640
Anal. Chem. 1985, 57,640-648
Mathematical Modeling of Temperature Programmed Capillary Gas Chromatographic Retention Indexes for Polychlorinated Dibenzofurans Michael D. Hale, Fred D. Hileman,* Thomas Mazer,l Traci L. Shell, Roy W. Noble, and Joseph J. Brooks Monsanto Company, Daytort Laboratory, 1515 Nicholas Road, Dayton, Ohio 45418
A model is developed describing the relationship between retenflon characterlstlcs on a gas chromatographic column and molecular structure of the chlorinated dlbentofurans. Model varlabies account for (1) number and posltlon of chlorlnes present, (2) relations between chlorines on each aromatlc rlng, and (3) Interactions between the two rlngs. The model has proven to bB useful for validation and predlctlon, and evldence suggests that the general principles may be applied to structurally similar compounds.
A number of techniques have been used for the analysis of polychlorinated dibenzofurans (PCDFs) (1-4), but gas chromatography remains the primary separation technique used for their analysis. Gas chromatographic retention parameters are typically used to qualitatively identify which PCDF congeners are present in the sample being analyzed. Mazer et al. recently reported on the preparation of the 38 tetrachlorodibenzofurans (TCDFs) and the determination of their retention characteristics on two different chromatographic phases (5). On review of those results, the retention data showed definite patterns which could be interpreted from a molecular structure viewpoint (6). This kind of approach in chromatographic data interpretation has gained widespread use (7-9),and it is believed that a mathematical model that describes the pattern in these retention data would be of great analytical value. First, it could be used to examine the retention data for consistency. This is important because of the difficulties associated with preparing and then positively identifying any particular PCDF. Second, it could predict retention characteristics of PCDFs which had not yet been synthesized. With this method, it would be possible to greatly narrow the window of observation to search for and identify a given PCDF. The work described here involves the development of a model describing the relationship between the molecular structure of a PCDF and its retention index on a nonpolar capillary gas chromatography column. The descriptive structural parameters are, for the most part, directly related to conventional organic structural relationships, such as ortho, meta, and para orientation of chlorine atoms about an aromatic ring. Using the technique described here, one can predict the retention index for any given PCDF having only a knowledge of its chlorine substitution pattern. EXPERIMENTAL SECTION Congener Synthesis. Of the 135 different PCDFs, 110 were prepared by using the techniques described previously ( 5 , 6 )or obtained from commercial or government sources. Briefly, the synthesis involved the careful oxidative thermal degradation of specific polychlorinated biphenyl congeners to form PCDFs via the mechanisms proposed by Buser (10). Additional PCDFs were formed by either chlorination of a selected PCDF congener using Present address: Brehm Laboratory, Wright State University, Dayton, OH 45435.
antimony pentachloride or by dechlorinationof a selected PCDF congener using ultraviolet light. The various PCDFs formed in these reactions were continually cross correlated with each other to ensure that the proper identification had been made. Data Collection. A Hewlett-Packard 5985B GC/MS system with a 30 m J & W DB-5 fused silica capillary column operated in the splitless injection mode was used to analyze the various PCDF congeners. The column was temperature programmed from 175 "C with an initial hold of 1 min to 300 "C at a rate of 6 "C/min. Helium was used as a carrier gas at 7 psi head pressure and the column was directly coupled to the ion source of the mass spectrometer. The mass spectrometer was operated in the selected ion monitoring mode following ions characteristic of monochlorothrough octachlorodibenzofuran. In addition, a mixture of normal ~ H ~ ~ tetra) hydrocarbons, including octadecane ( ~ I - C ~ through tricontane (~Z-C~H,,,), was obtained from Supelco and coinjected with each sample. Normal hydrocarbons lower than octadecane were readily observed as trace impurities in the dodecane used as the splitless injection solvent. The mass spectrometer also monitored an ion characteristic of these normal hydrocarbons. Thus, a scale was built into every GC/MS analysis allowing a hydrocarbon number (retention index) to be assigned to a PCDF as shown in Figure 1. In some situations, the identification of a specific PCDF congener on the DB-5 column was ambiguous,requiring independent confirmation. These cases were most often resolved by analyzing the PCDFs in question on 8i highly polar SP-2330 fused silica capillary column also obtained from Supelco. Though providing better discriminating power in most cases, this type of column had difficulty in eluting the entire range of congeners being studied without going to extended isothermal temperature holds at the column's upper temperature limit. The temperature programmed retention index system was chosen for this work (11). This retention index (RI) is calculated by using the following equation:
where T,(PCDF) is the retention time of the PCDF in question and T,(C,) and Tr(CZ+Jare the retention times of the normal hydrocarbons bracketing the PCDF with carbon numbers z and z plus one. The temperature program used with the DB-5 column was chosen so that the monochloro- through octachlorodibenzofurans would elute without experiencing excessively long retention times. In addition, the linear temperature program that was employed had the hydrocarbons eluting linearly in the retention range for all PCDFs. Data Analysis. Regression analysis was the primary method for examining the retention data. PROC GLM and PROC NLIN of the Statistical Analysis System (SAS) were used to perform the required analyses (12).
RESULTS AND DISCUSSION First Steps. In order to describe the retention index of a chlorinated dibenzofuran molecule as a function of structure, it is helpful to consider that diben~ofuransare symmetrical about a plane dividing the molecule into two parts. Those two parts are the two aromatic rings (arbitrarily labeled as ring 1 and ring 2), with the plane dividing the molecule through the biphenyl and ether linkages. If each half of the
0003-2700/85/0357-0640$01.50/00 1985 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 57, NO. 3, MARCH 1985
1247 - TCDF (22691
2400
2300
e 6
7
8
9
1
11
12
RETENTION TIME (minutes)
Flgure 1. Determination of the retention index for 1247-TCDF. CI
CI
CI
2,4,6,8-TCDF or 2,4,2',4'-TCDF
Figure 2. Comparison of the conventional numbering scheme (a)and the numbering scheme used in this work (b) for chlorinated dibenzofurans.
molecule (each aromatic ring) contributes to the retention index in an additive manner, then a very simple mathematical model retention index = ring 1 effect ring 2 effect + random variation (Model 1) can be used to describe the retention index as a function of structure. This approach has been used before in an independent work, where a similar model was used to describe the retention behavior of chlorinated biphenyls (13). The random variation term includes instrument, sampling, and all of the other variabilities involved in making a measurement and will ,hereafter be referred to simply as "variation". The numbering scheme normally used for PCDFs with one ring numbered 1-4 and the other ring as 6-9 obscures the similarity of the two aromatic rings as shown in Figure 2a. Thus, throughout this work the positions numbered 1,2,3,4 were retained and positions 6 through 9 were renumbered using 1',2',3',4'. With this notation, 2468-TCDF becomes 242'4'-TCDF as shown in Figure 2b. Every congener has a unique representation using this convention, in which the numbers are listed in ascending order, with nonprimed numbers before primed numbers. This notation makes it easy to identify the inter- and intra-ring relationships used in the mathematical model described here and emphasizes the similarity of the two rings. The TCDFs provide a good starting place for testing this simple model since their complete retention characteristics have already been established (5). Additionally, they yield retention indexes which cover the middle of the range of the retention indexes for the PCDFs. More importantly, the TCDFs have representative examples of all possible substitution patterns for an individual ring. That is, eve& possible substitution pattern for an aromatic ring in a PCDF can be found in one (or more) of the TCDFs. The first step in the search for patterns linking retention behavior with structural characteristics was the construction of Table I, which organizes the 2:2 substituted TCDFs (those
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641
Table I. Observed Retention Indexes for the 2:2 Substituted TCDFs ring 2
substitution pattern
12
13
ring 1 substitution pattern 14
23
24
34
12 13 14 23 24 34
2406 2341 2364 2322 2281 2329
2341 2273 2296 2263 2226 2272
2364 2296 2314 2290 2242 2288
2322 2263 2290 2338 2298 2354
2281 2226 2242 2298 2254 2304
2329 2272 2288 2354 2304 2362
grand mean
2302
Table 11. First Residuals for the Decomposition of the Retention Indexes for the 2:2 Substituted TCDFs ring 2
substitution pattern
12
13
12 13 14 23 24 34 column mean
104 39 62 20 -21 27 38
39 -29 -6 -39 -76 -30 -24
ring 1 substitution pattern 14 23 24 34 row mean
62 -6 12 -12 -60 -14 -3
20 -39 -12 -36 -4 52 8
-21 -76 -60 -4 -48 2 -35
27 -30 -14 52 2 60 16
38 -24 -3 8 -35 16
isomers which have two chlorine substitutions on each of the aromatic rings). The table was structured so that the retention index for a given 2 2 substituted TCDF isomer could be found at the intersection of the column and row which have the appropriate substitution patterns for ring 1 and ring 2, respectively. Note that this organization makes the table symmetrical about the principal diagonal, simplifying the calculations and aiding in the identification of patterns. First, a grand mean was computed for Table I. This value, 2302, was then subtracted from each entry in Table I, leaving residuals as shown in Table 11. This operation corresponds to fitting a model retention index = constant (grand mean) variation (residual)
+
Row and column averages were computed for those residuals and are listed in the right-hand column and bottom row of Table 11. (Note: column i mean = row i mean, by construction.) Substitution patterns with comparable row mean values are grouped together in Figure 3, revealing a definite pattern. The substitutions with the greatest average negative residuals from the grand mean (13-substitution and 24-substitution) contain neither vicinal chlorines nor hydrogens (Figure 3a). Likewise, substitutions with the greatest average positive residuals have both vicinal chlorines and hydrogens (Figure 3c). Finally, substitutions with average residuals near zero have only vicinal chlorines or vicinal hydrogens (Figure 3b). One interpretation of these data is that vicinal-like atoms increase the retention of a TCDF. To refine the model, a methodology was employed (14) in which new residuals were calculated for each entry in Table I by adding the appropriate row and column mean from Table I1 to the grand mean and subtracting this from the observed values in Table I. Table I11 shows the results of these calculations. This procedure has now actually yielded an implementation of model 1. The row (or column) mean for a particular substitution pattern provides an estimate of the retention effect for that pattern. retention index = constant (grand mean) ring 1 effect (column mean) + ring 2 effect (row mean) + variation (residual)
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 3, MARCH 1985 H
CI
-35 c
-24
CI
i
H
a
0
H
CI
la) Negative means No like vicinal atoms
H
Cl
-3 H
CI
0
CI
H
0
(b) Moderate means Single pair of like vicinal atoms
CI H
H
@
38
O
H
CI
IcI Positive means Two pair of like vicinal atoms
Flgure 3. Comparison of the row (column)means from Table I1 for the various chlorine and hydrogen substitution patterns.
Table 111. Decomposition of Retention Indexee for the 2:2 Substituted TCDFs” ring 2
substitution pattern
12
12 13 14 23 24 34
28 25 21 -21 -24 -21
ring 1 substitution pattern 13 14 23 24
25 19 21 -23 -17 -22
21 21 18 -17 -22 -21
-21
-24
-23 -17 20 23 28
-17
-22 23 22 21
~
34 -21 -22 -21 28 21 28
“Note: Rows and columns do not add to zero because of rounding. During actual calculations,carry decimal places instead of rounding to whole numbers. A serious deficiency in the simple additive model 1 can be observed in Table I11 by the obvious pattern of 3 X 3 blocks of residuals of the same sign. The pattern results from arranging the table so that the first three columns and first three rows all have a chlorine atom in the 1position, while the other rows and columns do not. This positive-negative pattern indicates an interaction between the two rings involving the 1 and 1’ positions, a feature not explained by the simple additive model 1. Thus, model 1does not provide an adequate description of the data because it assumes that the two halves of the PCDF behave independently with no provision for interaction between the two rings. Perhaps even more striking is that the residuals in Table I11 are all of approximately the same magnitude. This implies that model 1 would be significantly improved by including one additional factor to describe this inter-ring interaction. This simple refinement yields a better model retention index = constant ring 1 effect + ring 2 effect ring interaction effect + variation (Model 2)
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+
At this stage of the analysis, it was clear that there were at least two explanations for the interaction phenomenon. If both the 1and 1’positions were chlorinated, then steric crowding could distort the molecule from planarity, giving an increased retention index as shown by the positive residuals in the upper left quadrant of Table 111. This type of steric crowding has been suggested in other compounds containing the five-
member ring structure (15). Another possibility is that when only one of the 1 or 1’ positions is chlorine substituted, a hydrogen bonding effect between the hydrogen and the chlorine atoms (16) occupying those positions decreases chromatographic retention shown as the negative residuals in upper right and lower left quadrants of Table 111. Either of these two possibilities would cause the same pattern of residuals that is observed in Table 111. An independent method is needed to determine which type of interaction is actually occurring, because the type of mathematical model developed here cannot distinguish between those two cases. (In statistical parlance, the models are said to be linearly equivalent.) The question of which possibility is “correct”will not be addressed and both interactions will be illustrated by the working models presented here. Model Development. The methods developed in the previous section serve as good diagnostic tools. In fact, a spreadsheet program on a microcomputer may be used to good advantage, implementing this tabular approach in a “hands on” interactive environment. Further model development requires more powerful methods, however, and is better done using regression techniques. The regression analog of the tabular approach is implemented by using dummy variables to represent the possible substitution patterns. Each variable takes the values 0, 1,and 2, depending on whether that pattern does not occur, occurs once, or occurs twice, respectively, for the given PCDF. B y
creating variables this way, t h e regression coefficient for a variable is used as a n estimate of t h e effect for the corresponding chlorination pattern. The notation Ax is used to denote the dummy variable for the pattern x. As an example for 122’4’-TCDB, the variable A12 = 1,the variable A24 = 1, and all of the other variables = 0. To implement model 2, a variable representing the ringinteraction must also be included. The variable HBOND is created to model the latter possibility of inter-ring hydrogen bonding. This variable is -1 when only one chlorine has been substituted in either the 1or 1’position, and +1 otherwise. For the above case of 122’4’-TCDF, HBOND equals -1. The values 1and -1 were chosen for simple agreement with Table I11 so that the average size of the residual would yield the coefficient for HBOND. The values 1 and 0 work as well, doubling the size of the coefficient but are not intuitive from the table. Later in this description, one of the other possible explanations for interaction will be illustrated in a more refined model using the variable STERIC, which is 0, unless both the 1and 1’ positions are chlorine substituted, in which case STERIC is 1. Table IV shows regression coefficients for the various substitution patterns for all PCDFs using model 2. The size of the HBOND coefficient, -23 when present, 23 when absent, is quite notable and accounts for a good deal of the fine detail in the model. Without this term, the model degrades dramatically, showing little discriminating power among congeners with the same number of chlorine substitutions. A second ring-interaction variable, FOURFOUR, having a much smaller effect, i.e., smaller coefficient, has been introduced in Table IV. The need for this ring interaction variable only became apparent after all of the gross effects were identified and the residuals carefully examined. The variable FOURFOUR is 1when both 4 positions are chlorine substituted, and 0 otherwise. The variable FOURFOUR was constructed to correspond to the effect of the two chlorines blocking the interaction of the ether linkage oxygen with the chromatographic liquid phase. The decrease in this interaction then results in decreased retention (i.e., negative coefficient as shown in Table IV) of the PCDFs with this substitution pattern.
ANALYTICAL CHEMISTRY, VOL. 57, NO.3, MARCH 1985
Table V. Determination of Primary Intra-ring Interactions
Table IV. Regression Coefficients for Model 2 dummy variable
substitution pattern coefficient
constant (intercept)
none
A1 A2 A3 A4 A12 A13 A14 A23 A24 A34 A123 A124 A134 A234 A1234 HBONDb FOURFOURb
1 2 3 4 12 13 14 23 24 34 123 124 134 234 1234
a
1540 206 189 182 195 419 357 384 387 350 401 602 565 561 591 799 f23 -9
643
aV
A2 A3 A4 substitution A1 pattern (206) (189) (182) (195)
av A
model 2
A
388 384
357 350
-31 -34
395 371 377
419 387 401
+24 +16 +21 +24
401
384
-17
Meta 193
A13 A24
x
X X
X
-32
Ortho 383
A12 A23 A34
x
x x
x x
x Para
572 799
See text. *See text for an explanation of HBOND and FOUR-
FOUR.
To use the coefficients in Table IV, one identifies the pattern on each of the rings, selects the corresponding estimates of effect from the table, and adds those values together. To predict a retention index for 122’4’-TCDF, one would begin by taking the constant term (1540), add the coefficient calculated for the pattern 12 (419), add the coefficient calculated for the pattern 24 (350), and finally include the appropriate interaction terms. In this case, the HBOND coefficient (-23) is included and the FOURFOUR coefficient (-9) is excluded since only one 4-chlorine is present. The calculated value of 2286 compares well with the observed value of 2281. The agreement between observed and fitted indexes, the R2 value of 0.999, and the standard deviation of 6.3 all indicate that this model works quite well to describe the retention indexes of all analyzed PCDFs. Several of the congeners had been prepared via alternate routes so that several replicate values were used for the regression analysis. Thus, for model 2 there were 246 data values for 110 different congeners. No simple mathematical model will account for the variation observed between replicates and certainly would not be useful for prediction if it could. Although a “lack-of-fit”test indicates significant model inadequacy, many of the repeated values were not true replicates, causing the estimate of “pure error” to be much too small: The standard deviation of 6.3 is anticipated given the broad range of values present in this set of data. The initial data set covered retention indexes from approximately 1880 to 3140. A 95% confidence interval for a single future reading is approximately 25 units wide, so that the window of observation for a new PCDF can be considerably narrowed using model 2 and the values in Table IV. This model has proven useful in validating the structural assignments given to a PCDF. One 3 1 substituted TCDF did not fit the model and an error in structure assignment was detected and reported (17). This approach provided yet another check on the structural assignment given to a specific PCDF in addition to the cross correlations mentioned in the section on congener synthesis. Improving the Model. Although model 2 works well, it lacks the simplicity and unifying theme of a parsimonious description. A model with few simple descriptors, which could be identified with structural characteristics of the molecule, is highly desirable. To accomplish this, the coefficients in Table IV were carefully examined for patterns which might allow the further consolidationof parameters. One of the red advantages of the approach in the preceding section is that it effectively provided for the reduction of information from
A14
X
X
-17
110 congeners to a much smaller set of numbers, clearly associated with different chlorine substitution patterns. First, a simplification of the data in Table IV was obtained by calculating an average coefficient value for each degree of chlorine substitution. For example, the average for the four monochloro substitution pattern coefficients is 193, while the average value for the dichloro coefficients is 383, approximately two times that of the monochloro coefficients. This pattern is consistent through the trichloro and tetrachloro averages, with values of 572 and 799 being approximately three times and four times that of the monochloro coefficients, respectively. Thus, each chlorine substitution adds an increment of approximately 190 retention index units to the constant value. This increment may be used to reduce the number of structural parameters under consideration for a model. The magnitude of this increment is consistent with comparable values reported by Haken et al. (18) in their evaluation of the increase in retention indexes with increasing number of chlorines present in chlorobenzenes on a comparable chromatographic phase. The consistency in these data indicates that this model may also be used to describe the retention behavior of compounds similar in nature to the chlorinated dibenzofurans. Second, the values in Table IV were much simpler to deal with than the raw data in identifying interactions of chlorines on the same aromatic ring. Much of the “noise” in the data was stripped away by the initial data analysis so that underlying relationships, that were otherwise obscured, were made more apparent. The analysis process began by comparing various combinationsof the structural effect estimates. For example, one might compare the coefficient for A13 (357) with the sum of the A1 (206) and A3 (182) coefficients. This sum, 388, is 31 retention units greater than the coefficient for A13 (357). Likewise, when the coefficient for A2 (189) is added to the coefficient for A4 (195), their sum of 384 is 34 units greater than the coefficient for A24 (350). A tabulation of all the various combinations of two chlorines is given in Table V showing the summed results (E)vs. the directly calculated regression coefficients from model 2. A delta value is also provided which serves as a measure of decreased or increased retention of a particular substitution pattern as compared to the sum of its components. Table V is organized so that the patterns with similar delta values are grouped together. In the first two cases a meta relationship is introduced, while in the next three cases the chlorines are placed in an ortho relationship. The final case represents the only way to introduce two chlorines in a para relationship. The delta values provide a quantitative restatement of the ordering expressed in Figure 3 in which ortho chlorine substitution (vicinal chlorines) increased retention of the PCDFs relative to meta chlorine substitution (nonvicind chlorines). Again these same relationships may be observed in the data shown by Haken et al. (18). In that work, the
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 3, MARCH 1985
Table VI. Determination of the Effect of Intervening Chlorine8 on Intra-ring Interactions substitution pattern
ortho meta para corrected model
A1 A2 A3 A4 A12 A13 A14 A23 A24 A34 (206) (189) (182) (195) (419) (357) (384) (387) (350) (401)
(21)
(-32) (-17)
2
av A
A
“Closed” A123 A123 A123 A234 A234 A234
X
X X
X X
X
X
X X
X X
X
593 546 601 590 532 582
x XX X X XX X
x X X X
582 588 590 579 574 571
602 602 602 591 591 591
560 562 565 558 555 556
565 565 565 561 561 561
-20 -14 -12 -16 -12 -17 -20
“Open” A124 A124 A124 A134 A134 A134
X
X X
x
X
x
X
X X
X X
X
m-dichlorobenzene eluted prior to the o-dichlorobenzene. Also, the 1,3,54richlorobenzenewas the first trichlorobenzene to elute and 1,2,3-trichlorobenzene was the last. The consistency in these data again indicates that the models developed in this work can be applied to similar systems. Following these ideas, a new model was developed which had simple parameters to account for (1) the number and position of the chlorines present, (2) the structural relationships between the chlorines on each ring (intra-ring effects), and (3) the interactions between the rings (inter-ring effects). The hypothesis was evaluated by regressing the coefficient values from Table IV on dummy variables ONE, TWO, THREE, and FOUR (position variables) followed by ORTHO, META, and PARA (intra-ring variables). The variable ONE takes the values 0, 1,and 2, by simply counting the number of chlorine substitutions in the 1 and 1’ positions. TWO, THREE, and FOUR are defined similarly for the other positions. The intra-ring variables ORTHO, META, and PARA were originally defined to count the number of ortho, meta, and para relationships, respectively. While the regression worked fairly well, it was clear that there were intra-ring factors at work that were not accounted for by the simple ORTHO, META, and PARA effects observed here. This is illustrated in Table VI, which is in the same format as Table V except that now an attempt is made to calculate the trichloro substitution coefficients using combinations of the monochloro and dichloro substitution coefficients from Model 2. In Table VI, the intra-ring coefficients determined in Table V are included as modifiers to the summed value (E) resulting from the relationships that are introduced. As an example, if A123 is formed by adding A1 plus A23, an ortho relationship (1 to 2) and a meta relationship (1 to 3) are introduced. The coefficients for these intra-ring parameters are used to correct the summed (C)value in Table VI. No consideration is given to the ortho relationship present in A23 since this is already included implicitly in the original coefficient for A23; i.e., the regression coefficient for A23 estimates a net effect for the 23 substitution pattern. The trichloro substitutions in Table VI fdl into two groups. The groups defined as “closed” groups consist of substitution patterns with two pairs of vicinal chlorines (123 and 234) while the “open” groups contain patterns with one pair of vicinal chlorines separated from a lone chlorine by a hydrogen (124 and 134). The delta values for “closed”group substitutions are all comparable with an average value of -16. What is postulated as occurring is that the strength of the meta effect is reduced by an interveningchlorine. For example, in making the META modification for the case of A1 plus A23, this meta effect is mediated by the intervening 2-position chlorine and
556 573 614 607 566 552
X
x X X
X
x
X X X
X X X
-5 -3 0 -3 -6 -5
-4
Chart I ONE : TWO: THREE: FOUR: intraORTHO: ring META: variables PARA: interSTERIC: ring FOURFOUR: variables position variables
chlorines in the 1 position chlorines in the 2 position chlorines in the 3 position chlorines in the 4 position ortho relationships meta relationships para relationships steric hindrance ether linkage blocking
the correction should be cut in half (i.e., from -32 to -16). A similar situation occurs for A13 plus A2 except that the META effect inherent in the A13 coefficient is reduced, again by the insertion of the intervening 2-position chlorine. Similarly, uopenngroup substitution patterns have similar delta values and can be considered as having para-substituted chlorines which have been mediated by an intervening chlorine. Pursuing this idea, the regression mentioned in the previous paragraph was applied again, but this time the META and PARA variables were adjusted if intervening chlorines were present, to account for the disruption of the usual META and PARA relationships. The dummy variable META now takes the value 0 when there are no meta relationships present, 1 when there is one meta relationship, and 0.5 when there is a meta relationship with an intervening chlorine present. (Note: A1234 is considered to have two meta relationships, each with an intervening chlorine, so that META = 1 for A1234.) In a similar fashion, the dummy variable PARA was redefined to be zero when no para relationships exists, 1when one para relationship exists, 2f 3 when there is one intervening chlorine, and 1f 3 when there are two intervening chlorines. Finally, the variable ORTHO remains a count of the number of ortho relationships present, which cannot be modified by intervening chlorines. On identification of these intraring effects by examination of the regression coefficients of model 2, it was possible to postulate the following model to describe the relationships between molecular structure and retention index present in the original data retention index = constant + position effect + intra-ring effect inter-ring effect variation (Model 3) To fit model 3, the original PCDF retention indexes were regressed against the variables shown in Chart I. The resulthg coefficients in Table VI1 gave a good regression fit using these new variables, The variables are grouped to correspond to the three objectives stated previously. The position effect variables (ONE, TWO, THREE, FOUR) which take on the
+
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 3, MARCH 1985
CONSTAN
ION!
-
1M5.I
.21 x162.3 .
imo .21 x I85
645
STERIC correction-
324.6
F
-
310.8
Position
!mum. 21 x 180.1 .
w.2
IFWR 21 x 195.7 *
391.4
!ORTHO . 6 i x 23.0
1?8.0
0
=
1
2
value of ONE
Y
0
h Intra-ring
IMRA . 4 x O . S a i x - 3 2 . 5 . -65.0
HBOND correction
VI0
!PARA . 2 x 0.33bi x -22.0. -14.5
ISTERIC.IIx91.9.
-45.0
-
1
2
Value of ONE
Figure 5. Comparative interpretation of the STERIC and HBOND interactions.
91.9
~nter-ring{ I
corresponding coefficients to the constant value. AS an example, consider again 122'4'-TCDF. Take the constant term of 1565.3 and add to it 162.3 (one 1position) plus 370.8 (two 2 positions) plus 195.7 (one 4 position) plus 23 (the ortho relationship for 12) minus 32.5 (the meta relationship for 2'4') to obtain 2285 which again compares well with the observed value of 2281. To further illustrate the intra-ring and inter-ring parameters in Table VII, consider the extreme case of predicting the retention index of octachlorodibenzofuran (OCDF). The calculation of this index using Table W is illustrated in Figure 4. The portions of the molecule involved in a particular calculation have been designated by a dot. In this example, all the parameters in Table VI1 are used and provision has been made for intervening chlorines in the calculation of the META and PARA variables. The position and intra-ring variables used in model 3 are familiar to structural organic chemistry. The inter-ring variables are considerably less familiar, especially the variable STERIC, and its counterpart HBOND used in Model 2. Figure 5 is presented so that the nature of these inter-ring variables and their relationship to each other may be more clearly understood. A special regression analysis was performed on a second set of dummy variables to better delineate the effects of the dummy position variables already introduced (i.e., ONE, TWO, THREE, and FOUR). Each of the position variables was decomposed into three variables, corresponding to 0,1, and 2 chlorine substitutions for that position. Figure 5 plots the coefficients for these dummy variables for ONE against 0, 1,and 2. Similar plots for TWO and THREE, not included here, reveal that the three points nearly fall on a straight line. For ONE, however, there is a severe nonlinearity. The same is true for FOUR, but to a lesser extent. The
CI
i \
0' Observed * 3147 Predicted *
3152.8
'MCIA is multiplied $0.5 slnce there Is one Intervening chlorlne
)PARA
1s mult~pliedty 0.33 slnce there are tul intervanlng chlorlnes
Flgure 4. Prediction of the retention index for octachlorodibenzofuran using model 3.
Table VII. Regression Coefficients for Model 3 dummy variable
coefficient
,
constant
1565.3 162.3 185.4 180.1 195.7 23.0 -32.5 -22.0 91.9 -9.9
ONE
TWO THREE FOUR ORTHO META PARA STERIC FOURFOUR
values 0,1, and 2, account for both the number of chlorines present in the congener, as well as the positional effects. Recalling the differences in coefficients for Al, A2, A3, and A4 as shown in Table IV, the model suffers greatly if these positional differences are ignored. The intra-ring variables account for the relationships between chlorines on a given ring. The inter-ring variables represent the interactions between the rings as described during the development of model 2. Illustration of Model 3. To use the coefficients in Table VII, one identifies the appropriate chlorinated positions, intra-ring parameters, and inter-ring parameters, adding the Table VIII. Analysis of Variance for Model 3" source model residual
DF
sum of squares
mean square
F value
prob
