Anal. Chem. 1906, 58, 2072-2077
2072
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20
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40
60
60
70
TYE, Y M L O
Flgure 4. Chromatogram of linear fraction of an acid-cataiyzed ptert-butylphenol-formaldehyde condensate.
tween 1950 and 1961(6-8). The separation of the calixarene fraction into calix[4,5,6,7, and 8larenes is shown in Figure 3. There was also a major component and two or three minor components whose structures are unknown, but these did not appear to be linear condensates judging from retention times. The linear condensate fraction, which had eluted from the silica gel with an equivolume mixture of methanol, acetone, and chloroform, was separated into about 40 components on the C18reversed-phase HPLC column. Its chromatogram is shown in Figure 4. There were several very weak peaks that eluted between the major peaks a t about 5,11,15,18,21, etc. minutes. These might belong to a second homologous series of unknown structure. The retention times and area percents of the major peaks that correspond to methylene-bridged
p-tert-butylphenol-formaldehydelinear condensatesare given in Table 11. The maximum area percent at 6.58 min corresponds to the p-tert-butylphenol-formaldehydelinear hexamer. Future syntheses are planned to obtain pure compounds and to determine the effect of reaction conditions upon the amount of calixarenes produced and the distribution of linear condensates in either base-catalyzed or acid-catalyzed p-alkylphenol-formaldehyde condensations. The availability of additional linear condensates of known structure would make feasible the development of a quantitative HPLC analytical procedure. ACKNOWLEDGMENT The authors thank C. D. Gutsche of Washington University, St. Louis, MO, and J. H. Munch of Petrolite Corp. for providing the calixarenes and p-tert-butylphenol-formaldehyde condensates used in this study. Registry No. (p-tert-Butylphenol).(formaldehyde) (copolymer), 25085-50-1;p-tert-butylphenol-formaldehydedimer, 102699-84-3;p-tert-butylphenol-formaldehydetrimer, 10269985-4; p-tert-butylphenol-formaldehydetetramer, 102699-86-5; p-tert-butylphenol-formaldehyde pentamer, 102699-87-6; p tert-butylphenol-formaldehydeheptamer, 102699-88-7; calix[4]arene derivative, 60705-62-6; calix[5]arene derivative, 8147522-1; dx[61arene derivative, 78092-53-2; &[7]arene derivative, 84161-29-5;calix[8]arene derivative, 68971-82-4. LITERATURE CITED (1) Ludwig, F. J.; Baiile, A. 0. AM/. Chem. 1984, 56. 2081-2085. (2) Gutsche, C. D. Acc. Chem. Res. 1983. 16, 161-170. (3) Gutsche, C. D. In Toplcs in Current Chemishy: Springer-Verlag: Beriln, 1984 Voi. 123, pp 6-9. (4) Bwiks, R. S.; Fauke, A. R.; Munch, J. H. US. Patent 4259464, 1981. (5) DeQroote, M.; Kelser, B. US. Patent 2499370, 1950. (6) Finn, S. R.; Lewis, G. J. J . SOC. Chem. Ind., London 1950, 69, 132- 133. (7) Hiyes,-B. T.; Hunter, R. F. J . Appi. Chem. 1958, 6 , 743. (8) Foster, H. M.; Hein, D. W. J . Org. Chem. 1961, 26, 2539-2541.
RECEIVED for review March 20,1986. Accepted April 23,1986. This work constitutes a partial fulfillment of Petrolite Corporation's commitment under National Science Foundation Industry/University Cooperative Research Grant No. CHE8216719 awarded to C. D. Gutsche of Washington University, St. Louis, MO.
Multivariate Relationships between Gas Chromatographic Retention Index and Molecular Connectivity of Mononitrated Polycyclic Aromatic Hydrocarbons Albert Robbat, Jr.,* Nicholas P. Corso, Philip J. Doherty,' and Durwood Marshall
Department of Chemistry, Tufts University, Medford, Massachusetts 02155
P-r of gas chm&o@raphic nt.nlkn Ind0x.e ( I ) was made by uae of a muttlvarlate h e a r rdatlarwhip between I and mX' -( dmuwrsI lXV,'XV, 5XvlOx, 1 x, and ' x ) for monoMcatad polycycHc aromatic hydrocarbons, on an SE-52 drtkrrcuy phme. A multkarlste hear cdbration model (Le., Inverse eatlmate) betweon predicted I and "'xt (Oxv,'x', 'xV, Ox, 'x, 2x, and ax)to predlct x estlmatcrr was used to vdldate the pr.dlct.d I . The differencenchd x value8 and prodktd x was approxknately 1%
.
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0003-2700/86/0358-2072$01.50/0
The facile nitration (NOz) of polycyclic aromatic hydrocarbons (PAH) in the presence of nitrating agents and literature reporta of the mutagenic (1-3)and carcinogenic activity (4,5) of some of the nitrated PAH have produced considerable interest in the development of analytical methods for the determination of these compounds in environmentally significant samples. Recent reviews (6, 7) have summarized current state-of-the-art analytical methods. We have shown that high-resolutioncapillary gas chromatographic separation on fused silica SE-52 column has unmatched separation capability for nitrated PAH in complex environmental samples
0 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 9, AUGUST 1986
Table I. Retention Indexes and no."
1 2* 3 4* 5 6* 7* 8* 9* 10 11 12* 13* 14* 15 16* 17* 18 19* 20* 21* 22* 23 24 25* 26* 27* 28* 29* 30* 31* 32* a
2073
Values of Mononitrated PAH
m ~ t
compounds 5-nitroindan 1-nitronaphthalene 1-nitro-2-methylnaphthalene 2-nitronaphthalene 2-nitrobiphenyl 3-nitrobiphenyl 4-nitrobiphenyl 7-nitro-1-tetralone 4-nitrophenyl phenylether 3-nitrofluorene 3-nitrodibenzofuron 5-nitroacenaphthene 2-nitrofluorene 3-nitro-9-fluorenone 4-nitrophenanthrene 9-nitroanthracene 9-nitrophenanthrene 2-nitro-9-fluorenone 1-methyl-9-nitroanthracene 3-nitrophenanthrene 1-methyl-10-nitroanthracene 1-methyl-9-nitrophenanthrene 9-methyl-10-nitroanthracene 1-nitrofluoranthene 7-nitrofluoranthrene 2-nitrofluoranthene 3-nitrofluoranthene 4-nitropyrene 1-nitropyrene 2-nitropyrene 4-nitro-p-terphenyl 6-nitrochrysene
4402
oX v
lXV
2Xv
OX
'X
*X
3x
187.52 200.00 205.38 207.30 217.20 239.64 244.47 244.64 248.12 260.30 265.93 269.97 284.69 287.74 288.44 289.54 300.00 300.45 305.23 307.66 311.06 320.57 321.60 342.45 344.85 350.17 351.97 354.26 360.78 363.02 377.01 400.00
6.617 6.805 7.728 6.805 7.960 7.960 7.960 7.525 8.368 8.512 8.213 8.065 8.512 8.713 8.960 8.960 8.960 8.713 9.883 8.960 9.883 9.883 9.883 9.960 9.960 9.960 9.960 9.960 9.960 9.960 11.269 11.244
4.034 3.910 4.327 3.904 4.577 4.571 4.571 4.488 4.729 5.111 4.812 4.951 5.111 5.108 5.321 5.320 5.321 5.108 5.737 5.315 5.737 5.737 5.743 6.071 6.071 6.065 6.071 6.065 6.065 6.059 6.642 6.865
3.064 2.755 3.188 2.789 3.140 3.173 3.170 3.375 3.166 3.932 3.536 3.841 3.932 3.862 3.919 3.925 3.920 3.862 4.376 3.949 4.373 4.368 4.321 4.666 4.666 4.700 4.663 4.701 4.698 4.734 4.747 5.113
8.552 9.259 10.130 9.259 10.673 10.673 10.673 10.129 11.380 11.121 11.121 10.414 11.121 11.991 11.828 11.828 11.828 11.991 12.698 11.828 12.698 12.698 12.698 12.983 12.983 12.983 12.983 12.983 12.983 12.983 14.656 14.234
5.771 6.288 6.698 6.271 7.288 7.271 7.271 6.682 7.754 7.754 7.754 7.271 7.754 8.181 8.271 8.271 8.271 8.181 8.682 8.254 8.682 8.682 8.698 9.271 9.271 9.254 9.271 9.254 9.254 9.237 10.237 10.381
5.268 5.547 6.053 5.622 6.254 6.329 6.317 6.149 6.765 7.185 7.185 6.755 7.185 7.616 7.462 7.474 7.464 7.616 8.011 7.527 8.001 7.992 7.905 8.606 8.606 8.684 8.597 8.682 8.670 8.756 8.992 9.010
4.273 4.486 5.046 4.523 4.986 5.015 5.076 4.997 5.226 6.155 6.155 5.881 6.148 6.714 6.378 6.300 6.374 6.714 6.718 6.458 6.779 6.853 6.859 7.736 7.722 7.743 7.772 7.723 7.778 7.742 7.559 8.203
Asterisk indicates compounds found in model I.
analytical reference standards are commercially available making positive identification extremely difficult. Over loo0 isomeric nitrated PAH having between two and five condensed rings and one and four nitro substitutions theoretically exist. Retention in gas chromatography is the result of competitive solubility of the solute between mobile and stationary phases. The interaction (partition) of solute between these two phases is determined by the molecular structure and chemical properties of the solute. These differences govern the retention behavior through the column. Molecular connectivity, m ~ t devised , by Randic (10)and expanded by Kier and Hall (11) is a topological description of molecular structure based on a count of skeletal atom groupings, weighted by the degree of branching. Molecular connectivity has been used to correlate molecular structure with a number of physical and biological properties such as solubility parameters (12),boiling points (12),densities (13),and partition coefficients (14). In the past few years, numerous investigators (15-21)have shown good correlation between observed retention indexes and molecular connectivitiy descriptors. In an earlier paper, we reported the relationship between retention index and molecular connectivity for nitrated PAH (21). A linear relationship was obtained between I and lxV for mononitrated PAH. For dinitrated PAH, on the other hand, statistically equivalent single descriptor linear relationships were obtained for both I vs. 3xvand I vs. 4 ~ c v . I t must be emphasized t h a t even though the correlation coefficients of the univariate linear least-square regression lines of retention index vs. molecular connectivity are high, the accuracy of the predictions is not high enough to make the technique of practical use for identification purposes. Some of the predicted retention indexes are 10 or more index units away from the observed value. Some isomers have nearly the same predicted retention index even though their observed
retention indexes are markedly different. However, the single variate molecular connectivity descriptor serves as a useful screening tool for obtaining approximate retention behavior. For example, the relationship can be used as an aid to the mass spedrometrist in narrowing potential nitrated PAH candidates with respect to a n elution time domain. A more useful relationship is one that successfully predicts both the retention index and the experimental elution sequence. A single topological descriptor does not encode sufficient information with respect to gaslliquid partitioning of the analyte. In this paper, we report two multivariate approaches based on molecular connectivity that describe the retention characteristics of nitrated PAH on SE-52 stationary phase. The first approach (model I) uses stepwise linear regression to obtain an equation between I and m ~ that t successfully predicts both the retention index and elution sequence. A six-descriptor equation was obtained that fits this criteria. Six descriptors encode sufficient information about mononitrated PAH such that 19 out of 23 compounds in the data set were predicted within f2.1 index units of their observed value. The second multivariate approach (model 11) uses the linear-calibration (Le,, inverse regression) framework to estimate a vector of molecular connectivity descriptors as a function of retention index. The linear-calibration model depends on a multiple-linear regression and a multivariate estimation of location and scale for the joint distribution of the molecular connectivity descriptors. Corrected inverse estimates based on the model Oxv, 'xV, 'x", Ox, 'x, 2x, and 3x for a given I is within f l % of the actual molecular connectivity descriptors.
EXPERIMENTAL SECTION Relevant molecular connectivities and retention indexes for mononitrated PAH are listed in Table I. Some of the Z values in Table I were calculated from a plot of ZNO2 vs. ZPAH, based on
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 9, AUGUST 1986
common compounds. The following relationship was obtained: INop= 1 . 0 8 I p -~ 95.00, ~ where N = 32, r = 0.999, and s = 1.24. INo2values were taken from ref 8 where 1-nitronaphthalene, 9-nitrophenanthrene, and 6-nitrochrysene were the bracketing standards used to calculate the retention index in the Van Den values, taken from ref 9, Do01 and Kratz relationship. The IpAH were calculated in a similar fashion using naphthalene, phenanthrene, chrysene, and picene as the bracketing standards. Molecular connectivities, x, are calculated by considering various fragments of a molecule where "'xtis used to designate a particular fragment. The superscript t describes the fragment type. Fragment types can either be paths of one or more bonds, clusters (c) where three atoms are adjacent to a single common atom (occurs in condensed ring systems and points of branching), or path clusters (pc), which are similar to a cluster with the addition of one more atom that is connected to any of the three atoms forming the cluster. Path terms describe the general bulkiness of a molecule, while cluster and path cluster terms describe more specifically branching in the condensed ring system. A valence value (AV) equal to the valence of an atom minus the number of hydrogen atoms attached to that atom or a nonvalence value (a), the number of non-hydrogen atoms connected to the atom of interest, is assigned to each atom in a molecule excluding hydrogen atoms. Nonvalence molecular connectivity indexes are calculated by using the 6 values where all non-hydrogen atoms and bonds between them are considered to be identical. Valence molecular connectivity indexes are calculated by using 6v values, which account for the nature of non-hydrogen atoms and the differences in bond orders. Detailed examples on calculating x terms have been published elsewhere (8,11,21-23). Molecular connectivity terms encode informationabout the number of atoms, type of atoms, bonding of atoms, and cyclization in a molecule. The molecular connectivity terms were calculated by programs supplied by Lowell Hall (Eastern Nazerene College) and computed on a Digital Equipment Corp. Series 10 computer. As described in ref 21, a total of 18 x descriptors, Oxvto 'x', 3xcv,'xpCvand Ox to 6x,3xc,'xpCwere generated for each nitrated PAH in the data set. RESULTS AND DISCUSSION The best single descriptor equation correlated I with lxV, viz., INo2= 67.71'~'- 63.52, r = 0.988 and x = 8.8 (eq 1). The slope was similar to that reported earlier (21). The intercept, however, was shifted approximately -56 index units and a decrease in the standard error of estimate of the line (s = 8.8 vs. 11.6) was observed, as a result of using nitrated PAH bracketing reference standards rather than the PAH brackets. The large standard error did not predict the correct experimental elution sequence. Nevertheless, using nitrated bracketing standards for calculating Z improved the residuals, i.e., the experimental minus predicted values. Two multivariant models that better differentiated isomeric nitro-PAH and elution order were developed. The first model used molecular connectivities to calculate retention index. The second model used inverse estimates for calculating x's based on experimental I values. Model I. Stepwise multiple linear regression was used to generate molecular connectivity equations which predicted the retention characteristics of mononitrated PAH. The goal was to minimize the number of descriptors required to obtain statistically significant I values and to determine the experimentally obtained elution sequence. The multivariate regression of I with six m ~ descriptors t is presented in Table 11, eq 2. The predicted I values and their respective residuals for this equation are shown in Table 111. The identity of the 23 mononitrated PAH used in this model is denoted in Table I. It is evident from Table 111, that 19 out of 23 mononitrated PAH are predicted within f2.1 index units of their observed Z values. Only two compounds, 7-nitro-1-tetralone and 2nitrofluoranthene had elution sequence misplacements. The descriptors are in the order of statistical significance. The first descriptor lx' is the x term most highly correlated with retention indexes of mononitrated PAH (as discussed
Table 11. Multivariant Relationship for Predioting Retention Characteristics of Mononitrated PAH
constant 'X"
3x OX 4 v
X
SxV
'X
S F statistic DF MR N
eq 2
eq 3
21.345 135.646 60.544 -31.902 124.650 56.330 -33.096 2.281 22204 (6, 16) 0.999 23
12.750 100.461 42.153 -33.407 -93.890 51.130 3.533 1108 (5, 17) 0.998 23
Table 111. Predicted Z and Residual for Mononitrated PAH six descriptors (eq 2)
five descriptors (eq 3)
compound
pred I
residual
pred I
residual
2 4 6 7 8 9 12 13 14 16 17 19 20
200.13 211.04 239.31 244.25 242.58 247.93 270.13 284.22 286.21 289.91 295.31 305.20 307.38 314.53 319.81 345.70 354.06 350.29 356.21 359.61 361.89 378.00 398.94
-0.13 -3.74 0.33 0.22 2.11 0.19 -0.16 0.47 1.53 -0.37 4.69 -0.05 0.28 -3.47 0.76 -0.85 -3.89 1.67 -1.95 1.17 1.13 -1.00 1.06
203.00 210.58 241.26 244.50 236.85 250.92 269.18 283.48 284.82 293.22 297.15 306.21 305.08 312.86 316.69 348.70 354.35 351.80 356.32 358.41 359.87 375.02 402.48
-3.00 -3.28 -1.62 -0.03 7.85 -2.80 0.79 1.21 2.92 -3.67 2.85 -0.98 2.58 -1.80 3.88 -3.85 -4.18 0.16 -2.06 2.37 3.15 1.99 -2.48
21 22
25 26 27 28 29 30 31 32
above). lxV, a valence descriptor, describes the general bulkiness, since its numerical value is proportional to the number of bonds, as well as information about the nature of the atoms in the molecule. Equation 2 depends mostly on the lxVdescriptor and, thus, is fine-tuned by the remaining five descriptors. 3x encodes information about two adjacent bonds and the number of atoms bonded to each atom in the fragment. Ox contains information about the number of atoms each atom is bonded to but contains no information about the types of atoms. 4xvand 5xvdescribe all the paths containing five and six adjacent atoms, respectively, and information about the nature of the atoms in each fragment. Some of the paths of five and six adjacent atoms contain information about the cyclization in the nitrated PAH. 'xVdescribes the fivemember rings in fluorenes, fluorenones, and fluoranthenes b
while
A
5xvdescribes six-member rings
'x, like lxV,encodes information about the number of bonds; on the other hand, it does not consider the nature of the atoms in the molecule. This equation consists of a combination of
ANALYTICAL CHEMISTRY, VOL. 58, NO. 9, AUGUST 1986
valence descriptors and nonvalence descriptors. Equations based solely on either xv or x did not meet the criteria set for the model. Cluster x descriptors, which describe the vertices of a ring system and nitro-substitution branches, did not add sufficient information to warrant inclusion into the equation. Retention of mononitrated PAH on SE-52 is a function of the solute vapor pressure (and thus the boiling point (8)), the dipole moment (8),the size of the compound, and is slightly dependent on fragment type and position of the nitro group attached to the parent PAH. The best five-descriptor combination, eq 3, is shown in Table 11. Table I11 illustrates that an additional compound is misplaced in the elution sequence as well as a general increase in residuals. Relationships of less than five descriptors are statistically equivalent to the single variate case. The best seven x descriptor relationship produced residuals no better than eq 2. Nine nitro-PAH did not fit the criteria set forth for the model. Equation 2 predicts later elution for 2-nitrobiphenyl, 4-nitrophenanthrene, and 1-nitrofluoranthene than experimentally observed. The 2-nitrobiphenylis markedly separated from 3-nitrobiphenyland 4-nitrobiphenyl. We have previously shown that retention of nitrated PAH on SE-52 is a function of solute vapor pressure and, thus, of the boiling point (8). The reduced retention time of 2-nitrobiphenyl is attributed to the nitro group’s forcing of the two phenyl rings out of a common plane, and thus a reduced A electron overlap. This additional steric hindrance between the two phenyl ring systems is not present in either 3-nitro- or 4-nitrobiphenyl. As a result, the vapor pressure of 2-nitrobiphenyl is higher than that of 3-nitro- and 4-nitrobiphenyl. In 4-nitrophenanthrene the nitro group is in the bay region and twisted out of the plane with the parent phenanthrene structure. Thus, the 4-nitrophenanthrene has a higher vapor pressure than the other isomeric nitrated phenanthrenes and is eluted sooner than predicted. The nitro group in 1-nitrofluoranthene is also in the bay region resulting in elution prior to other isomeric nitrated fluoranthenes. We suspect that the magnitude of the residuals for these compounds can be interpreted as a measure of the steric interaction in the molecule. The residual for 2-nitrobiphenyl, where phenyl rings are twisted out of plane, is approximately twice the magnitude of the compounds that have the nitro group in the bay region. The experimental I for 1-nitro-2-methylnaphthalene is 205.38 while I predicted is 224.05. The predicted value implies that 1-nitro-2-methylnaphthaleneshould elute after 2-nitronaphthalene due to its higher molecular weight. The nitro group in the f i t compound experiences steric interaction from both the adjacent methyl group and the peri hydrogen atom. This causes the nitro group to twist out of the naphthalene ring system plane. Consistent with the above findings, the vapor pressure of 1-nitro-2-methylnaphthaleneis higher than expected resulting in elution from the chromographiccolumn sooner than predicted. One might also expect that the added twisting of the nitro group in this compound as compared to 1-nitronaphthalene, where the nitro group is twisted from the naphthalene plane by peri hydrogen, should lead to 1-nitro2-methylnaphthalene eluting prior to 1-nitronaphthalene. Experimentally this elution order is not observed. The inductive nature of the methyl substituent ortho to the nitro group adds to the electron density of the aromatic ring system. The nitro group attracts the added complement of electrons. The resulting greater electron density of NOz in 1-nitro-2methylnaphthalene as compared to NOz in 1-nitronaphthalene leads to a lower vapor pressure and, thus, a longer elution time. Two other nitro compounds in the list, 5-nitroindan and 3nitrodibenzofuran, are not condensed polyaromatic hydrocarbons. The fact that molecular connectivity is a topological
2075
descriptor of molecules in two dimensions and that steric interactions as well as electronic properties play a role in the retention on SE-52 of some nitro-PAH, limits the applicability of eq 2 to predict I for all nitro compounds within the criteria set forth in the model. Model 11. An alternative to describing the retention index as some function of x descriptors was provided by a linear calibration approach; i.e., the retention index was used to predict the x descriptors. This was accomplished by generalizing an inverse estimator known as the linear compound estimator (24). This generalization to the multiple variable case x ’ = (xl,xz,..., xkl)was given by X = X(Ii) = R (constant)Z@
+
where X is a k X 1vector of estimated x variables, Ii is the retention index for compound i = 1,2, ...,n, f is a k X 1 vector of sample means for the k x variables (used in the regression) given by f’ = (ave (xl),ave (x2),..., ave (xk))and ave the sample mean operator, is a k x k sample covariance matrix of the x variables, is a k X 1vector of regression coefficients, Po is the regression estimate for the fitted constant term, Sz is the mean squared error for regression, n is the sample size, and prime denotes the matrix transpose. The regression referred to was the retention index regressed on kx predictors. The sense of linear calibration assumed that y = ( f ( X )+ error term) that was linear in the p’s. Our choice of f ( X )was a hyperplane (i.e., linear in the x predictors). Initially, the choice of x descriptors that “best” predict the retention index in model I seemed to be the natural candidates to include in the inverse estimator. However, it turned out that these x descriptors possess a steplike structure different from what would have been expected if one sampled from a multivariate normal distribution in k dimensions. Even though the inverse estimate, eq 4, was applicable for nonnormal distributions generating the x vectors, it was not designed for nonstocastically structured data. This steplike structure was evident in plots of the standarized x variable Cj(i),eq 5 vs. the retention index I i for i = 1, 2, ..., n
Cj(i)= (xj(i)- ave (xj(i)))/std dev (Xj(i))
(5)
where j = 1, 2, ..., k . The inclusion of x variables as given in model I for the inverse estimator yielded estimates with systematic over and under predictions. This systematic bias in the predicted x variables was related to the steplike structure standardized plot shown in Figure 1. The errors in the inverse estimates in eq 4 that corresponded to the compounds that were furthest from the regression line for CG)vs. Iiare typically f20-30%. The worst cases being over f100%. Notice of this prompted the “natural” set of x variables to be those whose standardized plots exhibited little step-structure (i.e., Oxv,‘xV,‘xV,Ox,‘x, 2x,and 3x with adjusted R2 2 0.9). This choice of It = 7x variables had a minimum of steplike structure and overlapping x values. The descriptors selected based on this criterion decreased the percent error by approximately a factor of 6. The typical error was f0.5-5%, while the worse case was less than 10% . Nevertheless, this accuracy was still insufficient to differentiate between some adjacent compounds or isomers. This problem of differentiation was further compounded by the fact that our data set contained only 32 compounds. In many cases eq 4 can narrow the 32 compounds to several candidates. On the other hand, since there exists isomers not in the set of 32 which can have similar retention indexes and x variables, this can cause indeterminices in the compound’s identity. In an effort to reduce the error, it seemed reasonable to take advantage of the minimum steplike structure in the
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 9, AUGUST 1986
Table IV. Percent Differences" for Corrected INOg
207.30 244.47 265.93 284.69 344.85 351.96 360.78 400.00 287.74 288.44 289.54 300.00 305.23 307.66 311.06 a%
o v
X
'XV
1.39 0.09 0.54 0.72 0.09 0.39 0.77 0.26 0.36 0.21 0.16 0.34 1.46 0.71 1.21
1.31 0.33 0.58 0.36 0.19 0.18 0.67 0.28 0.57 0.34 0.27 0.35 1.06 0.84 0.74
x Estimates of Some Mononitrated PAH Bared on Equation 4 z v
X
OX
'X
zX
3x
3.14 3.45 1.53 0.75 1.05 0.20 0.46 0.75 0.50 0.09 0.01 1.48 2.88 2.16 2.14
1.80 0.17 0.04 1.46 0.10 0.56 1.13 0.25 0.99 0.47 0.39 0.36 1.74 0.90 1.35
0.82 0.03 0.14 0.60 0.19 0.05 0.37 0.28 0.17 0.32 0.28 0.11 0.49 0.42 0.28
0.65 0.58 0.31 0.28 0.21 0.01 0.11 0.63 0.34 0.08 0.06 0.27 0.44 0.40 0.25
0.85 1.22 0.62 0.26 0.61 0.44 0.13 0.47 0.84 0.12 0.12 0.41 0.15 0.55 0.42
difference = I[(actual x value) - (corrected x estimate)]/(actual
x value11 x
100.
Table V. Sample Covariance Matrix, Z
OXV
'X" 2X v
OX
'X zX
3x
oXv
IXV
2Xv
OX
'X
zX
3x
1.440 00 0.918 36 0.756 43 1.69894 1.33690 1.26450 1.29133
0.60062 0.50442 1.07723 0.861 97 0.826 50 0.85837
0.440 90 0.87835 0.707 77 0.702 80 0.745 23
2.044 90 1.602 37 1.514 64 1.54113
1.276 90 1.21034 1.24570
1.18810 1.240 15
1.32250
Table VI. Predicted Z for Mononitrated PAH and Corresponding Predicted x's compound
Iprd
OXV
'XV
l-N02-2-CHS-naphthalene
224.05
7.88P 1.97b 8.270 2.84 8.344 1.97 8.419 2.50 8.973 0.14 8.993 0.37 9.013 0.59 9.082 1.36 9.763 1.21 9.782 1.02 9.785 0.99 10.021 0.61 10.022 0.62
4.626 6.91 4.983 2.62 5.010 2.09 5.037 4.68 5.315 0.11 5.300 0.28 5.281 0.63 5.321 0.23 5.780 0.84 5.766 0.59 5.763 0.54 6.036 0.48 6.036 0.48
4-NOz-fluorene
273.90
1-NO2-fluorene
276.53
3-NOz-dibenzofuran
279.11
l-NOZ-phenanthrene
295.91
1-NOz-anthracene
300.64
2-NOz-phenanthrene
305.15
2-NOZ-anthracene
310.01
3-CH3-9-N02-anthracene
311.03
2-CH,-9-NOz-anthracene
315.57
2-CH3-9-NOZ-phenanthrene
316.30
8-NOz-fluoranthene
357.28
9-NOz-fluoranthene
357.32
'Predicted
x estimates.
z Xv
3.140 1.50
3.922 0.51 3.931 0.95 3.941 11.46 3.887 0.74 3.862 2.35 3.863 2.18 3.917 1.78 4.467 0.88 4.440 0.27 4.436 0.29 4.698 0.04 4.698 0.04
OX
'X
zX
3x
10.483 3.48 10.859 2.36 10.962 1.43 11.067 0.49 11.836 0.07 11.875 0.40 11.913 0.72 12.001 1.46 12.525 1.36 12.563 1.06 12.569 1.02 13.099 0.89 13.100 0.90
7.221 7.81 7.472 3.85 7.552 2.82 7.633 1.56 8.267 0.05 8.281 0.33 8.286 0.39 8.349 1.36 8.657 0.09 8.671 0.07 8.673 0.09 9.277 0.25 9.277 0.25
6.272 3.62 6.905 3.03 6.978 1.86 7.052 1.85 7.478 0.35 7.490 0.65 7.533 0.08 7.621 1.05 7.983 1.54 7.987 1.49 7.988 1.36 8.677 0.06 8.677 0.06
5.01 1 6.94 5.968 1.76 6.016 1.76 6.064 1.48 6.364 0.87 6.408 0.58 6.463 0.19 6.567 2.59 6.766 1.97 6.817 2.63 6.824 1.71 7.731 0.83 7.732 0.83
* % difference = I[(actual x value) - (predicted x estimate)]/(actual x value)) X 100.
X vectors. Note that for the case of no structure in a data set, the inverse estimate (eq 4) was optimal and no correction term was needed (24). However, computation of the residuals from the univariate regressions of the stanx variables Cj(i)regressed on the retention index Zi resulted in the proper sign and magnitude necessary to lessen the structure in the inverse estimates. We defiie the k X 1 correction vector C,(Zi) as CJIi)' = (Cj(9Rl(Ii), Cj(i)Rz(Zi), * * * ) Cj(i)Rj(Zi), * * * I Cj(i)Rk(Zi)) (6)
where Cj(i)Rj(Zi)is the standardized x residual for x variable j = 1, 2, ...,k with retention Zb When the inverse estimate was close to the actual x vedor, the correction vector was small preserving the estimate. This correction approach yielded typical errors of estimated x variables
