702
Anal. Chem. 1903, 55, 782-707
Software for Quantitative Analysis by Carbon- 13 Fourier Transform Nuclear Magnetic Resonance Spectrometry Christopher H. Sotak, Charles L. Dumoulln, and George C. Levy* N I H Resource for Muiti-Nuclei NMR and Data Processing, Department of Chemistry, Syracuse University, Syracuse, New York 13210
Computational aspects of quantltatlve analysis by NMR are examlned. A simple mlxture of hBgh dynamic range and neat octanol-2-d were employed to lnvestlgate the accuracy and preclslon of results as functlons of resldual base llne abnormalltles,slgnal to noise, and spectral data point density. Components of a high dynamlc range mixture present at mole ratlos of 88.83%, 5.59%, 1.08%, 0.27%, and 0.13% were determlned, uslng a hlgh level spectroscoplc data analysls software package, as 87.50% (0.57), 5.43% (0.17), 1.04% (0.09), 0.28% (0.12), and 0.12% (0.07), respectively. The values in parentheses denote 1 1 standard devlatlon. Accuracy and preclslon of determination for mlnor components In the mlxture were found to decrease wlth slgnal to nolse. A slmilar trend was also observed for neat octanol-2-d spectra of decreasing signal to noise. Increasing the spectral data point denslty was found to Improve analytical results for neat octanol-2-d; however, satisfactory results were obtained even at relatively low digltal resolution.
The usefulness of I3C Fourier transform (FT) NMR spectrometry as an analytical technique stems from the potential direct relationship between the area under an NMR peak and the number of the particular type of nuclei which give rise to that signal. In addition, the sensitivity of 13Cchemical shifts to small differences in molecular environment, coupled with a large chemical shift range (-220 ppm), gives a “chromatographic” separation of resonances of interest and has made 13C NMR an attractive method for analyzing complex mixtures (1-9). Unfortunately, extracting the desired quantitative information from a 13C NMR analysis is hampered by several experimental limitations. These complications have been discussed previously in some detail (7-10)and will therefore only briefly be recounted for clarity. To avoid partial saturation of 13C resonances, the relaxation period between successive observe pulses must be at least five times the longest spin-lattice time, TI,of interest in the sample. The nuclear Overhauser enhancement (NOE) of the carbon signal must also be nullified in order to obtain quantitative results. Spin-relaxation reagents (11-15) have been used to shorten and equalize 13C T(s and to suppress NOES by circumventing normal carbon-hydrogen dipolar relaxation. The use of paramagnetic sample additives, however, does not alone ensure effective NOE suppression (16)and the gated decoupling method (17‘) is generally chosen to eliminate the NOE. Software Considerations, In conjunction with experimental considerations, care must be taken to avoid subjecting the analysis to additional inaccuracies in the course of processing the data. The integrity of peak intensity measurements can suffer from: (I) inarrfficient spectral data point density; (2) poor base line characterization; and (3) improper integration limits. The spectrd data point density is dictated by the number of computer words used to digitize the free induction decay (FID) This ~kouidbe sufficient to resolve the narrowest featwe in the transformed spectrum. Additional improvements in peak definition may be obtained by zero6003-2100/89/0355-0182$01.50/0
filling. The precision and accuracy of peak integrations obtained on discrete data can be shown to increase with the number of points integrated. Analytical accuracy is also strongly dependent on the correct identification of the base line around and under the peak. Lengthy signal averaging and instrumental artifacts contribute to base line distortions which must be compensated for, through software, to make high accuracy integrations meaningful. Finally, an often overlooked aspect of integration in NMR is the choice of integration limits. If limits of integration are consistently chosen as a function of the peak line width, then the ratio of individual integrals within a spectrum will remain constant. This is an important consideration if the concentration of an unknown sample is determined from a calibration curve based on an internal standard. The above data processing considerations have been incorporated as part of a spectral data processing system developed for FT-NMR called ORACLE (Optimized Realtime Analysis in the Chemical Laboratory Environment). This software is designed to provide a high degree of flexibility and control over spectral data analysis while a t the same time minimizing the user’s ability to bias the data. The basic method used to achieve this goal is to use sophisticated algorithms which base decisions on statistical methods. A block diagram of the software modules employed in quantifying a spectrum is shown in Figure 1. The FID is zerofilled, multiplied by an appropriate exponentially decaying function, and Fourier transformed. The resulting spectrum is carefully phased, and base line distortions are removed through an iterative base line flattening algorithm. Resonances are identified by the peak analysis module and pertinent peak information, such as, position, intensity, and line width are tabulated. Peak areas are then calculated with a floating point Simpson’s rule algorithm, with integration limits chosen as a constant multiple of peak line width. An integration module allows the user to manipulate or repeat peak integrations obtained automaticallyin the peak analysis module. Partially resolved peaks can be integrated by modifying the number of peak line widths used to determine the limits of integration or may be deconvoluted by using a curve fitting algorithm. In addition, the integration module allows the data to be corrected for signal attenuation due to nonuniform excitation and filter roll-off. Two chemical systems were chosen to investigate the analytical results that can be obtained by adhering to the above data processing considerations. The first system demonstrates the accuracy of determination for a minor component in a mixture of high dynamic range. Four-component mixtures having dynamic ranges varying in mole ratio from 101to approximately 600:l were employed for this purpose. The second system, neat octanol-2-d, having eight lines of equal area, was used to evaluate the effects of signal to noise and spectral data point density on the accuracy and precision of 13C analyses. EXPERIMENTAL SECTION Materials. Components of mixtures consisted of reagent grade tetrahydrofwan (Fisher Scientific),2-propanol (Fisher Scientific), 0 1983 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983 FID FROM
783
Table I. Carbon-13 TIData for an Equivolume Mixture of Tetrahydrofuran, 2-Propanol, 1,2-Dirnethoxyethane, and 1,3-Propanediol
EXPERIMENT
1 ZEROFILL
c-1
compound
1
"C Ti,s C-2 OCH,
14.9
17.1
4.5
9.2
APODIZE
1
1
2
CH,CH( OH)CH,
FOURIER
1
TRANSFORM
CH,OCH,CH,OCH, 1
HOCH,CH,CH,OH
PHASE
10.2
7.9
2
3.2
2.6
1 BASELINE FLATTEN
1 PEAK IDENTIFICATION
INTEGRATION
Figure 1. Data processing protocol.
1,2-dimethoxyethane (Eastman),and 1,3-propanediol (Aldrich Chemical Co.). These compounds were used without further purification. Mixtures were prepared from the neat components by weighing out the appropriate quantities of each compound (to the nearest 0.1 mg) corresponding to the desired mole ratio = (approximately THF:2-PrOH:1,2-DME:1,3-Pr(OH)2 100.00:10.00:1.000.30). Samples were prepared just prior to obtaining their 13Cspectra and were quickly stoppered to minimize evaporation. Acetone-d6was added to the mixture in a volume ratio of approximately lo%, to serve as field frequency lock solvent. Spectra were acquired from samples in 10-mm tubes fitted with Teflon vortex plugs. Spectra of neat octanol-2-d (Aldrich Chemical Co.) were obtained in 10-mm sample tubes fitted with a 3-mm coaxial inner tube containing DzO as the field frequency lock solvent. Instrumentation. All spectra were obtained on a Bruker WM-360 wide bore (lacfrequency equal to 90.56 MHz) NMR spectrometer equipped with an Aspect 2000 minicomputer. Spin-Lattice Relaxation Time Measurements and Suppression of the Nuclear Overhauser Effect. Spin-lattice relaxation times for an equivolume mixture of tetrahydrofuran, 2-propanol, 1,f-dimethoxyethane, and 1,3-propanediol were measured by using the fasbinvenion-recoverytechnique (18). This method was also employed to measure the T, values of neat octanol-2-d. Chemical shifts of spectral resonances were assigned from published spectra (19). The spin-lattice relaxation times were calculated, using ORACLE, from a least-squares, three-parameter fit to an exponential function of intensity vs. T values. The TI values for each component in the mixture are given in Table I. The TIvalues (in seconds) for neat octanol-2-d are as follows: 1.1
1.2
1.2
1.4
1.9
2.4
2.7
3.6
CH,-CH( OH)-CH,-CH,-CH,-CH,-CH,-CH, To avoid saturation of the nuclear spins, pulse intervals were chosen to be greater than five times the longest TIin the sample. The nuclear Overhauser effect was suppressed by the gated decoupling method. The criterion for suppreasion, a waiting period between pulses in excess of five times the longest TIin the sample, was previously satisfied by the relaxation delay employed to avoid saturation. Furthermore, the FID acquisition periods employed, 1.64 s for the high dynamic range mixture and 1.36 s for neat octanol-24, resulted in decoupling duty cycles of less than 5%. Acquisition and Processing Parameters. For the mixture analysis, 16384 (16K) data point FID's spanning a spectral region
J ' ' ' 1 ' ' 1 ' 1 1 1 1 1 1 1 1 1 1 I I I I I I I I I I I I I I I J I I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
70.'0
60.'0
50.'0
40.'0
30.'0
20.'0
PPM Figure 2. Carbon-I3 spectrum of a high dynamic range mixture of
= 330:41.5:3.98:1.00. mole ratio THF:2-PrOH:l,P-DME:1,3-Pr(OH), Vertical expansions are X 3 2 scale and were obtained before (lower) and after (upper)base line flattening. Spectral assignments are given in Table 11.
of 5000 Hz were accumulated for a total of 64 scans. Analog filters were chosen as twice the spectral width. A pulse width of 12.5 ps was utilized corresponding to a 90° tip angle of the macroscopic magnetization. Each accumulated FID was transferred to a Data General MV-8000 32-bit computer running the ORACLE software package. The FID's were zerofilled to 32K and multiplied by an exponentially decaying function, equivalent to 1.0 Hz line broadening,prior to Fourier transformation. The resulting spectra had four to five data points above half maximum intensity for each resonance of interest. Neat octanol-2-dspectra of relatively high signal-to-noiseratio (approximately 800:l) were obtained from the sum of 64, 16K FID's covering a spectral width of 6024 Hz. Neat octanol-2-d spectra of varying signal-to-noiseratio were derived from addition of a noise FID (multiplied by an appropriate scaling factor) to a 16-scan neat octanol-2-d FID. The coadded FID's were each of 16K and covered a spectral width of 6024 Hz;other conditions were as above except that 2.0-Hz line broadeningwas applied prior to Fourier transformation. To determine the effect of spectral data point density on the accuracy of integration, a set of five replicate neat octanol-2-d BID'S was obtained at each of the following spectral sizes: lK, 2K, 4K, 8K, 16K, and 32K. Each replicate was composed of 64 scans. All other conditions were as above. RESULTS AND DISCUSSION Mixture Analysis. A high dynamic range mixture of tetrahydrofuran, 2-propanol, 1,2-dimethoxyethane, and 1,3propanediol was investigated to evaluate the accuracy to which the minor component in the sample could be determined. The 13C spectrum of this mixture, Figure 2, having a mole ratio of THF:2-PrOH:l,2-DME:1,3-Pr(OH), = 330:41.5:3.98:1.00, shows eight resonances, each component giving rise to two peaks. It should be noted that because of molecular symmetry,
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
Table 11. Quantitative Results of Seven Replicate Analyses of a Sample of Mole Ratio THF:2-PrOH:1,2-DME:1,3-Pr(OH), = 330:41.5:3.98:1.00 expected peak no. shift, ppm component SIN (RMS) integrala 1
2 3
4 5 6 7, 8
72.5 68.0 63.5 60.3 58.7 36.4 26.2125.7
1,2-DME THF 2-PrOH 1,3-Pr(OH), 1,P-DME 1,3-Pr(OH), THF/ 2-PrOH
38:l 2260:l 156:l 5: 1 25:l 3:l 2030:l
mean integral ORACLE^
1.08 88.83 5.59 0.27
re1
% error
1.04 (0.08)b 87.50 (0.57) 5.43 (0.17) 0.28 (0.12) 1.04 (0.11) 0.12 (0.07)' 100.00
1.08
0.13 100.00
-3.7 -1.5 -2.9 3.7 -3.7 -7.7
' Based on six
Values in parentheses denote t 1standard deviation. a Percent of the peak areas of THF/2-PrOH. replicates (due to insufficient S/Nin remaining spectrum). Table III. Quantitative Results of Eight Replicate Analyses of a Sample of Mole Ratio THF: 2-PrOH:1,2-DME:1,3-Pr(OH), = 298 :31.4:3.26:1.00 peak no. 1
2 3 4 5 6 7, 8 a
shift, ppm
component
72.5 68.0 63.5 60.3 58.7 36.4 26.2/25.7
1,2-DME THF 2-PrOH 1,3-Pr(OH), 1,S-DME 1,3-Pr(OH), THF/2-PrOH
Percent of the peak areas of THF/2-PrOH.
SIN (RMS)
expected integrala
mean integral ORACLE^
28:l 1.00 0.99 (0.12)b 2500:l 90.3 1 89.3 (1.0) 144:l 4.85 7.76 (0.23) 0.22 (0.11) 6: 1 0.31 1.00 1.01 (0.08) 24:l 5:l 0.16 0.08 (0.06) 100.00 100.00 2840:l Values in parentheses denote r t l standard deviation.
Table IV. Quantitative Results of Five Replicate Analyses of a Sample of Mole Ratio THF:2-PrOH:1,2-DME:1,3-Pr(OH), = 263 :26.1:4.22:1.00 expected peak no. shift, ppm component SIN (RMS) integrala
-1.0 -1.1
-1.9 - 29 -1.0 -50
mean integral ORACLE^
re1 % error
1.45 1.38 (0.09)b 1,2-DME 40:l 2250:l 91.03 89.22 (0.63) THF 113:l 4.48 4.19 (0.07) 2-PrOH 0.35 0.39 (0.06) 1,3-Pr(OH), 7:l 1.33 (0.09) 37:l 1.45 1,2-DME 0.17 0.09 (0.04) 1,3-Pr(OH), 4:l 100.00 100.00 2380:l THFI2-PrOH Percent of the Deak areas of THF/2-PrOH. Values in Darentheses denote t l standard deviation. 1
2 3 4 5 6 7, 8
a
re1 % error
-4.8 -2.0 -6.5
72.5 68.0 63.5 60.3 58.7 36.4 26.2125.7
carbon-2 of 2-PrOH and 1,3-Pr(OH)2are present at only half the molar concentrationof other carbons in the sample. Thus, the true dynamic range of the mixture (carbon-1 or carbon-2 of THF:carbon-2 of 1,3-Pr(OH)J is 660:l. The integrated intensity of each peak was normalized to the area of the partially resolved THF/2-PrOH pair at 26.2 and 25.7 ppm, respectively. Seven consecutive replicate analyses were performed on the sample to facilitate estimation of the precision of the analysis. The results are shown in Table 11. The mean integral for all components is in good agreement with the actual composition. In addition, excellent internal precision is observed for peaks 1 and 5. Component peaks with lower signal-to-noise ratio exhibit a concomitant decrease in accuracy and precision of determination. The above analysis was repeated for two additional mixtures of similar mole ratios to estimate the precision between runs. These data are given in Tables I11 and IV. Although the number of replicates and concentration ratios is not identical, the data still illustrate good run-to-run reproducibility for Components of even moderate signal-to-noise ratio. However, the precision is seen to decrease with signal to noise. The authors are aware that an additional significant figure is not warranted in some of these data. Neat Octanol-1-d. Neat octanol3-d, Figure 3, was employed to probe the accuracy and precision of 13C analyses as a function of signal-to-noise ( S I N ) ratio and spectral data point density. To evaluate to the effect of SIN, a preliminary
11
-8.3 -4 7
I
i
I , , $ ,I
60.' 0
I
I
I
, I #I
40.'0
I
I
I
,
I
I
I I I
I
I
I
I
I
I
20.'0
PPM Carbon-13 spectrum of neat octanoi-24. Spectral assignments are given in Table V. Figure 3.
experiment was conducted which had sufficient S I N (approximately 8W1) so as to minimize its effect on the analysis. Five successive replicate analyses of neat octanol-2-d were performed. Each spectrum was integrated with and the resulting peak areas were normalized to the unresolved carbon-l/carbon-7 pair at 23.4 and 22.9 ppm, respectively. These data are given in Table V. The accuracy and precision of the
785
ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
Table V. Quantitative Results for Five Replicate Analyses of Neat Octanol-2-d peak carbon shift, ppm no. no. 1 2 67.3 2 3 39.6 32.3 3 6 4 5 29.8 5 4 26.2 8 8 14.1
mean integral
re1 % error
ORACLE a
48.76(0.59)‘ 49.26(0.36) 48.69(0.35) 49.24(0.53) 48.83(0.42) 48.57 (0.37)
-2.5 -1.5 -2.8 -1.5 -2.3 -2.9
The “true a Percent of peak areas of C-1and C-7. value” of the integral is assumed to be 50.00%. Values in parentheses denote t 1 standard deviation. Table VI. Effect of Signal-to.NoiseRatio on Accuracy and Precision of Integrations of Neat Octanol-2-d mean integral
re1
SIN (RMS)
0R A CLE a
% errorb
303 :1 202:l 123:l 66:l 45:l 27:l 14:l 7:l
48.0 (0.8)‘ 47.1 (1.2) 46.2 (1.7) 45.1 (3.0) 43.4 (4.2) 40.7 (6.3) 36.2 (11) 29.2 (19)
-3.4 -4.0 -5.7 -7.6 -9.8 -13 -28 -42
a Percent of peak areas of C-1and C-7. The “true value” of the integral is assumed to be 50.00%. Values in parentheses denote f 1 standard deviation.
mean integral for each carbon show good consistency. The internal precision of the mean integral is also observed to be slightly better than the precision between runs. If all carbons in each replicate are considered equivalent, then the mean integral for the analysis, based on 30 values, is 48.88 f 0.49% (1standard deviation) which yields a relative percent error of 2.2. Neat octanol-2-d spectra of decreasing S I N were obtained from an FID to which was added noise of progressively increasing magnitude. Each spectrum was integrated and the resulting peak areas were normalized to carbon-l/carbon-7 pair as before. The mean integral for the six peaks in each spectrum was calculated and is tabulated in Table VI with the corresponding spectral SIN. Precision of determination decreases with S I N , corroborating the trend observed in the analysis of the high dynamic range mixtures. It is not clear why the relative accuracy of the data in Table VI shows a systematic trend to lower values with decreasing SIN. This may be due to the use of a noise spectrum which was not white, or to some degree of misphasing in the spectra of lower S I N , or to a combination of the two effects. The combined integral of the carbon-l/carbon-7 peaks is used as an arbitrary standard; this integral, at twice the size of the remaining integrals, is less prone to error. Neat octanol-2-d was also studied to ascertain the effect of spectral data point density on analytical results. The results obtained for each spectral density were calculated from 30 values, five spectra of six peak areas each, and are tabulated in Table VII. Analytical accuracy increases with spectral resolution since the accuracy with which the integral is approximated is enhanced as the number of data points defining the line shape increases. The precision of the analysis also increases with spectral density. It is noteworthy that the case of 2K real data points accuracy is good (albeit with poor precision) with individual peaks being defined by as few as two points above half height. Quantitative Determination of Overlapping Component Peaks by Deconvolution. In spite of the high spectral
Table VII. Effect of Spectral Data Point Density on the Accuracy and Precision of Integrations of Neat Octanol-2-d spectral dataset no. of size (K) real ptsa
Hz/
512 1024 2048 4096 8192 16384 32768
11.8 5.9 2.9 1.5 0.7 0.4 0.2
1 2 4 8 16 32 64f
re1 pt
mean inte ral
5
ORACLE
d
47.1 ( l l ) e 49.4 (4.3) 49.1 (0.6) 48.9 (0.5) 48.8 (0.4) 48.8 (0.4)
%
errore d
-5.8 -1.1 -1.8 -2.2 -2.4 -2.3
av fwhh,
Hz
d
7.6 3.7 4.0 3.8 3.8 3.8
a The number of displayed points (Reals) is half the Percent of peak areas of C-1and spectral data set size. (3-7. The “true value” of the integral is assumed to be Unable to determine due to spectral distor50.00%. tions. e Values in parentheses denote f 1 standard deviation. f Derived from 32K spectral data sets which were zerofilled to 64K.
dispersion of resonances in 13CNMR, peaks of quantitative interest frequently overlap. This situation presents little difficulty if resolved resonances for the overlapping components exist elsewhere in the spectrum which can be used in the determination. However, this is not always the case, and methods must be employed to separate the individual contributions to an overlapping line shape in order to quantify the components. Resolving integrals for closely spaced peaks is achieved in ORACLE by either manipulating the integration limits using the line width criterion or curve fitting. If the peak overlap occurs only near the base line, limits of integration for the overlapping bands are chosen such that they do not overlap. This method is limited in accuracy since the wings of the normally Lorentzian NMR line shape extend a considerable distance and can contribute significant area to adjacent peaks, as in the case of a small peak which lies on the wing of a much larger peak. A more accurate approach is to determine the individual components of a composite profile by fitting the curve based on an appropriate function. Lorentzian curves approximate most NMR line shapes adequately (except near the base line, in practice). The coefficients of the individual Lorentzians contained in the composite profile are then determined by a least-squares optimization method. The least-squares method was applied to deconvolute the the partially resolved THF/2-PrOH pair in the high dynamic range mixtures. For the data in Table 11, a value for 2-PrOH (normalized to THF as 100%) of 11.7 f 0.1% (1 standard deviation) was obtained resulting in a relative percent error of -7.3 (actual value 12.55%). The procedure was repeated for the data in Tables I11 and IV, yielding values for 2-PrOH of 9.75 f 0.07% (relative percent error of -9.1) and 9.61 f 0.12% (relative percent error of -2.4), respectively. Although the precision of the above values is generally better than those obtained for carbon-2 of 2-PrOH, the accuracy is lower. This may be due to a poorer least-squares fit of the minor 2-PrOH component relative to the large THF component of the composite profile. The curve fitting procedure was also used to resolve the carbon-l/carbon-7 pair of neat octanol-2-d. From the data in Table V, the area of the carbon-7 peak (normalized to carbon-1 as 100%) was found to be 99.0 0.7% (1 standard deviation) resulting in a relative percent error of -1.0. The peaks are of approximately the same intensity and line shape and hence should be fit to similar degrees of accuracy. This would account for the improvement in the relative percent error over that obtained in the high dynamic range mixtures.
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 4, APRIL 1983
Details of Processing Methods. By observing the appropriate experimental conditions, 13C NMR spectra can usually be measured such that individual peak intensities are proportional to the concentration of the corresponding component in the sample. If all of the spectral line widths and line shapes are identical, then the relative peak heights will represent the relative concentrations of the species giving rise to each signal. However, line widths and line shapes within a spectrum frequently vary due to a variety of chemical and instrumental influences, and therefore relative peak areas must be measured to obtain quantitative results. Several numerical integration methods are available for approximating the area under a peak defined by discrete data points. The use of a particular technique depends on the degree of accuracy desired in determining the integral. The simplest numerical integration technique is to sum all the data points contained within the peak. The total area under the curve is simply the combined area of a set of rectangles, each having equal width. The “rectangle rule” gives a fairly crude approximation of the integral, especially in cases where the number of data points on the peak is relatively small. As the number of points defining the peak increases, however, the integration errors of individual rectangles (rectangle area true area) are decreased. Thus, the accuracy of the integral increases with the number of data points defining the peak. A more accurate method of approximating the area under the peak is to linearly interpolate between adjacent data points. The total area is now given by the combined area of a set of trapezoids of equal width. This technique is called the “trapezoidal rule”. The integration errors for individual trapezoids are less than those obtained by summation; therefore, the accuracy of integration is increased. Further increases in accuracy are possible if three data points rather than two are used for interpolation. A common three-point interpolation scheme for integration is “Simpson’s rule”. With this method, a parabolic function is defined by three consecutive points and the area underneath the function is calculated. The integration errors using “Simpson’s rule” are smaller than those obtained by either the ”rectangle” or “trapezoidal” rule because a parabola is a better approximation to the peak curvature (over small ranges) than rectangles or trapezoids. Regardless of the method of integration employed, two aspects of the data can introduce large systematic errors in integrals: base line distortion and base line offset. Base line distortions can be caused by a variety of instrumental and processing considerations. Lengthy experiments containing a large number of acquisitions frequently suffer from broad distortions that are not apparent in shorter experiments. Minor anomalies such as imperceptible pulse breakthrough, probe background, and amplifier distortions are additive and become serious only after many acquisitions. Formidable distortions can also occur where the data overflow or are truncated prior to Fourier transformation. Correction of these distortions is possible with a method proposed by Pearson (20), as long as they are broad with respect to the spectral information which is to be retained. The algorithm fits a function to the signal-free portion of the spectrum. The function is then subtracted from the data set to give a partially corrected spectrum. The process if repeated until the base line is flat. Almost any function can be used for the fitting procedure; however, since most distortions manifest themselves as continuous curves, fitting a multiorder polynomial is appropriate. We have extended Pearson’s method to be more versatile by allowing the user to specify the order of the correction and further to select multiple spectral regions to be independently corrected. Very complex distortions can be removed by ap-
Table VIII. Accuracy of Integration for Lorentzian Lines as a Function of Integration Range accuracy (fraction of total integral)
integration range (in units of line width)
99.99% 99.90%
6366 636 63.6 6.31 3.08 1.00
99.00% 90.00% 80.00% 50.00%
plying the fitting procedure to small portions or blocks of the data set. Thus, a fourth-order correction on four individual blocks of data can be as effective as a sixteenth-order correction on the entire spectrum. Perhaps the most important aspect of method, however, is that line shape and integrals are preserved for narrow features and, thus, the integrity of quantitative data is maintained. Another consideration in processing quantitative data is the proper choice of integration limits. Their importance can be demonstrated by evaluating the theoretical line shape of an NMR signal. The line shape can be derived from the Bloch equations and has been shown to be a Lorentzian function
in which a is the full width at half maximum height and A is the peak intensity times a. Integration between the limits -vl and +vl gives
When v1 is infinite, the total integrated intensity becomes
It is unreasonable to integrate digitally over all frequencies, so an approximation must be made by choosing limits. Fortunately, if this approximation is consistently implemented, the ratio of individual integrals within a spectrum will remain constant. Table VI11 shows how the limits can be chosen as a function of peak width to obtain various accuracies of integration for a Lorentzian line. Since most of the integrated intensity lies within a few multiples of the line width, small errors in the determination of the line width or in the line shape will have very little effect on the integral if the limits are consistently chosen as several line widths. Integration as a function of line width facilitates automatic integrations. The default limit criterion can be changed, however, in order to integrate partially resolved peaks. The number of integrated pointa can be increased without changing the limit criterion by zerofilling the data set while in the Fourier codomain. This action increases the digital resolution, or number of points per peak, thus increasing the accuracy of a peak’s integration. Convolution techniques, such as line broadening, also increase digital resolution, but will not affect integrals when the integration limits are chosen as a function of line width. Although the above data processing considerations have been applied to NMR, they are in no way limited to NMR. Modular design provides for easy adaptation of pertinent features to other spectroscopies as well as chromatography. The ORACLE software package is currently configured for the Data General MV-8000 and DEC VAX 32-bit computers and will soon be implemented on other laboratory computers.
Anal. Chem. 1983, 55, 787-790
is available a t no charge to academic laboratories. Inquiries may be addressed to the corresponding author.
ORACLE
ACKNOWLEDGMENT The authors thank Ani1 Kumar for technical assistance.
LITERATURE CITED (1) (2) (3) (4)
(5) (6) (7) (8) (9) (10) (11)
Carman, C. J.; Wllkes, C. E. Macromolecules 1974, 7 . 40-43. O’Nelll, I. K.; M. A. Pringuer Org. Mag. Reson. 1974, 6 , 398-399. Sarneskl, J. E.; Reilley, C. N. Anal. Chem. 1978, 48, 1303-1308. Levy, G. C.; Hewitt, J. M. J . Assoc. Off. Anal. Chem. 1977, 60, 241-243. Abldl, S. L. Anal. Chem. 1982, 54, 510-516. Forsyth, D. A.; Hediger, M.; Hahmoud, S. S.; Glessen, B. C. Anal. Chem. 1982, 54, 1896-1898. Shoolery, J. N. Prog. NMR Spectros. 1977, 1 1 , 79-93. Marecl, T. H.; Scott, K. N. Anal. Chem. 1977, 49, 2130-2136. Thiault, B.; Mersseman, M. Org. Mag. Reson. 1978, 8 , 28-33. Blunt, J. W.; Munro, M. H. G. Aust. J . Chem. 1978, 29, 975-988. LaMar, G. N. J . Am. Chem. SOC. 1971, 93, 1040-1041.
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RECEIVED for review October 12,1982. Accepted January 3, 1983. Financial support from the National Science Foundation (Grant CHE 81-05109) and the Division of Research Resources, National Institutes of Health (Biotechnology Research Resource, RR 01317), is gratefully acknowledged.
CORRESPONDENCE Study of Chromatographic Retention with Spectroscopic Bandwidths Sir: An improved understanding of chromatographic retention behavior has been provided from statistical thermodynamic descriptions of solute distribution coefficients (1-3). Also, spectroscopic measurements have been developed to obtain specific structural information about bonded stationary phases ( 4 , 5 ) . The link between these two areas of study is the determination of the relationship between distribution coefficients and the structures of the chromatographicphases. The purpose of this work is to establish a first step toward developing an understanding of the intermolecular interactions giving rise to the molecular shape dependence of solute retention. The fundamental mechanisms that determine the structure dependence of chromatographic retention are presently not well understood. It is widely accepted that the polar mobile phase gives rise to the retention selectivity for polycyclic aromatic hydrocarbrons using n-alkylsilane stationary phases (6). The selectivity is thought to arise from hydrophobic exclusion in the mobile phase, with the stationary phase being inert. This idea is consistent with the observations that the selectivity decreases markedly upon changing mobile phase composition and less upon changing the nonpolar stationary phase. A mobile phase origin of PAH selectivity is also consistent with the inverse correlation between retention time and molecular size (7). On the other hand, there are recent chromatographic data that are more consistent with the stationary phase as the origin of selectivity. These data show that the retention indexes of polycyclic aromatic hydrocarbons (PAH)correlate with length-to-breadthratio in LC separations employing a C18stationary phase with an acetonitrile-water mobile phase (8). The fact that solutes having higher length-to-breadthratios are retained longer by the n-octadecyl stationary phase suggests that the shape selectivity originates from the stationary phase rather than the mobile phase. Chromatographic retention measurements alone are insufficient to identify unambiguously the contributions to selectivity from each phase because retention inherently senses the difference between two phases. Spectroscopy allows study of the phases individually. It can potentially provide information pertinent to chromatographic retention mechanisms
because the intermolecular interactions that determine retention also control the positions and widths of spectroscopic bands. The relation between spectroscopic and chromatographic measurements is thus the attractive and repulsive intermolecular interactions that influence both measurements. The observed molecular shape selectivity of chromatographic retention suggests that steric interactions between the solute and its environment are controlling selectivity among isomers. Steric interactions for nonpolar molecules occur through short-range repulsive forces between molecules. Spectroscopic bandwidths are known to be affected by repulsive and attractive forces that cause modulations of the quantum states of the molecule. The relation of these interactions to vibrational line shapes is the subject of vibrational dephasing theory (9-11). Vibrational spectra are not studied in this application because a large, unknown contribution from lifetime broadening diminishes the reliability of such measurements. Since the same interactions operate on electronic spectra, the ideas developed in vibrational dephasing theory are adapted to this discussion. Both the temporal properties and the strengths of the attractive and repulsive interactions contribute to spectroscopic bandwidths. For nonpolar molecules, where the interaction strengths are weak, it is likely that the repulsive, steric interactions largely control the bandwidths; for polar phases, both attractive and repulsive interactions contribute. The fact that the steric interactions are evident in the spectroscopic bandwidths is the principle behind this work. The purpose of this work is 2-fold: first, the existence of a relationship between spectroscopy and chromatography is explored, and second, the spectroscopic measurements are applied to the study of the origin of retention in the reverse-phase separation of polycyclic aromatic hydrocarbons. To study the shape dependence of the solute-solvent steric interactions, the retention times and the electronic spectral bandwidths of solutes having varying shapes were measured. The bandwidths for the solute in each solvent should thus reveal the extent to which the corresponding phase contributes to shape selectivity in chromatographic retention. A series of dimethylnaphthalene derivatives was chosen to achieve a
0003-2700/83/0355-0787$01.50/00 1983 American Chemical Soclety
