Anal. Chem. 1998, 70, 3712-3716
A Comparison of Gas-Hexadecane and Gas-Apolane Partition Coefficients Jeff D. Weckwerth,† Peter W. Carr,* Mark F. Vitha,‡ and Asad Nasehzadeh§
Department of Chemistry, University of Minnesota, Smith and Kolthoff Halls, 207 Pleasant Street SE, Minneapolis, Minnesota 55455
Gas-apolane partition coefficients (L87) of 157 nonpolar and polar organic solutes, spanning a wide range of functional groups, dipolarities, and hydrogen-bonding capabilities, are measured by open tubular capillary gas chromatography at 40 °C. The experimental values compare well (R2 ) 0.999) with literature values of log L87 from packed-column gas chromatography. The log L87 values are also compared with gas-hexadecane partition coefficients (log L16) at 25 °C. A strong linear relationship exists between log L87 and log L16 for all solutes (R2 ) 0.994), as well as for chemically relevant subsets of the data. Therefore, unknown L16 values at 25 °C can be predicted from the corresponding L87 values, which can be measured on open-tubular or packed columns at higher temperatures, due to the low volatility of apolane. Predicted values of L16 would be extremely useful, since log L16 is often the major explanatory parameter in many linear solvation energy relationships. Gas chromatography (GC) is a well-established method for the measurement of gas-liquid partition coefficients (L).1,2 A key underlying assumption in these measurements is that GC retention depends only on the interaction of the solute with the bulk liquid stationary phase; that is, GC retention is a pure partitioning process. The gas-liquid partition coefficient of a solute is therefore related to the experimentally determined GC capacity factor (k′) by the equation,
VM k′ ) VS φ
L ) k′
(1)
where VM and VS are the volumes of the mobile and stationary phases, respectively, and φ is the phase ratio (VS/VM). Packed-column GC is frequently used to measure gashexadecane partition coefficients (L1625, packed) of many nonpolar
and polar organic solutes at 25 °C.3 However, packed-column GC measurements, especially for polar compounds, are known to be subject to gas-liquid, gas-solid, and liquid-solid interfacial adsorption.4-6 To minimize the effect of interfacial adsorption, Carr et al. measured L16 by open-tubular GC with fused-silica capillary columns (L1625,capillary).7 A comparison of the packedcolumn and open-tubular L16 values showed a considerable decrease in interfacial adsorption for open-tubular GC.7 This decrease is apparently due to a much larger ratio of hexadecane volume to surface area, as well as a more uniform coating of hexadecane on the solid surface, and to the avoidance of the chemically complex surface of diatomaceous earth.8 The motivation for the accurate measurement of L16 (at 25 °C) centers on the importance of these values in many linear solvation energy relationships (LSERs). The well-known Kamlet-Taft LSER method9-12 has been very useful for the study of chemical interactions in various solution processes. This work focuses on the application of LSERs to GC retention, which has been studied in detail.8 Li et al. showed that, for several GC stationary phases, capacity factors can be correlated with the probe solutes’ solvatochromic parameters using the following LSER:8
log k′ ) log k′0 + l log L16 + sπ*2C +RR2C + dδ2
(2)
In this equation, the parameters π*2C and R2C represent a solute’s dipolarity/polarizability and hydrogen bond acidity, respectively.8 The δ2 parameter is a polarizability correction factor taken as 0 for aliphatic solutes, 1 for aromatic solutes, and 0.5 for polyhalogenated solutes. The L16 value represents a composite measure of the ability of a solute to interact with a solvent by dispersion forces and to some extent by induction, but it also incorporates the cavity formation process.13 The variables, k′0, l, s, a, and d, are determined by a leastsquares regression of log k′ against log L16, π*2C, R2C, and δ2. Since
Present address: Hutchinson Technology Inc., Hutchinson, MN 55350. Present address: Chemistry Department, Drake University, 2507 University Ave., Des Moines, IA 50311. § Present address: Department of Chemistry, University of Shahid Bahonar, P.O. Box 76175-133, Kerman, Iran. (1) Conder, J. R.; Young, C. L. Physicochemical Measurements by Gas Chromatography; Wiley: New York, 1979. (2) Laub, R. J.; Pecsok, R. L. Physicochemical Applications of Gas Chromatography; Wiley: New York, 1978.
(3) Abraham, M. H.; Grellier, P. L.; McGill, R. A. J. Chem. Soc., Perkin Trans. 2 1987, 797. (4) Martire, D. E. Anal. Chem. 1966, 38, 244. (5) Pecsok, R. L.; Gump, B. H. J. Phys. Chem. 1967, 71, 2202. (6) Urone, P.; Parcher, J. F. Anal. Chem. 1966, 38, 270. (7) Zhang, Y.; Dallas, A. J.; Carr, P. W. J. Chromatogr. 1993, 638, 43. (8) Li, J.; Dallas, A. J.; Carr, P. W. J. Chromatogr. 1990, 517, 103. (9) Kamlet, M. J.; Taft, R. W. J. Am. Chem. Soc. 1976, 98, 377. (10) Kamlet, M. J.; Taft, R. W. J. Am. Chem. Soc. 1976, 98, 2886. (11) Kamlet, M. J.; Taft, R. W. J. Am. Chem. Soc. 1977, 99, 6027. (12) Kamlet, M. J.; Taft, R. W. J. Am. Chem. Soc. 1977, 99, 8325. (13) Abraham, M. H.; Fuchs, R. J. J. Chem. Soc., Perkin Trans. 2 1988, 523.
3712 Analytical Chemistry, Vol. 70, No. 17, September 1, 1998
S0003-2700(97)01370-X CCC: $15.00
† ‡
© 1998 American Chemical Society Published on Web 07/28/1998
the solvatochromic descriptors (log L16, π*2C, R2C, δ2) are based on known chemical behavior of the solutes, the coefficients (l, s, a, d) provide chemical information about the stationary phase. The magnitudes of the s and a coefficients represent the dipolarity/polarizability and the hydrogen bond basicity of the stationary phase, respectively. The d coefficient is related to the polarizability of the stationary phase. Note that there is no solute hydrogen bond basicity parameter (β2C) in eq 2 because none of the stationary phases studied by Li have any appreciable hydrogen bond acidity. The l coefficient arises from two conceptually independent but covarying processes: the favorable intermolecular dispersive attractive forces (London forces) and the unfavorable solute sizedependent cavity formation.13 It can be shown that both of these processes scale nearly linearly with solute size (volume or area) and thus are difficult to separate empirically.13 A positive l coefficient in gas-liquid chromatographic systems indicates that attractive (favorable) dispersion interactions between the solute and the stationary phase dominate over unfavorable cavity formation processes. Without exception, L16 is the dominant term in LSERs of gas-liquid chromatographic systems, and it is extremely important in many other types of LSERs. Because of the importance of the log L16 parameter, accurate values are crucial in LSERs. While values of π*2C, R2C, and β2C can be easily estimated based on knowledge from other compounds, values of L16 can be difficult to estimate. Direct measurements of L16 at 25 °C are impossible for large, nonvolatile compounds. Some very valuable semiempirical methods may be used to calculate L16, often quite accurately,14,15 but these methods also depend on the availability of good experimental data for their parametrization. Another option is to use a different gas-liquid partition coefficient as a measure of dispersion and cavity formation. Theoretically, any nonpolar liquid could be used. Kova´ts et al. have developed a highly branched nonpolar hydrocarbon, apolane87,16 that has for present purposes several advantages over hexadecane.16 Apolane-87 is a stable, nonvolatile phase that can be used over the temperature range of 30-280 °C. The thickness of the coated apolane-87 stationary phase will remain virtually constant over a long period of time at elevated temperatures. In contrast, hexadecane is a volatile liquid and cannot be used at temperatures higher than ∼25 °C. Kova´ts et al. have already measured many gas-apolane partition coefficients (L87) at high temperatures by packed-column GC.16 In this work, we used open-tubular GC to measure L87 values of 157 nonpolar and polar organic solutes. These solutes span a wide range of functional groups, dipolarities, and hydrogenbonding capabilities. The L87 values are compared with those measured by packed-column GC to assess the presence of any interfacial adsorption effects, and they are compared with the corresponding L16 values. A new LSER approach is proposed, in which log L87 is substituted for log L16. (14) Havelec, P.; Sevcik, J. G. K. J. Chromatogr., A 1994, 677, 319. (15) Giesen, D. J.; Storer, J. W.; Cramer, C. J.; Truhlar, D. G. J. Am. Chem. Soc. 1995, 117, 1057. (16) Defayes, G.; Fritz, D. F.; Gorner, T.; Huber, G.; De Reyff, C.; Kova´ts, E. J. Chromatogr. 1990, 500, 139.
EXPERIMENTAL SECTION Unless otherwise stated, all solutes were of general laboratory or analytical grade in the highest purity available. A 15 m × 0.32 mm × 0.25 µm deactivated fused-silica apolane-87 capillary column (Alltech) was used throughout this work. A Hewlett-Packard 5880A GC with heated on-column injector and flame ionization detector was used. Both the injector and the detector temperatures were kept at 250 °C. Data were collected at 39.2 °C. The column oven temperature was stable to within 0.02 °C. All of the L87 values in this study were measured or calculated at 39.2 °C. To convert these values to 40 °C, multiply by 0.997. Helium (carrier gas) flow was adjusted as necessary to make the retention times adequate. All data were collected on a Hewlett-Packard 5880A GC terminal integrator. Corrected retention times and capacity factors were calculated referenced to the retention of methane. We estimate that the k′ of methane is ∼0.005; therefore it is quite reasonable to use methane as a dead time marker. All partition coefficients reported here are based on at least three to five replicate measurements. Preliminary injection tests were performed to elucidate the interfacial adsorption and sample size dependency of the solutes. These tests showed that in some cases the retention time of the solute was dependent on sample size. To avoid this problem, usully 0.1-0.2 µL of the sample vapor (at room temperature) of the volatile solutes was injected, and/ or very dilute solutions of nonvolatile solutes were prepared in carbon disulfide, and the needle of a 10-µL syringe was then just wetted (zero volume) by the solution and subsequently injected into the capillary. Average values of the replicate retention times were used to determine the capacity factors, with standard deviations not exceeding 1%. The gas-apolane partition coefficient of n-decane from Kova´ts’16 data was used as a reference to calculate the film thickness of the coated apolane, which was found to be 0.295 µm. RESULTS AND DISCUSSION All partition coefficients measured in this work are given in Table 1 along with values for log L16 at 25 °C and log L87. Kova´ts’ log L87 values are measured at higher temperatures (110-190 °C) and are extrapolated to 39.2 °C using the equations given in his work.16 Although we obtained excellent peak shapes and constant retention times for most solutes, there is some evidence that the column is not completely inert. Adsorption-producing tailed peaks was observed for some protic compounds with high boiling points, including alcohols, phenols, and carboxylic acids. Poor peak shapes and changes in retention time with the amount injected were also observed for carboxylic acids. The retention time of acetic acid, which is known to be very highly self-associated in both the gas phase and in nonpolar solvents,7 was not reproducible, even for very small injection volumes. Therefore L87 values of carboxylic acids are not reported in Table 1. We also paid special attention to both aromatic and aliphatic amines and alcohols. The retention times of these compounds were found to be dependent upon the amount injected, presumably due to solute-solute associations and to incomplete deactivation of the fused-silica capillary column. The latter problem could be significantly reduced by a better deactivation procedure.7 Equation 3 shows the correlation between open-tubular log L87 data at 40 °C and the packed-column data at the same temperature. Analytical Chemistry, Vol. 70, No. 17, September 1, 1998
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Table 1. Gas-Hexadecane and Gas-Apolane Partition Coefficients no.
compound
log L16a
log L87b
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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79
n-pentane n-hexane 2-methylpentane n-heptane 2,4-dimethylpentane n-octane 2,5-dimethylhexane 2,3,4-trimethylpentane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane cyclopentane cyclohexane cycloheptane cyclooctane ethylcyclohexane cyclodecane 1-pentene 1-hexene dichloromethane trichloromethane tetrachloromethane 1,2-dichloroethane 1,1,1-trichloroethane 1,1,2,2-tetrachloroethane tetrachloroethene chloropropane chlorobutane chloropentane dibromomethane tetrabromomethane diethyl ether tert-butyl methyl ether dipropyl ether diisopropyl ether dibutyl ether isobutyl ether pentyl ether hexyl ether tetrahydrofuran tetrahydropyran p-dioxane methanol ethanol 1-propanol 2-propanol 1-butanol 2-butanol 2-methyl-1-propanol 2-methyl-2-propanol 1-pentanol isopentanol 1-hexanol 2-hexanol 1-heptanol 1-octanol cyclopentanol cyclohexanol 2,2,2-trifluoroethanol 1,1,1,3,3,3-hexafluoro-2-propanol acetaldehyde propionaldehyde butyraldehyde isobutyraldehyde valeraldehyde hexanal heptanal octanal nonanal acetone butanone 2-pentanone 3-pentanone 3-methyl-2-butanone 2-hexanone 2-heptanone
2.163 2.668d 2.507 3.173d 2.812 3.677d 3.309 3.401 4.176 4.685 5.191d 5.696d 6.200d 6.705d 2.426 2.906 3.543
1.714 2.176 2.055 2.628 2.322 3.071 2.754 2.876 3.529 3.977e 4.422 4.863 5.302 5.759 2.092 2.477 3.092 3.619 3.262 4.430 1.646 2.101 1.760 2.132 2.412 2.229 2.302 3.141 3.359 1.846 2.291 2.769 2.500 4.047 1.616 1.908 2.449 2.056 3.325 2.941 4.199 5.077 2.171 2.520 2.390 0.936 1.231 1.660 1.485 2.137 1.936 2.018 1.595 2.605 2.479 3.018 2.815 3.491 3.925 2.723 3.157 0.942 1.018 0.936 1.441 1.904 1.801 2.363 2.821 3.264 3.710 4.156 1.357 1.982 2.356 2.357 2.176 2.772 3.203
3.767 2.571 1.997 2.478 2.822 2.572 2.695 3.802 4.022 2.212 2.716 3.232 2.836 2.066 2.363 2.971 2.561 3.954 3.485 4.845 5.745 2.521 2.926 2.788 0.975 1.356 1.975 1.750 2.539 2.322 2.381 1.994 3.057 2.885 3.550 3.340 4.067 4.569 3.107 3.594 1.315 1.370 1.230 1.770 2.270 3.370 4.293 4.750 1.766 2.269 2.726 2.779 2.560 3.262 3.760
log L87c (Kova´ts) 1.720 2.178 2.631 3.082 2.890 3.530 3.977 4.422 4.866 5.300 2.106 2.482 3.094 3.631 4.434 1.644 2.103 1.794 2.156 2.411
2.292 2.746 2.483
2.445 3.328 4.196 2.144 2.384 1.675 2.151 1.946 2.610 3.024 2.842 3.498 3.924
2.003 2.376 2.818 3.238
no.
compound
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157
2-octanone 2-nonanone cyclopentanone cyclohexanone acetonitrile propionitrile butyronitrile isobutyronitrile valeronitrile hexanenitrile ethylamine propylamine butylamine diethylamine hexylamine triethylamine methyl formate ethyl formate propyl formate methyl acetate ethyl acetate propyl acetate isopropyl acetate butyl acetate isobutyl acetate pentyl acetate isopentyl acetate hexyl acetate ethyl propionate ethyl butyrate isobutyl isobutyrate dimethylformamide dimethylacetamide nitromethane nitroethane nitropropane dimethyl sulfoxide benzene toluene ethylbenzene o-xylene m-xylene p-xylene propylbenzene butylbenzene 1,2-diethylbenzene 1,4-diethylbenzene 1,2-dipropylbenzene 1,4-dipropylbenzene 1,2-dibutylbenzene 1,4-dibutylbenzene 1,2-dipentylbenzene 1,4-dipentylbenzene naphthalene fluorobenzene chlorobenzene o-dichlorobenzene p-dichlorobenzene bromobenzene iodobenzene anisole benzyl alcohol 2-phenylethanol 3-phenylpropanol 4-phenylbutanol phenol o-cresol m-cresol p-cresol benzaldehyde acetophenone benzonitrile benzyl cyanide aniline N-methylaniline N,N-dimethylaniline pyridine nitrobenzene
log L16a 4.755 3.093 3.580 1.537 1.978 2.456 2.258 2.972 3.472 1.646 2.083 2.575 2.386 3.612 3.008 1.454 1.894 2.411 1.946 2.359 2.861 2.612 3.361 3.168 3.852 3.675 4.341 2.880 3.320 2.922 3.357 1.839 2.313 2.773 3.110 2.792 3.343 3.785 3.947 3.868 3.867 4.239 4.714
2.785 3.630 4.453 4.405 4.022 4.505 3.916 4.162 4.552 4.800 5.049 3.641 4.183 4.187 4.254 3.935 4.458 3.913 4.570 3.934 4.492 4.753 2.969 4.433
log L87b 3.623 4.081 2.706 3.161 1.185 1.662 2.139 1.956 2.573 3.017 1.452 1.815 2.299 2.030 3.111 2.523 0.861 1.425 1.975 1.530 1.990 2.402 2.205 2.853 2.673 3.287 3.122 3.722 2.398 2.792 3.284 2.609 2.977 1.663 2.023 2.416 2.690 2.480 2.926 3.297 3.463 3.378 3.372 3.678 4.099 4.154 4.154 4.794 4.894 5.586 5.764 6.380 6.557 4.631 2.422 3.218 3.924 3.980 3.597 4.070 3.450 3.753 4.043 4.263 4.487 3.407 3.739 3.764 3.761 3.504 3.956 3.479 3.989 3.506 3.998 4.223 2.620 3.992
log L87c (Kova´ts) 3.646
1.657 2.067 2.567 3.010
1.993 2.406 2.846 3.287 3.719
1.665 2.024 2.390 2.928 3.307
3.664 4.090
4.608 2.372 3.204 3.574 4.017
2.605
a Measured at 25 °C by open-tubular GC, from ref 7 unless otherwise indicated. b Measured at 39.2 °C by open-tubular GC. c Calculated at 39.2 °C by packed-column GC, from ref 16. d Reference 3. e Used as reference compound for calculating phase ratio.
3714 Analytical Chemistry, Vol. 70, No. 17, September 1, 1998
log L8740,capillary ) (1.000 28 ( 0.000 81) log L8740,packed N ) 63
R ) 0.9998
sd ) 0.019
(3)
Since the packed-column L87 value for decane was used to calculate the film thickness for the capillary measurements, the slope of nearly 1 and the intercept of 0 are not surprising. However, the goodness of the fit would be the same, regardless. The quality of this regression suggests that these two data sets are essentially identical and that the open-tubular data offer no improvement in interfacial adsorption over the packed-column data for the 63 solutes tested. However, this does not necessarily indicate the complete absence of interfacial adsorption effects, as we have clearly demonstrated that problems with the compounds discussed above do exist. Since precise measurements of L87 for several compounds (such as the carboxylic acids) could not be obtained, we have already removed the compounds that would undergo the strongest adsorption. For the limited data compared here, packedcolumn GC gives equally good L87 values as open-tubular GC. Closer inspection of the data reveals the existence of real deviations between the packed-column data and the open-tubular data. In Figure 1A, we plotted the difference of log L8740,packed log L8740,capillary versus log L8740, capillary. If log L8740, capillary and log L8740, packed are equivalent; the plot will be scattered randomly around zero. Figure 1A shows that there are some small but distinguishable trends in the scatter. The apparent partition coefficients for polar compounds, especially hydrogen bond acids, are slightly higher for the packed-column measurements than for the capillary measurements. This is presumably due to larger contributions of interfacial adsorption effects to the apparent partition coefficients for the packed columns. Figure 1A also shows that L87 values for nonpolar aromatic solutes are generally less for packed columns than for open tubes. We have no explanation for this behavior, since these solutes should not adsorb on packed beds or capillary surfaces. The deviations mentioned above are in agreement with those observed by Carr et al. for the comparison of log L16packed against log L16capillary.7 Figure 1B shows the residual plot for these data sets. We see the same trends in the residuals for the polar, protic, and aromatic solutes, but the deviations are larger. On the basis of this residual analysis, we conclude that there are slight differences, due to interfacial adsorption, in log L8740,packed and log L8740, capillary. However, these differences are extremely small, and either method is suitable for the measurement of L87. The decrease in interfacial adsorption from log L16 at 25 °C to log L87 at 40 °C likely exists because Kova´ts’ log L87packed data were measured at high temperatures (where interfacial adsorption is less important) and extrapolated to 40 °C. A strong correlation also exists between the open-tubular log L87 data at 40 °C (log L8740, capillary) plotted against the log L16 data at 25 °C (log L1625). Linear regression gives the following correlation:
log L1625 ) (0.175 ( 0.024) + (1.1004 ( 0.0082) log L8740, capillary N ) 139
R ) 0.996
sd ) 0.093
(4)
This regression analysis shows a fairly good agreement between
Figure 1. Deviations between packed-column (A) and open-tubular (B) log L87 measurements. The solutes are represented as follows: nonpolar aliphatic (O), polar aliphatic (0), protic aliphatic (4), nonpolar aromatic (b), polar aromatic (9), and protic aromatic (2).
the two data sets. The differences in the intercept from zero and the slope from unity are not unexpected. They are most likely due to the difference in temperature, or possibly to a different blend of dispersion interactions and cavity formation in the two hydrocarbons. If we break down the data into subsets of similar compounds (e.g., toluene, ethylbenzene, propylbenzene, etc.), we Analytical Chemistry, Vol. 70, No. 17, September 1, 1998
3715
find even better linear agreement between log L8740,capillary and log L1625. Whether we use eq 4 for all compounds or the equation for a specific class of compound, we can obtain excellent estimates of log L1625 via the much simpler measurement of log L87. We have already shown that the values of log L87packed that were extrapolated to 40 °C compared remarkably well with the log L87capillary values that were measured directly. If we continue the extrapolation down to 25 °C, we can compare L87 and L16 at the same temperature. This comparison is given by eq 5. By
log L1625 ) (1.062 ( 0.011) log L8725, packed N ) 58 L16
R ) 0.998
sd ) 0.066
(5)
L87
comparing and at the same temperature, the intercept disappears, and the slope is much closer to unity. The goodness of fit also improves. We should therefore be able to measure log L87 at high temperatures, extrapolate the values to 25 °C using Kova´ts’ model, and obtain a good estimate of log L16 for use in LSERs. Use of L87 in LSERs. Although an empirically based prediction of log L16 would be a great advantage to LSER studies, a better possibility might be to substitute log L87 for log L16 in the general LSER equations. For example, eq 2 could be replaced by eq 6. In these equations, both log L16 and log L87 should model
log k′ ) log k′0 + l log L87 + sπ*2C +RR2C + dδ2
quality of these fits compares favorably with any other reported fits of GC data. The coefficients (l, s, a, d) of the correlations using the two different LSERs must also be comparable. For these data, the values of the coefficients are not statistically equal for the two LSERs. However, this is not unusual, since L87 could be made up of a different blend of dispersion and induction interactions and cavity formation processes than is L16. More importantly, the trends in the coefficients with changing polarity and temperature are similar. The strong relationship between the two sets of coefficients provides further support for the replacement of L16 with L87 when convenient.
(6)
dispersion and cavity formation, and as stated earlier, the measurement of L87 at elevated temperatures is much simpler than the measurement of L16 at 25 °C. Equation 6 was used to correlate Li’s GC retention data. Although the quality of the fits, as judged by the standard error (sd), is slightly better when eq 2 is used, very good fits (R2 > 0.994) are also obtained with eq 6, and the
3716 Analytical Chemistry, Vol. 70, No. 17, September 1, 1998
CONCLUSION Apolane-87 is proposed as a new nonpolar reference to be used in LSERs. Both packed-column and open-tubular GC can be used to obtain accurate measurements of L.87 Although interfacial adsorption takes place in both techniques, the effects are not as severe as they are for the measurement of L16. Additionally, L87 can be measured at any temperature between 30 and 280 °C with higher accuracy and presumably for a much wider array of solutes. The L87 values compare fairly well with L16, especially for various subsets of the data, and L87 can be used to predict L16. We can also substitute L87 for L16 in a LSER without sacrificing goodness of fit or the chemical relevance of the coefficients. ACKNOWLEDGMENT This work was supported by grants from the Chemical Analysis Program of the National Science Foundation and from the Graduate School of the University of Minnesota.
Received for review December 20, 1997. Accepted June 11, 1998. AC971370H
