LITERATURE CITED H. D. Beckey. lnt. J. Mass Spectrom. /on Phys., 2, 500 (1969). H. U. Winkler and H. D. Beckey. Org. Mass Spectrom., 6,655 (1972). R. J. Beuhler, E. Flanigan. L. J. Greene, and L. Friedman, Biochem. Biophys. Res. Commun., 46, 1082 (1972). R. J. Beuhler, E. Flanigan, L. J. Greene, and L. Friedman, J. Amer. Chem. SOC.,96,3990 (1974). M. A. Baldwin and F. W. McLafferty, Org. Mass Spectrom., 7, 1353
(1973). J. D. Baideschwieler, Science, 159,263 (1968). J. M. S.Henis, Anal. Chem., 41 (lo),22A (1969). M. M. Bursey, T. A. Elwood, M. K. Hoffman, T. A. Lehman, and J. M. Terarek, Anal. Chem., 42, 1370 (1970). M. M. Bursey and M. K. Hoffman, Can. J. Chem., 49,3393(1971). M. M. Bursey, J. Kao, J. D. Henion, C. E. Parker, and T. S. Huang, Anal. Chem., 46, 1709 (1974). M. L. Gross, P-H Lin. and S. L. Franklin, Anal. Chem., 44,974 (1972). J. A. Hipple, H. Sommer, and H. A. Thomas, Phys. Rev., 76, 1877
(1949). H. Sommer, H. A. Thomas, and J. A. Hipple, Phys. Rev., 82, 697 (1951). C.E. Berry, J. Appl. Phys., 25, 28 (1954). D. Alpert and R . S. Buritz, J. Appl. Phys., 25, 202 (1954). D. Lichtman, J. Appl. Phys., 31, 1213 (1960). R. W. Lawson, J. Sci. Instrum., 39,281 (1962). H. B. Niemann and B. C. Kennedy, Rev. Sci. Instrum., 37, 722 (1966). R. T. Mclver, Jr., Rev. Sci. Instrum., 41,555 (1970). R. T. Mclver, Jr., and A. D. Baranyi. Int. J. Mass Spectrom. /on Phys.,
14.449 (1974).
L. k. Anders. J. L. Beauchamp, R. C. Dunbar. and J. D. Baldeschwieler, J. Chem. Phys., 45, 1062 (1966). J. L. Beauchamp and J. T. Armstrong, Rev. Sci. Instrum., 40, 123
(1969).
(23)R. T. Mclver, Jr., Rev. Sci. Instrum.. 44, 1071 (1973). (24)R. T. Mclver, Jr., J. A. Scott. and J. M. Riveros, J. Amer. Chem. SOC., 95,2706 (1973). (25)S.E. Buttrill, Jr., J. Chem. Phys., 50, 4125 (1969). (26)M. B. Comisarow, J. Chem. Phys., 55, 205 (1971). (27)W. T. Huntress, J. Chem. Phys., 55, 2146 (1971). (28)E. B. Ledford and R. T. Mclver, Jr., unpublished results. (29)M. S. B. Munson and F. H. Field, J. Amer. Chem. SOC.,88, 2621 (1966). (30)F. H. Field, Accounts Chem. Res., 1, 42 (1968). (31)M. S.B. Munson. Anal. Chem.. 43,(13).28A (1971). (32)G. W. A. Milne, H. M. Fales, and T. Axenrod, Anal. Chem., 43, 1815 (1971). (33)G. P. Arsenault in "Biochemical Applications of Mass Spectrometry," G. R. Waller, Ed., Wiley, New York, N.Y., 1972,pp 817-832. (34)D. P. Beggs in "Applications of the Newer Techniques of Analysis," I. L. Simmons and G. W. Ewing. Ed.. Plenum, New York, N.Y.. p 345. (35)E. C. Horning, M. G. Horning, D. I. Carroll, I. Dzidic, and R. N. Stillwell, Anal. Chem., 45,938 (1973). (36)K. Jager and A. Henglein. Z.Naturforsch. A,, 22, 700 (1967). (37)R . T. Mclver. Jr., and J. S. Miller, J. Amer. Chem. Soc., 96,4323 (1974). (38)M. B. Comisarow, private communication. (39)J. A. McCloskey, A. M. Lawson. K. Tsuboyama, P. M. Kruger. and R . N. Stillwell. J. Amer. Chem. SOC.,90,4182 (1968). (40)D. F. Hunt, C. E. Hignite. and K. Biemann, Biochem. Biophys. Res. Cornmun., 33,378 (1968). (41)G. P. Arsenault in "Experimental Methods in Biophysical Chemistry," C. Nicolau, Ed., Wiley. New York, N.Y.. 1973.
RECEIVEDfor review August 5, 1974. Accepted December 9, 1974. We wish to thank the National Science Foundation (GP-38170X) and the Alfred P. Sloan Foundation for partial support of this work.
Computer-Aided Quantitative Analysis by Proton Magnetic Resonance for Compounds with Complex Spectra Osamu Yamamoto and Masaru Yanagisawa National Chemical Laboratory for Industry, 1- 1-5Honmachi, Shibuya-ku, Tokyo, Japan
In order to make a PMR quantitative analysis of a mixture of proton-containing substances which show nonseparable complex patterns in PMR spectra, a method is presented in which the observed and the calculated spectra of the mixture are compared in a computer, and a least squares calculation is made for residuals between the intensities of both spectra at a number of frequency points to obtain the relative concentrations of each component in the mixture. The calculated spectrum of the mixture is generated in the computer from known PMR parameters and assumed relative concentrations. The matched filter technique Is applied to the spectra to avoid the difficulty arising from the dlsagreement in intensity due to small frequency deviation between the observed and the calculated spectra. The degree of accuracy of the relative concentrations obtained by this method is comparable to that of the usual integration technique. The present method has a general applicability to a variety of proton-containing compounds showing complex signal patterns for which the usual integration method cannot be applied.
High resolution PMR spectrometry has been proved to be very useful as a tool for quantitative analysis of protoncontaining organic compounds ( I ) . For this purpose, the electronic integrator has been developed and conveniently used in a variety of quantitative applications. A large number of studies have been published on the use of the integrator ( 2 - 5 ) . It has been thought, however, that the inte-
grator method includes some complex instrumental factors which affect the precision obtained, so that a considerable degree of operator skill is required a t the present stage of development of the NMR instrument. Recently, the use of the PMR spectrometer controlled by a dedicated computer for quantitative analysis has been demonstrated, which makes it possible to acquire data, numerically integrate appropriately selected regions, and print out the results as a teletype record (6). But all earlier methods required that PMR spectra exhibit a t least two separable absorption bands, and only then was it possible to obtain some quantitative information from the ratios of the areas under the separated signals. Many organic compounds show complicated signals in a limited region of the PMR spectra. A mixture of such compounds often gives a PMR spectrum in which very complicated patterns are superposed in a narrow region and not easily separable into any definite groups. In this case, PMR spectrometry has not been able to supply any quantitative information. Typical examples are mixtures of mono- or polysubstituted benzenes. In principle, chemical shifts and coupling constants can be extracted exactly from any complicated spectrum by a conventional spectral analysis and, conversely, if the exact values of these PMR parameters are known, the PMR spectrum, including intensities, can be accurately reproduced. This seems to indicate that it is possible to separate the individual signals from the superposed spectral pattern, and thus possible to make a PMR quantitative analysis, without integration, of compounds that show complicated signals in a narrow region. In the present study, we A N A L Y T i C A L CHEMISTRY, VOL. 47, NO. 4, APRIL 1975
697
L
Observed Spectrum
Frequency and intensity table
1
Specifying the calculation mode and necessary parameters
].east squares calculation for the mole fraction Changing the broadening function
Plotting the spectrum
Figure 1. Flow diagram of the program QUANTNMR
attempt such a quantitative analysis using a computer program QUANTNMR, and apply this method to disubstituted benzene derivatives.
COMPUTATION METHOD In the method reported here, it is necessary at first to have accurate PMR parameters, i.e., the chemical shifts, 6, and the coupling constants, J , of the components which are expected to exist in a given sample to be analyzed. These PMR parameters can be obtained for a sample containing a single component by a conventional procedure, e.g., by the LAOCOON program (7, 8). Once the PMR parameters for each component are obtained, we can synthesize any PMR spectra of the mixtures by superposing the calculated spectra of each component weighted by the mole fractions. The line shape of the calculated spectra is assumed to be Lorentzian. An observed spectrum of a mixture containing such components is measured on a spectrometer, and fed into the computer via a suitable input medium. Then the observed spectrum, after the appropriate data procession, is compared with the synthesized spectrum generated in the manner described above. The program proceeds to a least squares calculation for the residuals between the calculated and the observed intensities a t a number of frequency points as a function of the mole fractions of the components, and finally the most reliable relative concentrations are obtained. A schematic flow diagram of the calculation is shown in Figure l. The data procession of the observed spectrum includes the calibration of frequency, the base-line correction, and the normalization of intensity. The calibration of frequency is particularly important when the frequency axis is not exactly linear as in the Varian HA-100 spectrometer. The calibration of the frequency scale can be made by a curve corresponding to a quadratic equation ( 9 ) , but, in this case, the data points obtained are not equidistant. For the reason to be given later, it is necessary to carry out the Fourier 698
* ANALYTICAL C H E M I S T R Y ,
VOL. 47, NO. 4 , APRIL 1975
transformation of the spectrum, which requires equidistant data points. Non-equidistant data points are converted to equidistant by means of the Lagrangean interpolation (IO). The base-line correction and the normalization of the intensity for the observed spectrum are performed in the usual way. When the observed spectrum is compared with the calculated one, a serious problem arises because of the sharpness of the PMR signal and of the solvent and the concentration dependence of the chemical shift. Since the full-line width of a single line in usual PMR spectra is of the order of 0.5 Hz, a deviation in the chemical shift of this order of magnitude, leads to a drastic change in intensity at a frequency near the signal peak. Thus, we can never hope to have any reliable results. To avoid the difficulty, it is necessary to adjust the total concentration of the unknown sample in the NMR solvent as closely as possible to the concentration with which 6 and J values of each component in the sample have been determined. Of course, the same solvent should be used. Actually, this procedure is important and effective in obtaining approximate agreement of the observed spectrum with the calculated one. However, even if the total concentration of the sample could be exactly adjusted to the concentration used in the spectral analysis of the components, the observed and the synthesized spectra would not exactly coincide, because a particular component sees an environment in the mixture different from that when a single component is in the same solvent. The second remedy will be a modification of 6 values used to synthesize the spectrum of the mixture. If the exact values of 6’s were obtained for a particular mixture, the exact agreement in frequency would be achieved. But this is an impractical procedure at the beginning of the calculation, since it is necessary to distinguish which signals belong to a particular component in the complex patterns of the mixture, and this distinction can be made only after the approximate synthesized spectrum has been obtained. In other words, this procedure will be effective after the approximate relative concentrations are obtained. It is essential, therefore, to devise a method which permits the calculation leading to a reliable result even if some deviations in frequency exist between the observed and the calculated spectra. One of the promising procedures is the use of the broadening function. Because of the sharpness of the PMR signal, there is less agreement in intensity a t a particular frequency, the more the deviation in frequency increases. Conversely, if the line width of the signal is broad, the agreement in intensity will be much improved under conditions which are, in other respects, the same. Usually the line shape of the high resolution PMR signals is of a Lorentzian type, so that the signal can be broadened by means of so called “matched filter technique” ( 1 1 ) .This can be conveniently made by Fourier-transforming the spectrum to get the free induction signal, multiplying the weighting function of the exponential decay type, and inversely Fourier-transforming the signal to obtain a broadened CW spectrum ( I I , 1 2 ) . In a step of the program before the least squares calculation, the matched filter technique is applied to the observed spectrum. The extent o f the broadening depends on the deviation of the frequency in the observed spectrum. While the broadening function is effective in offsetting the disagreement in intensity due to the frequency deviation, as broadening progresses, the broadened spectrum will continue to lose its features which characterize the quantitative relationship. Thus a compromise should be made between the two conflicting effects. The quantity A = ZAi will be a good measure of the reliability of the relative con-
Table I. Effect of the-Broadening Function on a n Improvement in the Obtained Values of Relative Concentrationsa R e l a t i v e concentration obtained, mole
True v a l u e
1.o
0.0
Broadening function (Hz)* Ortho isomer Meta isomer
2 .o
3.0
4.0
5.0
...
6.0
21.5 23.4 26.8 28.6 29.4 29.8 30.0 30.0 51.0 42.2 40.6 40.1 39.9 39.7 39.6 40.0 Para isonier 27.5 34.4 32.6 31.3 30.7 30.4 30.3 30.0 a For a synthesized spectrum of a bromoanisole mixture with frequency deviations (see text). Given as the full half-line width. ~
Table 11. Relative Concentrations Obtained for Bromoanisole a n d / o r Dichlorobenzene Mixtures Bromoanisole mixtures
R u i So.
Ortho isomer Meta i s o m e r
Para isomer
2
1
4
3
'Taken
Found
Taken
Found
Taken
48.1 9.9 41.3
49.4 (48.7)" 7.7 (13.0) 42.8 (38.3)
80.1 9.9 10.0
81.5 (80.4) 7.1 (10.3) 11.4 ( 9.4)
31.3 37.6 31.2
Found
31.2 (30.8) 36.2 (36.8) 32.5 (32.4)
Taken
69.9 0.0 30.1
Found
69.4 (67.8) 0.0 (0.0) 30.6 (32.2)
Dichlorobenzene mixtures Run
Ortho isomer Meta i s o m e r Para i s o m e r
KO.
8
7
6
3
Taken
Found
Taken
Found
Taken
Found
Taken
Found
39.8 49.8 10.4
41.3 46.9 11.8
79.6 9.9 10.5
79.1 9.8 11.1
0.0 68.9 30.1
0.0 70.1 29.9
79.9 0.0 20.1
80.7 0.0 19.3
Bromoanisole and dichlorobenzene mixtures
Taken
Found
Taken
12
11
10
9
Run So.
Found
Taken
Bromoanisole 8.4 0.0 47.9 10.1 50.2 O r t h o isomer 11.6 10.0 10.8 10.1 Meta i s o n i e r 10.0 9.9 40.0 10.6 10.1 10.1 Para isoiiier Dichlorobenzene Ortho isomer 19.8 21.9 49.9 51.0 39.5 Meta isoiiier 9.9 8.9 9.9 10.5 10.5 Para i s o m e r 0.0 0.0 9.9 8.5 0.0 The figure in parentheses shows the value obtained from the integration of the methoxy group.
Found
Taken
0.0 8.9 38.7
0.0 50.3 0.0
0.0 49.2 0.0
43.3 9.2 0.0
0.0 49.7 0.0
0.0 50.8 0.0
Found
(L
centrations obtained, where A, is the standard deviation of the relative concentration for the component i obtained from the least squares calculation. Along this line, the program starts a t a predetermined value of the broadening function and, for the broadened spectrum, the least, squares calculation is made to obtain the relative concentrations and the A value is calculated. Then the broadening function is changed slightly, the above calculation repeated, and the new A value calculated. The calculation will be stopped when the smallest A value is obtained, and a set of the relative concentrations for the smallest 1 is regarded as the best values. The final stage of the program is to output the relative concentrations and to plot the calculated spectrum on the plotter. The observed spectrum may be also plotted superposed on the calculated one.
EXPERIMENTAL Apparatub. A Varian HA-100D spectrometer was used to obtain the PMR spectra. The observed spectrum was stored in a C-1024 time-averaging computer, and then punched onto paper tape with a Talley P-120 high speed tape puncher via an interface made by Apex Co., Ltd., Tokyo, Japan. The paper tape was fed to a FACOM 270/30 computer, in which all the necessary calculations were made. Reagents. A number of mixtures consisting of bromoanisoles and/or dichlorobenzenes were made by accurately weighing the components into CC1L. The total concentrations of the benzene derivatives in CC14 (and a small amount of TMS) were adjusted to about I O f 1 mole %. This concentration was the same as that of
the single-component sample with which the spectral analysis was made in order to ohtain the A and J values for the component. Bromoanisoles and dichlorobenzenes were obtained from commerical sources, and fractionally distilled and checked for purity by gas chromatography. Procedure. A Fortran program named QUANTNMR was written along the lines described above. The spectral analyses for obtaining the PMR parameters of dichlorobenzenes and bromoanisoles were carried out by the LAOCOON MBYH program (A modification of LAOCOON 11 (8) by the authors.) If desired, the frequencies and intensities calculated from the LAOCOON MBYH of the isomers may be tabulated, stored in the auxiliary memories, and called out to the cores a t the time of the concentration calculation, thus saving computation time. The computer used in this work is a small scale computer, FACOM 270/30, which has 32 K words of core memories with 265 K words of drum memories. The size of the program of QUANTNMR and its data area cover most of the core memories of this computer, so we used the latter version just mentioned. If one can employ a larger scale computer, the calculation of the frequencies and intensities of each isomer by the LAOCOON program can be carried out a t the same time as the QUANTNMR calculation. This requires an additional calculation time equivalent to about one fifth qf the total, if the accurate PMR parameters of the isomers are known.
RESULTS AND DISCUSSION At first, in order to demonstrate the effect of the broadening function, a fictitious observed spectrum of a mixture of ortho, meta, and para isomers of bromoanisoles was synthesized in the computer based on the known PMR parameters of the compounds, whose 6 values were intentionally ANALYTICAL CHEMISTRY, VOL. 47, NO. 4 , A P R I L 1975
699
I
760.0
740.0
720.0
700.0
I
750.0
720.0
730.0
740.0
680.0
'
660.0
640.0
1
700.0
710.0
690.0
I
760.0 Figure 2.
740.0
720.0
700.0
680.0
640.0
Obsewed (solid line) and calculated (dashed line) spectra obtained from QUANTNMR.
(a)Bromoanisole mixture (Run No. 13). (b)dichlorobenzene mixture (Run No. 6), and
shifted slightly from the actual values. Then the calculation was made for the fictitious'spectrum with the several values of the broadening function. The shifted 6 values were +1.0, -1.0, and -0.5 Hz for the ortho, meta, and para isomers, respectively. Results are shown in Table I. It is clear from the table that the broadening of about 4 Hz (given as the full half-width) is sufficient to obtain the true 700
660.0
A N A L Y T I C A L CHEMISTRY, VOL. 4 7 , NO. 4 , APRIL 1975
(c)bromoanisole and dichlorobenzene mixture (Run No. 12)
values of the relative concentrations for this example. The smallest A was obtained a t a broadening of 5 Hz. Some typical examples of the actual calculation are shown in Table I1 and Figure 2. Generally close agreement was obtained for the relative concentrations, within an error of about f 3 mole %. For the bromoanisole isomers, the relative concentrations obtained by integration of the
~~
~~~~
~
~
~~
~~~~~
Table 111. Effect of the Calculation .Mode a n d the Number of Calculation Points Run No. 13, Bromoanisole mixtures: true values:
Run S o . 14, Dichlorobenzene mixhxes: true values:
ortho isomer, 40.2 mole $4;meta isomer, 49.7; para isomer, 10.1
ortho isomer, 10.1 mole
i;
meta isomer, 79.7; para isomer, 10.2
Equidistant mode Points
Ortho isoilier Meta isomer P a r a isomer
19s
49
23
11
95
49
25
11
4 1 .O 48.2 10.8
40.8 48.4 10.8
41.4 48.1 10.5
42.2 47.4 10.4
9.7 80.5 9.8
9.7 80.5 9.8
10.0 80.5 9.5
14.7 76.2 9.1
198
4s
23
12
50
21
12
41.0 48.1 10.8
40.8 48.1 11.1
41.2 47.7 11.1
41.0 47.3 11.7
9.9 80.2 9.9
10.5 79.9 9.6
12.3 78.0 9.8
101
46
21
12
4s
24
12
40.6 47.8 11.6
41.1 47.5 11.4
40.7 47.5 11.8
40.6 47.8 11.6
13.5 76.6 9.9
8.2 81.1 10.8
140
13a
bloximirm in die reqion n o d e
Points
Ortho isomer Meta isomer P a r a isomer
96
9.8 80.3 9.9
P e a l point n o d e Points
Ortho isomer Meta isomer P a r a isomer
SO
10.7 79.0 10.3
10.9 78.8 10.3
Specified data mode Points
2b
Ortho isomer 40.9 41.7 43.9 Meta isomer 48.1 48.3 46.0 P a r a isomer 11.1 10.3 10.1 In the two calculations, different data points were specified.
methoxy methyl signals are also shown, the errors being about the same. The general accuracy obtained in our procedure seems to be the same as that for the integration method. In Figure 2, it can be seen that the agreement in frequency is not always satisfactory by visual inspection, but the overall agreement of the relative concentration is good without the refinement by correcting 6 values. Furthermore, the relative intensities of the calculated spectrum without broadening do not always correspond to those of the observed spectrum. This is due to the fact that the line widths of the individual signals differ even in the same molecule. In bromoanisoles, the different long-range couplings with methoxy methyl protons may cause the line widths of the aromatic protons to be different. The improvement will be accomplished by giving different natural line widths to different spins in the molecule when calculating the spectra. The natural line width of a particular spin can be found by plotting the calculated spectrum and comparing it with the observed one by the trial and error method during the spectral analysis of the single component. If there are great differences in the line width for various spins, this procedure will provide a better result. But, in most cases, this is not very important in obtaining the relative concentrations within the above range of accuracy. The selection and the number of the calculation points over the whole region of the spectrum are important, and should be determined by taking account of the economy of the calculation time, the desired accuracy of the results, and the simplicity of the input labor. In principle, the best result would be obtained when all of the data points were used for the calculation of intensity. But it is clearly uneconomical from the standpoint of saving calculation time, so that the optimum selection of some discrete calculation points is desirable. The program has several modes for the selection of data points. The first and the simplest is the equidistant mode, which divides the whole range of the input spectrum into several small regions, and takes the dividing points as the calculation points. The second is the maximum in the equidistant region mode, a modification of the first, which
takes a point with the maximum intensity in each divided region as the calculation point rather than the dividing point. The third is the specified data point mode, in which the calculation points are specified and given as the input data by considering the feature of the spectral pattern. In this mode, the data points which characterize the spectrum, e.g., the peak points or shoulders of the signals, can be specified so that more reliable results may be expected, but obviously it requires much input labor. The last mode is the peak point mode, which uses the frequency points with the several largest intensities in the frequency-intensity tables generated for each component from the LAOCOON calculation. For all four modes, the total number of the calculation points should be given. A test was carried out to estimate the optimum number of the total calculation points. The results are shown in Table 111. It is obvious from the table that 20-40 of the calculation points are sufficient to obtain reasonable results in these examples, and larger numbers of the points do not improve the results, thus becoming timeconsuming. The total ranges of the spectra are 50 Hz for dichlorobenzenes, and 100 Hz for bromoanisoles, so the 2040 calculation points correspond to the calculation intervals of about 1.25-5 Hz. This is rather surprising for dichlorobenzene mixtures, because only a singlet signal is obtained from the para isomer. When the signal of this compound is between two neighboring calculation points, the information of para dichlorobenzene cannot be introduced into the calculation, if the signal is sharp. The broadening effect can solve this problem, as shown in Table 111. The selection of the calculation mode does not seem to be critical, so long as the proper number of calculation points is chosen. The determination of the calculation mode will depend on the spectral pattern. If the complex spectrum is spread more or less evenly over the entire calculation range, the equidistant mode or the maximum in the equidistant region mode is preferred. On the other hand, if the complex spectrum can be divided into some blocks between which no signal can be observed over a relatively wide region, the specified data point mode or the ANALYTICAL CHEMISTRY, VOL. 47. NO. 4, APRIL 1975
701
Table IV. Effect of the Frequency Deviations= Deviation, HZ
Ortho isomer Meta isomer Para isomer
-2.0
27.5 35 .O 37.5
-1.5
-1.0
23.3 36.3 40.4
19.4 37.4 43.2
True
-0.5 0.0 R u n No. 15, bromoanisole mixture
15.5 38.5 46.0
11.3 39.5 49.2
0.5
10.3 39.5 50.2
1.0
9.0 39.6 51.4
1.5
2.0
value
7.7 39.6 52.7
6.5 39.7 52.8
9.9 40.4 49.7
R u n Yo. 6 , dichlorobenzene mixture
Ortho isomer
77.3 77.3 77.8 79.1 80.0 81.3 82.5 79.6 Meta isomer 5.8 7.9 9.4 9.8 10.7 11.2 11.4 9.9 P a r a isomer 17.0 14.8 12.8 11.1 9.3 7.6 6.1 10.5 a Without correction of 6 values, So the samples show the original frequency deviation, and the test was carried out by giving further frequency deviation shown above.
peak point mode will be suitable. It is interesting to estimate how much deviation in frequency is permitted. A set of test results is shown in Table IV. In these examples, the 6 values are intentionally shifted to the same extent in the same direction for simplicity. From Table IV, it is apparent that a 1-1.5 Hz deviation is the limit. Of course, this limit depends upon the width of a single line. Much more deviation will be allowed for broader signals. The typical line width of a PMR single line is about 0.5 Hz. It seems that a frequency deviation as great as 2-3 times the full half-width can be permitted. Thus, as mentioned previously, it is still necessary to adjust the total concentration of the mixture in the same PMR solvent to the specified values. This is particularly true for those compounds with a strong concentration dependence on the chemical shift, such as benzene derivatives. Further refinement by correcting the 6 values after visual inspection of the calculated and the observed spectra is also recommended, when a large deviation in frequency is observed after plotting the spectra. Instead of changing the 6 values, one may correct a part of the frequencies in the frequency-intensity table in the auxiliary memories when this mode of operation is employed. These procedures can be most conveniently carried out by the TSS approach, if the TSS station has a graphic display which quickly shows the observed and the calculated spectra in a superposed relationship after one cycle of calculation. Since the aim of our method is the quantitative rather than qualitative analysis, the nature of the components in a given mixture should be known. On the basis of this knowledge, the operator selects the 6 and J values from the data file, and calculation is made for the relative concentrations of the sample. However, in practice, one or two components specified by the operator are often absent from the actual sample mixture. In this case, the results of the calculation should show that the components do not exist. This case was also examined for the sample containing only two or three components by specifying more components from the data file in the calculation. The calculation gave the correct answers in these examples (see Table 11). Naturally it is expected that we may have some wrong answers when many more species are specified. Probably up to 10 will be the limit, although not confirmed. On the other hand, when the number of specified components was less than that of the components actually contained in the sample, the calculation did not give even the approximate values, or often did not converge, as expected. Thus, if the sample contains some components other than those filed in the computer, i.e., "impurities," this method cannot be employed. The signal-broadening technique used in our method is equivalent to the S/N enhancement technique using the matched filter weighting function ( I I ) , so that our method offers the additional advantage that the quantitative analy702
ANALYTICAL CHEMISTRY, VOL. 47, NO. 4, APRIL 1975
sis can be performed for spectra with relatively poor S/N ratios. Actually, from the spectrum with an average S/N ratio of about 6 (S/N ratio for the peak with average intensity), the same answer for the relative concentrations was able to be obtained as from the spectrum with a much higher S/N ratio. From the above discussion, it is clear that quantitative analysis by PMR can be carried out by the method presented in this work for the spectra showing complex nonseparable PMR patterns, for which the usual integration technique cannot be adopted. The accuracy obtained by this method is comparable to that for the integration method. It is natural that similar precautions should be taken as in the integration method. For example, signal saturation should be prevented. But the complexities in the operation that arise from instrumental factors as in the integration technique can be avoided in our method. It might be felt that the resolution of the spectrum is not critical because the whole spectrum is broadened while carrying out this method. But poor resolution is undesirable since it often leads to the distortion of a single line from the Lorentzian shape. Similarly the line distortions due to other sources, e.g., too rapid sweep, are also undesirable. Thus, more care must be taken to prevent the signal saturation arising from the slower sweep than inthe integration method. In dichlorobenzenes, we have experienced easy saturation of each line, so that weaker HI had to be used for these compounds than for bromoanisoles. The difficulty due to saturation will be solved by use of the pulse method. In the above discussion, the present method has been illustrated for bromoanisole and dichlorobenzene isomers as examples, but our method can be generally applied to the quantitative analysis of all proton-containing substances, provided that their PMR parameters are known. Fields in which this method promises to be successful in practical application include various isomers of olefinic and aromatic compounds, polymers, partially deuterated compounds, etc. LITERATURE CITED (1) F. Kasler, "Quantative Analysis by NMR Spectroscopy." Academic Press, New York, N.Y., 1973. (2) Anal. Chem., 38, 331R (1966). 13) Anal. Chem., 40, 560R (1968). (4) Anal. Chem., 42, 418R (1970). (5) Anal. Chem., 44, 407R (1972). (6) J. N. Shoolery and L. H. Smithon, J. Amer. Oil Chem. SOC., 47, 153 t 1970). (7) A. A. Bothner-By and S.Castellano, J. Chem. Phys., 41, 3863 (1964). (8) "Computer Program for Chemistry." Vol. 1, D. F. DeTar, Ed., W. A. Benjamin, New York. N.Y.. 1968, Chapter 3. (9) K. Hayamizu and 0. Yamamoto, J. Mol. Spectrosc., 25, 422 (1968). (10) H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry," D. Van Nostrand Co.. Princeton, N.Y., 1943, Chapter 13. (11) R. R. Ernst, Advan. Magn. Resonance, 2, 1 (1966). (12) 0. Yamamoto. M. Yanagisawa, K. Hayamizu, and G. Kotowycz, J. Magn. Resonance, 9, 216 (1973).
RECEIVEDfor review July 23, 1974. Accepted November 18. 1974.
