convenient expression for this model. His equation was recast in a more useful form and is
-+) h(t) hm,
exp(
-x
5.0-
-
112 g
:[?
TIME (secs)
Figure 1. Example of the measurements required for the graphlcal method. The value of CY is 0.1 and r / u is equal to 1.6.
4.0
7
-
f
1)
erf
($[?
-
:I)
(4)
where h ( t ) is the height of the chromatographic peak as a function of time in arbitrary units, h , is the maximum height that the peak would have if it were a perfect Gaussian, and 7 is the time constant of the exponential peak modifier. This equation clearly indicates that all relative ( t o tR) temporal properties of a peak expressed in units of u will be a function only of the ratio u/r. This dependence on only the ratio of u / 7 is the basis of the graphical method, described here, for obtaining measures of N , u, 7,tR, ml,and m p The literature pertaining to the general properties of the exponentiallymodified Gaussian profiie and to the theoretical and experimental work that has been done to justify this model has been concisely reviewed by Pauls and Rogers (11, 12), Grushka (13), and Maynard et al. (14). By far the simplest method of quantifying the asymmetry of a peak is the graphical determination of the A / B ratio (2, 9) where A and B are measured as shown in Figure 1. Clearly the A / B ratio is merely an empirical figure of merit. The chief virtue of the A / B ratio is that it is easy to measure and can be used (9) to determine the 7 / u ratio. The r/u ratio, if the peak is well fit by an exponentially modified Gaussian model, is characteristic of the peak and is related to fundamental properties of the peak whereas A / B is not. 7 / u is therefore more useful than A/B for describing the relative peak asymmetry. The purpose of the present work was to develop and assess a simple, graphical method of calculating u, 7,and subsequently N , tR,ml, and riz2 which would circumvent the need for computerized data aquisition which is intrinsic to the measurement of ?/a from statistical moment analysis (15)or via Yau’s algorithm (16). The primary advantage of the present method is that it can graphically deconvolve u and 7 via measurement of the experimental width and the A / B ratio. It should be noted that the graphical method described here is not necessarily more accurate or precise than the use of Yau’s algorithm. It merely circumvents the need for a computer.
DEVELOPMENT OF THE METHOD A Data General Nova 2 computer programmed in Fortran IV was used to generate exponentially modified peak profiles
3.0
-
2.0
-
1.0
-
I
1.o
I
I
2.0
I
I
I
3.0
(A/B)-l
Figure 2. Universal calibration curves relating d u to ( A I B ) - 1. Each curve corresponds to the indicated fractional height CY. of exactly known values of 6,7 , and t R . In order to calculate t, and h,, the time and height of the peak maximum respectively, a second-order Gram polynomial (17) was fitted to the points near the apex of the profile, in the same manner described by Pauls and Rogers (12). In order to locate the exact times (tl and t2,see Figure 1) corresponding to a given fraction (e.g., CY = 0.10) of the peak height a linear interpolation was performed between the two points which bracketed the desired fractional height. The peak width is computed as tz - t l , A and B are computed (see Figure 1) as tz - t , and t , - t l , respectively. All relative temporal characteristics of the peak were then calculated at each u and 7 and plotted vs. r / u . These plots are given in Figures 2-4. Care was taken to ensure that a constant data density of 50 points/u was maintained when calculating the profiles.
DESCRIPTION OF THE METHOD A typical exponentially modified Gaussian peak profile is shown in Figure 1. The time scale in this figure has been expanded for better measurement precision. Implementation of the method is very simple. Four measurements A , B , t,, and W , are required (see Figure 1). Given A and B, 718 can be determined from the plots of r / u vs. ( A / B )- 1 (Figure 2). Now that ?/a is known, the ratio W,lu can be determined
ANALYTICAL CHEMISTRY, VOL. 53, NO. 12, OCTOBER 1981
00 00
I -
I
10
20
40
3.0
5.0
TAUlSlGMA
TAUlSlGMA
Flgure 3. Universal calibration curves relatlng the peak width ( W,) to ~ / u .Each curve corresponds to the indicated fractional height a.
I
00
10
20
I
I
30
,
I
40
-1
50
TAUlSlGMA
Flgure 4. Univer.sai calibration curve relating the relative displacement of the peak maximum to rlu. The relative displacement is given as (fP
1941
- fR)/Q.
from the plot of W,/a vs. T / U (Figure 3). The parameter u is easily calculated by dividing W , by WJu. r is delmmined by multiplying u by r / u . Given t,, t R can be determined from the plot of (t, - tR)/u vs,. r / u (Figure 4) once r / u is determined as described above, A reliable value for N is then calculated from eq 1 or eq 2, as discussed below. The validity of the exponentially modified Gaussian model can be checked by determining values for u, T , and t R ut several values of a! and comparing the results. Good agreement between these values would indicate that the experimental peak is well fit by Ibe exponentially modified Gaussian model.
RESULTS AND DISCUSSION Data on the overall accuracy and precision for the measured and derived parameters are given in Table I. These results were obtained experimientally from the measurement of a set of 14 synthetic exponentially modified Gaussian peaks by six colleagues who had read the preceding description of the method. These synthetic chromatograms were plotted on a scale of 1 cm per u, and a ruler of f0.2 mm resolution was employed for their evaluation. A detailed statistical evaluation of all results was carried out to assess the limitations of the method. The brevity of this paper does not permit a detailed discussion but the interested reader may write for details. In general we may say that, in decreasing order of accuracy and precision, it is easiest to measure t , and ml, u and WllZ,and then AIB and r / u . Specifically we do not advise the use of this graphical method unless AIB, measured at either a equal to 0.5 or 0.1, exceeds l.06.
NG,NK,and and the theoretical values for each method as a functlon of TIU. The theoretlcai curves (solid curves) for each method are piottsd as the percent deviation from the value each method would produce had T been equal to zero. Figure 5. Experimental values of plate numbers (Le.,
N,,d
One of the major reasons for assessing column asymmetry is to arrive a t a means of accurately estimating the numlber of theoretical plate13on a column. Thus we will present some of the details concerning the accuracy of estimating plate counts via our method. We have calculated the number of theoretical plates (N) by three different methods. The results for all three methods are given in Figure 5 and are presented as percent deviation from the value which would have been obtained had T been equal t o zero. The solid curves are the theoretical values for the three different methods and the individual points are the experimentally determined vallues plotted as percent deviation from their own theoretical values. NK designates the results obtained when N is calculated from the time of the peak apex (tp)and the second central moment (mZ)as advocated by Kirkland et al. (9) (see eq 1). Nw1, designatesthe value of N obtained via the half-height m e t h d (see eq 3). Finally, No designates that value of N obtained from estimation of the Gaussian components of the peak profile, i.e., from t R and c (see eq 2). This yields the value of N that would have been obtained had r been equal to zero, i.e., the value of N for the parent Gaussian. NG will be a valid characteristic of the column provided that all peak asymmletry is extracolumn in origin. In contrast, N K is fundamentally a system parameter since it does not discriminate against the contribution of asymmetry to estimators of the efficiency. No should not be used to compute resolution. The results given in Figure 5 and Table I indicate that NwlD can be measured with better accuracy and precision than either NG or N K at either a equal to 0.5 or 0.1. This does not mean we advocate the use of N,,, to evaluate asymmetric peaks. Indeed Kirkland et al. (9) have shown that N , , , is not an effective measure of column performance when the peaks are asymmetric. NK was calculated from eq 1,but the data needed were obtained via the graphicalmethod described above. The data indicate that NK is not significantly more precise than No. However, there appears to be small negative bias in the method proposed here for measurement of NG due to a positive bias in u. It should be noted that although the measurement of A/B, at T / u < 0.2, is prone to measurement difficulties, the derived values of NG and NK can stilll be measured with quite respectable accuracy and precision. The data obtained at a = 0.1 clearly show an advantage in terms of both accuracy and precision over those data obtained at a = 0.5 for both NK and NG. The precision for NK and NG tends to improve ass 7 / u increases although the accuracy does not. Table I also presents our results for the estimation of the
1942
Anal. Chem. 1981,
as NGand NKwithout recourse to automated data acquisition systems or on-line computers. We found that when the A / B ratio is 11.06, one can obtain u, 7, N , ml, and mipwith reasonable accuracy and precision. Specifically, if the chromatogram is recorded a t a rate of a t least 1 cm/u, u can be determined within *3%, 7 within *ti%,N within *5%, ml within f0.1%, and m2within *3%. This graphical method thereby provides a simple technique for obtaining fundamental peak parameters without any computer assistance. Data sets which were used to construct large scale plots of the calibration curves are available upon request from the authors as is a detailed analysis of the data whose summary is contained in Table I.
Table I. Overall Accuracy and Precision for Measured and Derived Parameters a = 0.5
parameter
1
Nw.5 NK NG
av % error
a = 0.1
av % dev
il.8 i1.3 i8.5b f 3.4 i 2.5 i9.2 r0.83 i 6.5 i4.7 i0.17 i 4.5
f
av% error
av% dev
r4.6 t9.2c f 2.3 f1.3 2.9
i1.3 +7.Oc t4.0 f 2.3 t4.6
f 2.4
f 5.0
i2.7 t0.12 21.5
i4.5 i 0.08 i 3.1
1.6
f 2.9
i16.6 f 6.4 f 3.3 i13.6 f 1.8 f12.1 i7.8 f 0.21 f 5.9
LITERATURE CITED
a Average gercent error and precision relative to the value of U. Average calculated for peaks with 710 > 0.6. Average calculated for peaks with r / o > 0.4.
peak centroid (ml)and the second central moment (m2), For an exponentially modified Gaussian they are given as follows (13):
m l= t R + T =
u2
+
(5)
(6) Interestingly these moments can be obtained with quite good accuracy in part because most of the variables are correlated to some extent resulting in an intrinsic cancellation of errors. T h u s the graphical method proposed here does a good job of recovering the first and second central moment without having to perform a n integration. rii.2
53, 1942-1944
72
CONCLUSIONS The main point of this work is to show that it is possible to obtain fundamental chromatographic parameters such as tR, ml, fi2,u, 7, and various estimates of system efficiency such
(1) Karger, B. L.; Snyder, L. R.; Horvath, C. "An Introduction to Separation Science"; Wlley: New York, 1973. (2) Snyder, L. R.; Klrkland, J. J. Introduction to Modern Liquid Chromatography", 2nd ed.; Wlley: New York, 1979. (3) Giddlngs, J. C. Anal. Chem. 1963, 35, 1999. (4) Kucera, E. J. Chromafogr. 1965, 19, 237. (5) Karger, B. L.; LePage, J. N.; Tanaka, N. "High-Performance LiquM Chromatography, Advances and Perspectives"; Horvath, C., Ed.; Academic Press: New York, 1980; Vol. 1, p 113. (6)Johnson, H. W., Jr.; Stross, F. H. Anal. Chem. 1959, 31, 357. (7) Sternberg, J. C. Adv. Chromatogr. (New York) 1966, 2 , Chapter 8. (8) McWllllam, I. G.; Bolton, H. C. Anal. Chem. 1969, 41, 1762. (9) Kirkland, J. J.; Yau, W. W.; Stoklosa, H. J.; Dliks, C. H. J. Chromatogr. Sci. 1977, 15, 303. (10) Klssinger, P. T.; Felice, L. J.; Miner, D. J.; Preddy, C. R.; Shoup, R. E. In "Contemporary Topics in Analytical and Cllnlcal Chemistry"; Hercules, D. M. et ai. Eds.; Plenum Press: New York, 1978; Vol. 2. (11) Pauls, R. E.; Rogers, L. B. Anal. Chem. 1977, 49, 625. (12) Pauls, R. E.; Rogers, L. B. Sep. Sci. 1977, 12(4), 395. (13) Grushka, E. Anal. Chem. 1972, 44, 1733. (14) Maynard, V.; Grushka, E. Anal. Chem. 1972, 4 4 , 1427. (15) See, For example, ref 4, 7, 11-13, 16. (16) Yau, W. W. Anal. Chem. 1977, 49, 395. (17) Davis, H. T. "Tables of Mathematical Functions"; Princlpia Press: San Antonio, TX, 1963; Vol. 11, p 307.
RECEIVEDfor Review January 2, 1981. Resubmitted and accepted June 16,1981. This work was supported in part by a grant from the National Institute of Occupational Health and Safety (No. R01-OH00876001) and by summer support for W.E.B. provided by the Proctor & Gamble Co.
Ion Chromatographic Separation of Anions on Silica-Coated Polyamide Crown Resin Manabu Igawa," Kimlko Saito, Jun Tsukamoto, and Masao Tanaka Faculty of Technology, Kanaga wa University, Rokkakubashi, Kanaga wa-ku, Yokohama, 22 1 Japan
A crown ether contains a cavity in which a cation can be encapsuled selectively (1) and crown ether complexes are of great analytical interest (2). Blasius et al. (3)were the pioneers of applying crown ethers to chromatography. They chromatographed inorganic and organic ions over the complex forming exchangers with crown ethers or cryptands as anchor groups (3,4). Smulek and Lada prepared diatomaceous earth immobilized dibenzo-18-crown-6 and separated alkaline metal ions with it (5). Kimura et al. prepared poly(crown ether)modified silica and also separated alkaline metal ions (6). Igawa et al. also chromatographed cations and anions over a polyamide crown resin which contains dibenzo-18-crown-6 in a polymer chain (7))but the resin could not be used in HPLC on account of its mechanical weakness. We then prepared silica-coated polyamide crown resin and chromatographed both cations and anions. It separated anions well but did not separate cations except under a slow elution rate. In the chromatography over silica-coated polyamide crown resin, a 0003-2700/8 1/0353-1942$0 1.25/0
suppressor column, which was usually used in ion chromatography (@, was unnecessary as the eluent was water. In this paper, we chromatographed cations and anions over silicacoated polyamide crown resin or polyamide crown resin alone and measured the adsorption capacity of these packing materials by batch test in order to discuss the separation mechanism.
EXPERIMENTAL SECTION Materials. Polyamide crown resin and silica-coated polyamide crown resin were used as the packing materials. Polyamide crown resin was provided by Nippon Soda Co., Ltd., and the grain size of polyamide crown resin used in this chromatographic method was 200-400 mesh. Figure 1 shows the constitutional formula of polyamide crown resin. Silica gel, which was the base of Hitachi gel no. 3050, was spherical (diameter 10 wm) and porous. The coating procedure was as follows; polyamide crown resin (0.6 g) was dissolved in N-methyl-2-pyrrolidone (4 mL) and the silica gel (1.5 g) was added t o this solution. Evaporating the solvent 0 1981 American Chemical Society
