nickel was not retained and passed directly through the column as shown in Figure 5. The eluent was then switched to 2M HC1 and lead eluted. Peak height and peak area were measured for each of the three sample solutions and compared with calibration curves. The calibration curves were prepared from measured peak heights and areas obtained from chromatograms of lead injections in 2M HCl. T h e lead stock solution was standardized by titration with standard EDTA solution. Results for the lead analyses are listed in Table 11. Peak area gave a much better quantitative measure of the amount of sample component present than peak height. This was apparently due to the greater band spreading in the sample chromatograms and, hence, lower peak heights than in the chromatograms used for the preparation of the calibration curves. Analysis of NBS 124d. Three solutions of NBS Standard Sample 124d ounce metal were prepared by dissolution in "03 and filtration as for 37d. Five-milliliter aliquots of the stock sample solution were then diluted to 25 ml and these solutions were analyzed for lead content by the same procedure as t h a t used for 37d and previously described. The same calibration curves were used to obtain the quantity of lead present. NBS 124d was also analyzed for nickel content by the separation method previously described. Portions (56.3 pl) of the 5 : l dilutions of' stock sample solutions were injected into 50% acetonitrile-0.5M HC1 and zinc was eluted immediately. All other components were retained. This separation is shown in Figure 8. Stock zinc solution used for the preparation of the calibration curve, was prepared by dissolving Baker Analyzed Reagent zinc metal (99.9%) in HC1 and diluting to volume. The calibration curves were obtained from triplicate injections of dilutions of the stock solution in 50% aceto-
nitrile-0.5M HC1. Peak height or area can be used in this case, b u t peak area gives better accuracy (0.8% error as compared to 2.2%) and precision (1.6 pph rsd compared to 2.7 pph). NBS 124d was also analyzed for nickel content by the acetonitrile-HC1 procedure. Portions (56.3 pl) of the original, undiluted sample solutions were placed on the column in a n eluent of 90% acetonitrile-0.2M HC1, and zinc, lead, and copper were eluted. Nickel was retained from this medium and was then stripped with 2 M HCl (aqueous). A calibration curve was prepared using dilutions of a stock nickel solution standardized by EDTA titration. The standard solutions were sorbed in 90% CH&N-O.BM HC1 and eluted in 2M HCl. The amount of nickel in the sample was then determined by comparison of peak areas. Results for the lead, zinc, and nickel analyses are given in Table 11.
CONCLUSIONS It is evident from the separations shown in this work t h a t conditions required for forced-flow ion exchange separations in strongly acidic media can be moderated by the use of low capacity resins without significantly affecting the resolution of the chromatographic separation. Because of the lower capacity, the permissible sample size for separations on a n analytical scale is somewhat smaller than with conventional resins. The new resins are particularly well suited for use in rapid chromatographic separations.
ACKNOWLEDGMENT We wish to thank John J. Richard for assistance in preparing the resins used in this work. Received for review November 19, 1973. Accepted January 28, 1974.
Factor Analysis of Retention Indices for Hydrocarbons Darryl G. Howery D e p a r t m e n t of C h e m i s t r y . BfOOk/yn C o l i e g e of The City U n i v e r s i t y of New Y o r k . B r o o k i y n . A! Y . 7 1270
A data matrix of retention indices for 25 aliphatic and aromatic hydrocarbons on 12 stationary liquid phases is
subjected to t h e mathematical method of factor analysis. Six abstract eigenvectors are required to reproduce essentially all the data points within the precision limits of f l index unit. Three solutes tested relatively high on uniqueness tests. Solute factors of physical significance which successfully rotated onto fundamental factors of the s p a c e include molecular weight, molar refraction, enthalpy of vaporization, heat capacity, heat of combustion, triple-bond uniqueness, aromatic uniqueness, carbonbond scale, and carbon number. Two sets of six physically significant vectors reproduced the data matrix with an average column average error of eight index units. One set of six columns of data representing solutes from the original matrix reproduced the matrix with an average column average error of less than one index unit.
New insights into such diverse areas as solvent effects on NMR chemical shifts ( I - 3 ) , retention indices in GLC (4-8), halfwave potentials in polarography ( 9 ) , biological activity in medicinal chemistry ( I O ) , and acidities of substituted benzoic acids ( I I ) have resulted from applications (1) P. H. Weiner. E. R . Malinowski. and A . R . Levinstone, J. Phys. Chem , 7 4 , 4537 (1970). (2) P. H. Weiner and E. R . Malinowski, J. Phys. Chem 75, 3160 (1971). (3) M . R . Bacon and G. E Maciel, J . Amer. Chem SOC.. 95, 2413 (1973). (4) P. H. Weinerand D . G.Howery, Can. J . Chem.. 5 0 , 448 (1972). ( 5 ) P. H. Weiner and D . G. Howery, Anal. Chem.. 44, 1189 (1972) (6) P. H Weiner, C. J. Dack, and D G . Howery. J Chromatogr. 69, 249 (1972). (7) P. H. Weiner and J. F Parcher. J Chromatogr S c i . I O , 612 (1972), (8) P H . Weiner and J. F. Parcher. Ana/. Chem.. 4 5 , 302 (1973). (9) D . G. Howery, Buli. Chem. SOC.d a p . . 45, 2643 (1972). (10) M . C. Weiner and P. H. Weiner. J. Med Chem.. 16, 655 (1973). (11) P. H. Weiner. J. Amer. Chem. SOC..95, 5845 (1973) A N A L Y T I C A L C H E M I S T R Y , VOL. 46,
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of factor analysis, a general mathematical technique for studying two-dimensional matrices of d a t a (12). The mathematical formulation of a factor analysis method especially suited to problems of chemical interest has been presented by Malinowski and coworkers (I). In a detailed analysis of r.i. data, Rohrschneider suggested t h a t r.i.'s could be expressed in a form appropriate for factor analysis (13)-i.e., as a sum of product functions. Weiner and Howery demonstrated t h a t factor analysis could be used to interpret d a t a matrices comprising the retention indices of solutes on stationary liquid-phase solvents ( 4 ) . Subsequent work on a wide range of both solutes and solvents (5-8) has clearly demonstrated the capabilities of factor analysis in particular for testing t h e nature of the physically and chemically significant parameters of solutes which influence the r.i. and for predicting missing r.i. d a t a and properties of molecules. As part of a continuing program to elucidate the nature of solute-solvent interactions in GLC by the method of factor analysis, we have factor-analyzed a matrix of retention indices involving 25 hydrocarbons. EXPERIMENTAL The 25 hydrocarbon solutes represented in the d a t a matrix are listed in Table I. Twelve stationary-phase solvents were e m ployed: Apiezon M , OV-17, trimethylolpropanetripelargonate, polyphenyl ether, Castorwax, Oronite NIW, bis(2-ethoxyethyl). phthalate, Tergitol NPX, XF-1150, tricresylphosphate. QF-1 and Citroflex A-4. McReynolds, who furnished the d a t a ( 1 4 ) , reports the precision of the d a t a points to be generally better t h a n f l index unit. A copy of the d a t a matrix will be sent upon request. Factor analysis was carried out with t h e FORTRAN IV program developed by Malinowski and coworkers ( I ) using a FACOM 230-60 digital computer.
RESULTS AND DISCUSSION Number of F u n d a m e n t a l Factors. The mathematical details of the factor analysis method employed here have been presented by Malinowski and coworkers ( I ) , and a similar approach for the reproduction portion of the scheme was discussed by Bacon and Maciel ( 3 ) .Briefly, a correlation matrix is formed from the original d a t a m a trix. Then a matrix which diagonalizes the correlation matrix is found and the resulting eigenvalue problem is solved by standard techniques involving t h e secular determinant. The calculated set of abstract eigenvalues and associated eigenvectors characterizes completely the original d a t a matrix-ie., t h e complete set of eigenvectors will reproduce exactly the original matrix. In problems of chemical interest, t h e reproduction scheme of factor analysis is used to determine if the matrix space can be characterized adequately by a smaller subset of the complete set of eigenvectors. An estimate of the true complexity of the d a t a space is given by the minimum number of abstract eigenvectors (factors) needed to reproduce the original matrix. In the reproduction scheme of the factor analysis program ( l ) ,the d a t a matrix is calculated (reproduced) using successively larger numbers of the abstract eigenvectors, the largest eigenvector being used in the one-factor case. For example, the reproduction in the three-factor space involves the three largest eigenvectors. The minimum number of abstract eigenvectors required to reproduce a large fraction of the d a t a points within experimental precision is taken as a n indication of the number of fundamental factors in the d a t a space. In this problem, the first (12) R J Rummel Applied Factor Analysis Northwestern Univ Press. Evanston, 1 1 1 1970 (13) L Rohrschneider J Chromatogr 22, 6 (1966) (14) W 0 McReynolds Celanese Chemical Co Bishop. Texas personai communication 1971
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seven eigenvalues had numerical values of 1.2, 1.4 x 2 x 3.6 x 10-4, 2.9 x 10-4, 4 x and 4 x 10-6, implying t h a t there is one dominant factor, perhaps one moderately important factor, and possibly several minor factors operative in the d a t a space which statistically influence t h e retention index values. In a six-factor space, the average of the column average error in the reproduction was 0.4, with only one point being predicted with a n error exceeding two index units. Since the reproduction with t h e five largest eigenvectors was considerably poorer and since the physically significant test vectors which successfully rotated (see below) gave essentially equivalent results for six and for seven factors, this hydrocarbon r.i. space appears to be a six-factor space. Only the set composed of t h e six largest eigenvectors is required to span all of the measurably significant effects in t h e original d a t a space; the physically significant space is said to be a 6-factor space. The number of factors in a given data space depends in part upon the degree of accuracy of reproduction one specifies ( * l index unit in our case) and also upon t h e size of the matrix (15). Especially for small matrices and for matrices in which certain interactions may be weakly represented in the space, the number of abstract factors serves only as a n estimate of the number of physically significant interactions in the space. Rotation of Physically a n d Chemically Significant P a r a m e t e r s , The most powerful aspect of the factor analysis method of Malinowski is the least-squares rotational scheme which allows one to test physically and chemically significant parameters of the row elements of the data matrix for possible identification with one of the abstract eigenvectors required in the reproduction. A generalized least-squares method for calculating the rotational matrix which can be used to test suspected parameters is described by Malinowski and coworkers ( 1 ) . As input information in a rotational test, one submits the values of a given property for the row elements (solutes here) of the data matrix. Several of the points on the test vector can be left blank, a procedure termed free-floating. This feature greatly increases the scope of the rotational scheme since so often we lack quantitative data for a large number of solutes. The program outputs a calculated vector which represents the best least-squares fit between t h e test vector and the abstract eigenvector which most nearly approximates the test vector, using a specified number of abstract eigenvectors. The agreement between the inputed and predicted vectors considered in light of a ) the expected error in the points in the d a t a matrix and b ) the estimated error in the points on the test vector enables one to decide whether a test vector has been successfully rotated onto a fundamental abstract eigenvector. In many tests, this decision is necessarily somewhat arbitrary. since no statistical rules are available. A dividend of the rotational scheme is that the missing free-floated points on the test vectors are predicted from the best-fit vector. Physically Significant Parameters. A large number of physical properties. mostly of a thermodynamic nature, were tested. Three parameters which have been rotated successfully onto abstract eigenvectors in previous r.i. problems (6, 8)-molecular weight, molar refraction (calculated from group contributions), and molar enthalpy of vaporization-also tested quite well in the hydrocarbon r.i. space. In addition, we find t h a t test vectors for heat capacity and heat of combustion give adequate fits (see Table I for results). Heat content (H" - Ho"), molecular volume (see Table I), and entropy of formation gave good fits for most of the test points, though a few points on (151 D G Howery P H Weiner and J S Blinder J Chrornatogr SCJ in press
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A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 7, JUNE 1974
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Table 11. Uniqueness Values for Solutes in
- - -- - -
--
6-li’nctnr- Snnce Solute
1. Methane 2. Ethane 3. Propane 4. Butane 5. Pentane 6. Hexane 7. Heptane 8. Octane 9. Decane 10. 2,4-Dimethylpentane 11. 2-Methylheptane 12. 3-Methylheptane 13. 2,2,3-Trimethylhexane 14. Cyclohexane 15. 1-Octene 16. 2-Octene 17. 2-Ethylhexene 18. 1-Octyne 19. 2-Octyne 20. Benzene 21. Toluene 22. Ethylbenzene 23. Mesitylene 24. Styrene 25. Ethynlbenzene
Uniqueness value
Additional valuesa
0.00
0.01 0.02
0.03 0.05
0.07 0.09
0.12 0.19 0.13
13 (0.22)
0.10 0.10
0.55 0.70
0.11 0.09 0.08 0.31 0.74 0.23 0.21
0.28 0.57
0.31 0.90
23 ( - 0 . 3 1 ) , 10 (0.22) 18 (-0.27)
14 ( - 0 . 2 7 ) , 19 (0.26) 18 (0.26)
24 (0.23),22 (0.22) 21 & 24 (0.21) 13 ( - 0 . 3 1 ) , 2 4 (0.24) 23 ( 0 , 2 4 ) ,21 (0.23)
Other solutes which gave predicted values greater than 0.20 on the given uniqueness test. Solutes numbered as shown in left-hand column with values in parentheses.
each vector were fairly erroneous. In nearly all of these tests the agreement was poorest for the three or four smallest n-alkanes. The poorer agreement may be attributable to the fact t h a t the r.i.’s for all the solutes except the five smallest n-alkanes lie in the range 600-1300, so that the lower alkanes may be relatively poorly anchored in the space. Those properties which gave such poor fits that the properties do not appear to be rotatable onto fundamental factors of the hydrocarbon r.i. space include: free energy of formation, enthalpy of formation. entropy of vaporization, log vapor pressure, freezing point, boiling point, compressibility factor, viscosity, refractive index, density, dielectric constant, and surface tension. Log vapor pressure also did not seem to be a factor in r.i. spaces representing a wider range of functional groups (4,1 5 ) : however, a quite good fit was obtained with a n alcohol-only matrix (6) and a n ester-only matrix (8).More work needs to be done before the physical parameters common to all r.i. spaces can be stated with certainty. Conclusions drawn from factor analyses of spaces involving quite similar solutes may be less valid than those deduced from matrices involving a wide range of solute functional groups. Chemical/), Significant Parameters. Factor analysis enables one, by means of test vectors having arbitrary limits based on chemical intuition, to group compounds having similar behavior (similar factors) and to predict the relative effects of a suspected factor for all the solutes in the matrix. a. Uniqueness. A simple test which helps to distinguish those compounds which are involved in unique factors or which involve relatively poor data, and to group similar compounds is termed the uniqueness test ( 5 ) .This type of test vector contains a one for the compound being tested for uniqueness and zero’s for all the other row compo832
.
A N A L Y T I C A L C H E M I S T R Y . VOL. 4 6 . N O
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nents. Results of the uniqueness tests for the solutes in a six-factor space, including a tabulation of other solutes which gave absolute numerical values greater than 0.20 in each test, are given in Table 11. Ethynlbenzene (uniqueness value, 0.90) appears to be the most unique solute; cyclohexane and 2-octyne also have fairly large uniqueness values. In general, those solutes which are most similar chemically to the solute being tested tend to be ranked higher than not-so-similar solutes. The significance of the two pairs of negative values in Table I1 is not known. b. Multiple-Bond Scales. In a series of uniqueness-type tests, certain relationships between alkenes, alkynes, and aromatics have been established. This type of somewhat arbitrary vector based on chemical insights appears to have considerable potential for predicting similarities between row entities of a d a t a matrix and for furnishing clues as to the chemical factors in the space ( 4 , 6). Unity. The solutes can be tested for a common property by inputing a test vector composed of one’s, a unity vector. All of the solutes except the five smallest n-alkanes and decane had predicted values within zk0.2 of one; the fit for the lower n-alkanes was much poorer, methane giving the lowest value (0.14). Even taking into account the poorer agreement anticipated for the lower n-alkanes discussed above, we do not feel that there is a common, equal-valued factor in the space. Multiple-Bond Lhiqueness. A test vector for multiplebond character comprised of one’s for the solutes containing a multiple bond of any sort and zero’s for the remaining solutes gave predicted values for the aromatics and alkynes reasonably near one but nearer 0.5 for the alkenes (see Table I). All other solutes were ranked quite low, indicating that multiple bonds are a source of uniqueness. The alkenes did not test at all uniquely using a test vector in which the alkenes were assigned a value of one and all other solutes a value of zero; in fact, the alkynes (average predicted value, 0.32) tested higher than the alkenes iaverage value, 0.25) on the alkene Uniqueness test. A uniqueness test for alkynes (see Table I) clearly indicated that the triple bond is a unique factor in the space. When all solutes having aromatic character were tested for uniqueness (see Table I) the phenyl-containing solutes have predicted values near one, while none of the other rotated values exceed 0.25. Factor analysis indicates that both the triple bond and aromatic character may introduce fundamental factors whereas the double bond is not responsible for any unique effect. Multiple-Bond Scale. In a n attempt to interrelate double, triple, and aromatic bonds, a crude vector of one’s for the double-bond containing solutes, three’s for the aromatics and free-floated values for the three alkynes was tested. The fair fit for this multiple-bond scale is shown in Table I. Styrene behaves as expected if we assume additivity for aromatic and double-bond properties. The predicted values for the alkynes indicate a rough equality between aromatic and triple-bond character in this factor. The predicted value for ethynylbenzene (5.7) nearly equals the sum of the assigned aromatic contribution and the average of the two values predicted for the other two alkynes (3.2). The high ranking of alkynes on this scale and the definite uniqueness for alkynes imply that the triple bond is associated with two factors in the r.i. space. Carbon-Bond scaie. A test vector intended to test total carbon-bond character gave a fair fit (see Table I ) . In this vector a carbon-carbon single bond was assigned a value of one; a carbon-carbon double bond, two; and the benzene ring, ten. Several other almost equivalent test vectors, in which the value for benzene was altered slightly, produced nearly as good agreement. The results of this test and the test for the multiple-bond scale suggest that
the three types of multiple bonds seem t o be related to each other through more t h a n one factor. c C a r b o n Number. A test vector anchored with t h e number of carbon atoms for the n-alkanes gave a perfect fit for one factor (see Table I). In addition, the predicted carbon numbers for t h e other solutes seem to be reasonable numbers. T h e fit for this test becomes progressively worse as the number of factors increases. This result may indicate t h a t the carbon number is associated with the dominant factor, consistent with the general value of r.i.'s in the matrix. If the predicted values for a one-factor space are incorporated in a test vector in a six-factor space, the agreement is essentially perfect, indicating t h a t the carbon number is a factor in t h e six-factor space if the vector is properly anchored. Using t h e results of the carbon number test in Table I, we can speculate t h a t the side chains reduce the carbon number on the average by about 0.4 unit per chain, that the carbon number of alkenes differs b u t slightly from the corresponding n-alkanes, t h a t a triple bond adds in excess of one carbon number, and t h a t t h e aromatic ring contributes a n excess of about two carbon numbers. If benzene is taken as 8.1 on this scale, then the increment per aliphatic carbon atom for t h e remaining three aromatics is almost exactly one. For example, the predicted carbon number per aliphatic carbon atom for mesitylene is (11.2 - 8.1)/3 = 1.0. Factor analysis thus predicts t h e existence of a n aliphatic carbon atom scale. Test vectors of a chemical nature which did not test well include: number of hydrogen atoms, ratio of carbon atoms to hydrogen atoms, number of methylene groups in a row, number of methyl groups. number of ethyl groups and number of side chains. Our conclusions should be fairly general since the d a t a matrix incorporates a diversity of solvents. Takacs and coworkers (16) have presented a promising method for predicting the retention indices of several types of solutes on the basis of molecular structure, and Castello and coworkers (17) have accurately pinpointed the dependence of the retention times of branched-chain paraffins on non-polar solvents in terms of variences in physical properties. T h e procedures used by both groups apply to one solvent or to a series of closely related solvents. Reproduction Using Physically Significant Vectors. The objective in reproducing the d a t a matrix with physically significant vectors is to ascertain those vectors which best span all of t h e fundamental interactions of the space. Since the original d a t a matrix can be reproduced within experimental precision with six abstract eigenvectors. then from the viewpoint of the solutes we hope to find six complete independent physically significant vectors which (16) J M Takacs E Kocsi, E Gararnvolgyi E Eckhart T Lombosi. S Nyiredy I Borbely and G Krasznai, J Chromarogr 8 1 , 1 (1973) (17) G Castello M Lunardelli and M Berg J Chromatogr 76, 31 (1973)
when used as a set of vectors will also reproduce the d a t a matrix with acceptable accuracy. If the point-by-point agreement is near f l index unit, then we have one solution t o the solute part of the solute-solvent interaction problem. Using Physically a n d Chemically Significant P a r a m . . eters. Many combinations of six of the following vectors (all of which gave good or fair fits when tested in the rotation part of the scheme)-molecular weight, molar refraction, enthalpy of vaporization. heat capacity, heat of combustion, heat content, molecular volume, entropy of' formation, unity, multiple-bond scale, aromatic uniqueness, triple-bond uniqueness, carbon-bond scale and carbon number-were used as vectors in the reproduction scheme. Two best sets: 1) carbon number, molecular volume, aromatic uniqueness, triple-bond uniqueness, carbon-bond scale and multiple-bond scale; and 2) the previous set except enthalpy of vaporization replaces molecular volume, reproduced the original d a t a matrix with a n average column average error (a.c.a.e.) of eight index units. This rather large a.c.a.e. indicates that a t least one factor has not been identified; nevertheless, considering the complexity of the problem, considerable progress has been made towards identifying the major factors [as was true in the alcohol-solute problem ( 6 ) ] .T h a t several of the chemically significant vectors are included in the best sets tends to substantiate the value of such vectors. In each of the vectors used in the reproductions, predicted values from the rotational scheme were used to complete the vectors which originally had some free-floated points. Using Columns of Retention I n d e x D a t a . If one transposes the original d a t a matrix, one can use sets of six columns of d a t a from the d a t a matrix (where each vector is associated with a particular solute) to reproduce the d a t a matrix ( 5 ) . Out of many combinations tried. four sets reproduced the d a t a matrix within a n a.c.a.e. of two. T h e set which seems to best include all of the significant interactions (a.c.a.e., 0.8 index unit) represented butane, 2,4dimethylpentane. cyclohexane. 1-octene. 1-octyne, and benzene. This set is typical of a set one would pick from chemical knowledge; however. other seemingly as logical sets gave much poorer reproductions. The accuracy of reproduction is extremely dependent upon the columns of data selected. At present. we cannot predict whether such differences are due to true chemical effects or to statistical effects. A C K S O WLEDGMENT W. 0. McReynolds kindly furnished the high-precision d a t a matrix. Funds for the computer computations were supplied by T. Fujinaga and T. Kagiya, both of Kyoto University. The use of the computer center of Kyoto University is gratefully acknowledged. Received for review July 31. 1973. Accepted .January 15, 1974.
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