1452
Anal. Chem. 1984, 56, 1452-1457
Correlation of Elution Orders in Different Liquid Chromatographic Systems without Analyte Identification Robert E. Synovec and Edward S. Yeung* Department of Chemistry and Ames Laboratory, Iowa State University, Ames, Iowa 50011
To optlmire the separatlon of unidentlfled components In a mlxture, or to obtain quantitative informatlon about the components prlor to ldentlflcatlon, lt Is necessary to determlne the elutlon orders of the components when different columns or different eluents are used. Such a procedure for correlating peaks In ilquid chromatographyIs presented for the refractlve index detector. The concept Is based on a conslstency test derived from three of these unldentlfled chromatograms. Wlth very few exceptions, one of the NI elution orders can be chosen based on a least-squares crlterlon. A confldence level can also be speclfled for the chosen elution order. This concept is demonstrated for a four-component mixture as eluted from two dlfferent columns using three dlfferent eluents.
The use of high-performance liquid chromatography (LC) for the analysis of complicated mixtures of organic substances is a common occurrence. In some cases the analyst has previously identified the analyte or analytes of interest before doing any chromatography. Usually, this is not the case. It is more likely that an analyst is faced with the task of providing qualitative and quantitative information on a sample with little or no initial information. The problem of providing the information of interest and optimizing the chromatographic separation for routine analyses can become a timeconsuming and expensive task. When analyte identification has not preceded the chromatography,implementingmethods such as standard additions or construction of a calibration curve are obviously impossible. In principle, the response of many LC detectors can provide quantitative information simultaneous with the identification. In practice, mass spectrometry ( I , 2 ) , infrared spectrometry (3, 4, or nuclear magnetic resonance applied to detection in LC can only provide semiquantitative information, because analyte identification using any of these techniques alone is still not completely unambiguous. Recently it was reported that both qualitative and quantitative information about a sample can be obtained from the chromatograms produced by two eluents with different refractive indexes (RI) using RI detection ( 5 ) . No prior identification of any solutes was necessary. If however an analyte mixture produces more than one peak in a chromatogram, it is necessary to know which peaks correlate with the peaks in the other chromatogram. The RI detector gives a response that depends on the RI of the eluent, so the peak areas alone cannot be used directly to determine elution orders. This is unlike the case of an absorption detector, where peak areas for the individual analytes are expected to remain constant for most nonabsorbing eluents. To obtain quantitative information (5),one must either be able to preserve the elution order for the individual analytes when changing eluents or be able to separately correlate the elution orders in the two eluents. The former approach works in certain cases, e.g., in gel permeation chromatography (6) and in ion chromatography (7, 8). The latter is the subject of this investigation. It is well known that predictions about elution orders from the same column for an arbitrary set of analytes in different eluents are difficult, except for the cases mentioned above.
If different columns are used, the situation is even more serious. This is especially true if the first chromatographic system performs a normal-phase separation and the second chromatographic system provides a reversed-phase separation. In the process of optimizing the chromatographic conditions for an unknown sample, the analyst is thus faced with the difficult task of following the retention behaviors of each of the unidentified components using different columns and different eluents. This then is another reason for developing some scheme to correlate the observed elution orders from different chromatographic systems without analyte identification. THEORY The correlation scheme for determining the elution orders for a sample consisting of N analytes in two different eluents (regardless of the columns used) is based on obtaining the chromatogram of the same sample in a third eluent (regardless of the column used). To use the RI detector, this third eluent should have an RI different from those of the first two eluents. It is assumed that the N chromatographic peaks obtained in each of the three eluents are resolved well enough to obtain areas for each peak, by deconvolution if necessary. If the peaks are poorly resolved in either one of the first two eluents, the particular eluent-column combination will not be useful for studying the sample anyway. If the peaks are poorly resolved in the third eluent, one can find a different eluent-column combination that gives better separation. We have already shown ( 4 6 ) that if the correct elution orders are known, then any two of these chromatograms will provide enough information to predict the peak areas in the third, or any other, chromatogram. This is because the two chromatograms can be used to calculate, without analyte identification, the concentration and the RI of each component in the sample. This information can then be used to predict the response of each of the components in an arbitrary third eluent. So, the unknown elution order in the third eluent can be determined simply by matching up the peak areas found experimentally and the predicted peak areas. However, if the elution orders are not known in either of the first two eluents, there can then be a total of N! possible combinations of predictions from the first two chromatograms. The idea then is to find the one combination out of the N! combinationsthat best predicts the third chromatogram with respect to the individual peak areas, with the help of some least-squares criterion. Thus, one relies on achieving consistency in the quantitative information among the assigned elution orders in the three chromatograms to arrive at the best choice, without requiring analyte identification. We note that peak areas are used rather than peak heights because they reflect the total amount of each component injected, independent of differences in the retention times, the efficiencies of the columns, and the flow rates used for the three separations. We have shown (6-8) that to obtain quantitative information, one does not even need to independently determine the RI’s of the eluents or the actual response factor of the detector, if two “calibrating” substances A and B of known amounts V (volume fraction) can be eluted from the chromatographic system under the same conditionsthat produced
0003-2700/84/0356-1452$01.50/0?2 1984 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8 , JULY 1984
the three chromatograms. The response SIAin arbitrary area units, of substance A in eluent 1 is given by (5)
SIAKl = V(FA- F J
K2
K1
SZA-S2’
(2)
and
K2
_ - S3A- S3’
K3
SZA
- S2B
for each of the N! combinations, one can calculate a relative standard deviation for the fit
(1)
where K1 is a constant that contains the RI and the flow rate of eluent 1and the conversion factor relating S to the real RI units, and Fj (n: - l ) / ( n t + 2 ) is a function of the RI of the material i. By successively obtaining the responses of the two calibrating substances in the three eluents, one has six expressions of the form in eq 1. One can show (6) that
_ - SIA-S1’
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(3)
1
4
/
where the sums are over all peaks in the chromatogram. The number of degrees of freedom is ( N - 2) because both S”‘ and Sx are estimates of the “true” values in the chromatogram. The subscript INST denotes this method of weighting, commonly called instrumental weighting. When experimental variances are not available, e.g., when only one trial is used, one can use statistical weighting (9))Le., the weights are the reciprocals of the squares of the individual areas. The implication is that the standard deviation of a given peak is proportional to its magnitude. Thus
We note that this “calibration” need only be performed once for a given eluent at a given operating condition. Now, for any one of the analytes in the sample, its concentration, V , (volume fraction), can be calculated from its peak areas in any pair of chromatograms (6))so
or
Combining eq 4 and 5, one can then predict the peak area observed for this analyte in the third eluent, S3,’,from its peak areas in the other two eluents, S1”and SZx. The result is
S3,’ = (as1” + bS,,)/c
(6)
where
The least-squares criterion simply requires that the sum of squares of the residuals, appropriately weighted, be minimized. In other words, the relative standard deviation calculated either from eq 7 or 8 should be minimized. This then produces a unique choice among the N ! combinations of elution orders as the one that is the most consistent for all three chromatograms. This set of elution orders can be taken as the “correct” set for describing the N analytes in the three chromatographic systems. Once the correct elution order is found, quantitation of each component following eq 4 or 5 is straightforward. One can also proceed to calculate the refractive index of each of the N components, as given by eq 17 in ref 5, since
F, = and For the N peaks in each of the first two eluents, there is then a total of W possible predicted areas according to eq 6, arranged in N! distinct combinations. These predictions are obtained without analyte identification, and without knowing any physical property of the analytes, the eluents, and the calibrating substances. Now, each of these N! combinations can be tested against the experimentally observed third chromatogram, with areas S3xfor each of the peaks, by a least-squares criterion. Since the choice is ordinal, each peak should contribute equally to the decision making regardless of its magnitude. It is thus appropriate to weigh the squares of the residuals, ISx‘- PI2,by the estimated variances for the predicted area, and for the experimental area, a$, before applying the least-squares criterion (9). These estimated variances can be obtained if multiple chromatograms are available for each eluent so that each S value has associated with it an experimentally determined variance. The variance for the predicted area can be calculated from the individual variances of each of the terms in eq 6 by considering the statistical propagation of errors (IO)and is given in the supplementary material available for this paper. (See paragraph at end of paper for ordering information.) The result is that, 2
9
8
,
[ ””) [/I.-
(““)
SzXK2 - 1 1
F2( SzXK2
(9)
Naturally, one needs to associate some level of confidence with the chosen elution order as derived from minimizing either eq 7 or 8. That is, how good is the choice with the minimum u compared to that with the next smallest a? One can think of u as simply a quantity derived from the set of all experimentalmeasurements-the 3N peak areas. It is thus possible to use standard techniques for propagation of errors to determine the variations in u as a function of uncertainties in the individual area measurements. To a first approximation, one can consider the weighting factors in either eq 7 or 8 to be constants. This greatly simplifies the calculations, so that one obtains an estimate of the uncertainty in the u values (see supplementary material) with
+
CW2(U2&y
a2y)
(S”’- SX)2
1
(CW2(SX‘- SX)2)(Cw2)(ClS”1)2
(10) The weighting factors, w ,are 1/(uzSz;+u2s=)for u(uINST)and 4/(ISx’( + lSx1)2for USTA TAT), respectively. Assuming normal Gaussian distribution of errors, this means that since each experimentalarea has a 68% chance for being within the range of Sx f us=in any particular trial, the value of u in either eq 7 or 8 also has a 68% chance for being within the range of u f u ( u ) . If the best choice and the second-best choice, ac-
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
cording to the summed least squares of the residuals, are separated by an amount equal to u(u), there is better than a 76% chance that this particular ranking of the u’s will be preserved in future experiments. The probability can be calculated explicitly following the procedure in the supplmentary material. We can say that our chosen elution order has associated with it a 76% or better confidence level. Similarly, if the two best choices have u’s separated by 2u(u), one can show that the confidence level of the choice is better than 92%. Finally, there are two additional checks for the chosen elution order, although neither can be used independently to provide a clear choice among the N! possible elution orders. First, the total area calculated from eq 6 for all peaks in the third chromatogram should equal the total experimental area so that the entire area is accounted for. One can define
system was operated at 1.09 mL/min. Mixtures, for the analysis, with specified volume fractions were made by pipetting welldefined volumes of each component. The output of the RI detector (10 mV full scale) is connected to a digital voltmeter (Keithley, Cleveland, OH, Model 160B), the analog output of which is in turn connected to a computer (Digital Equipment, Maynard, MA, Model PDP 11/10 with a LPS-11 laboratory interface). The computer takes readings every 0.05 s and averages each set of 10 before storing the information. The peaks in each chromatogram are, then, base line adjusted via software before integration to produce peak areas or integrated response values. All areas are determined by using multiple injections (three or more). Linearity of the detector had been previously confirmed over the attenuation settings used in this study (32X to 2X). The computer was used in all calculations pertaining to the correlation scheme used. Basically no time delay was observed for the calculations necessary to do the correlation with N = 4 peaks in three chromatograms,although the computer time should increase at roughly an N! rate.
RESULTS AND DISCUSSION Ideally, r should equal unity. Second, one can calculate the analogue of a correlation coefficient (IO)as the ratio of the experimental deviations from the best-fit to the combined variances of the predicted areas and the experimental areas.
so
where u2Nw is the unweighted sum of the squares of the residuals, given by
(13) Equation 12 tells whether the lack of fit is fully explained by uncertainties in the predicted and the experimental areas. If R is larger than unity, the fit is better than the experimental uncertainties. If R is smaller than unity, the fit is worse than is expected from the experimental uncertainties. These two additional checks can be used to decide whether there exists some unusual problems with the chromatograms, such as some components not eluting off the column or that one has N peaks for a sample with more than N components, some of which are not resolved.
EXPERIMENTAL SECTION All reagents and eluents used were reagent grade with no further purification. Three chromatographicsystems were employed. A high-pressure pump (ISCO, Lincoln, NE, Model 314), a 1-pL sample loop at a conventional injection valve (Rheodyne,Berkeley, CA, Model 7410), and a commercial RI detector (Waters Associates, Milford, MA, Model R401), with the reference cell used in the static mode filled with the eluent being used, were used in all three chromatographicsystems. The differences are in the columns and eluent composition. The first system used a 25 cm X 4.6 mm, 5 pm, silica column (Anspec,Warrensville, IL) and an eluent composed of 99.8%toluene and 0.2% methanol by volume. The second system used the same column as the first system while the eluent was composed of 99.8% n-butyl chloride and 0.2% methanol by volume. The third system used a 25 cm X 4.6 mm, 10 pm, CIS column (Anspec, Warrensville, IL) and an eluent composed of 60% distilled and deionized H20 and 40% CH&N by volume. RI values for the eluents are as follows: n(toluene/methanol) = 1.4936, n(n-butyl chloride/methanol) = 1.4009, and n(60% H20/40% CH,CN) = 1.3443. These RI values were calculated by considering literature naD values and the theory in ref 5. The first system operated at an eluent flow rate of 1.00 mL/min. The second system had a flow rate of 0.67 mL/min, and the third
To test this correlation procedure, the three eluent-column combinations were chosen to provide a range of RI’s and different separation mechanisms. The former is necessary to retain significance (7) in the subtractions in eq 4 and 5. The latter results in a t least some scrambling of the elution orders to simulate real situations. The first step is to use calibrating substances A and B to standardize the three chromatographic systems. We have chosen p-anisaldehyde (n20D= 1.5730) and ethyl acetate ( n 2 0 = ~ 1.3724) because of their very different RI values, again to assure significance in the calculations. The observed areas are tabulated in Table I of the supplementary material. QS values are determined from the individual set of repeated injections (23). Two mixtures of four components each are used to test the correlation scheme. The four components chosen are chloroform (nZ0D= 1.4458), benzaldehyde (n20D = 1.5463), 3pentanone (n20D= 1.3924), and tetrahydrofuran (n20D= 1.4050). These provide a large variation of RI’s, as well as a small difference in RI for two of the components. An equal-volume mixture, I, allows an evaluation of the case in which two components have similar RI’s and are at similar concentrations. Another mixture, 11, consists of the same components a t different concentrations, chosen so that the peak areas in one of the eluents are quite close to one another. These four components are well separated in each of the three chromatographic systems. To show the separation in the three chromatographic systems, we have reconstructed a set of chromatograms for mixture I1 from the experimental retention times and the experimental peak widths and have plotted these on the same scale expansion (SX). These are shown in Figure 1. As indicated, chromatogram 3 has peak areas that are quite similar for all four components. Also, the area of the fourth peak in chromatogram 2 was actually measured with reasonable precision because a different scale expansion was used during data acquisition. The experimentally determined areas for each peak in the three eluents are listed in Table I1 of the supplementary material for mixture I and in Table I11 of the supplementary material for mixture 11, together with the corresponding standard deviations for the multiple trials. These data are listed with the particular analyte because the true elution orders have been determined by independent experiments. Naturally, the areas are simply indexed 1 through 4 in their order of elution in that particular chromatographic system when they are used in the correlation calculations. With these results one can proceed to use eq 6 to form the IP possible area predictions arranged in N ! different elution orders. For each of these possible elution orders, one can calculate the relative standard deviation of the fit either with instrumental weighting, eq 7, or with statistical weighting, eq
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
1455
Table I. Results of Least-Squares Fit for Mixture I rank of fit
1
eluen tsa
'INST
X X X Y
1
1.43
2 3
1.80
Y
2 3
Y Z Z Z
1 1
2 3
4.23 1.70 1.74 4.94 2.64 2.71 13.4
~(OINST 1"
OSTAT~
0.35 0.37 0.33 0.41 0.41 0.39 0.60 0.62 0.55
0.91 1.76 3.68g 0.68 0.71 1.76g 2.98 2.98 12.8
1"
'('STAT
0.31 0.59 0.35g 0.77 0.60 O.3lg 0.69 0.69 0.53
Re
rf
0.64 0.47 0.17 0.64 0.63 0.17 0.63 0.63 0.15
1.04 1.04 1.04
0.95 0.95 0.85 0.96 0.96 1.08
a "he eluents are, respectively, (1)toluene, (2) n-butyl chloride, and (3) 60:40 H,O:CH,CN; (X) areas in 1 and 2 used to predict area in 3; ( Y ) areas in 1 and 3 used to predict area in 2. and (Z) areas in 2 and 3 used to predict area in 1. Equation 7. Equation 10. Equation 8. e Equation 12. f Equation 11. g A different combination compared to 'INST was chosen.
Table 11. Results of Least-Squares Fit for Mixture I1 eluentsa X X X Y Y Y Z Z Z
rank of fit 1
2 3 1
2 3 1
2 3
'INST
1.82 2.65 6.16 1.83 1.97 2.28 1.53 1.60 1.91
'('INST)"
0.37 0.37 0.46 0.58 0.62 0.58 0.46 0.49 0.46
'JSTAT~
1.52 4.91g 5.20g 1.62g 1.82g 1.91g 0.57 0.75g 1.06g
'("STAT
1"
0.58 0.4gg 1.44g 2.33g 2.079 2.07g 0.23 0.2!jg 0.4gg
Re
rf
0.96 0.31 0.29 0.66 0.77 0.51 0.65 0.76 0.51
1.06 1.06 1.06 1.07 1.04 1.07 0.93 0.96 0.93
a The eluents are, respectively, (1)toluene, (2) n-butyl chloride, and (3) 60:40 H,O:CH,CN; (X) areas in 1 and 2 used to gredict area in 3; ( Y ) areas in 1and 3 used to predict area in 2; and ( Z ) areas in 2 and 3 used to predict area in 1. Equation 7. " Equation 10. Equation 8. e Equation 12. f Equation 11. g A different combination compared to OWST was chosen.
8, and even the corresponding uncertainties in these relative standard deviations, eq 10. The two additional parameters F and R from eq 11 and 12 can also be calculated. These are tabulated in Table I for mixture I and in Table I1 for mixture 11. Since any two of the three chromatograms for each mixture can be used to predict the areas in the third chromatogram, results are included for the three possible choices for the first two eluents. Only the three best fits in each case, as dictated by UINST and by CTSTAT, of the N! combinations are tabulated. Since there is only one correct set of elution orders, it should not matter in principle which two of the three chromatograms are used to predict the third. This is evident in Table I, where the choice giving the smallest uINST for each eluent combination refers to the identical set of elution orders. In fact, regardless of whether uINST or "STAT was used as the leastsquares criterion, the identical set of elution orders is always chosen. This is also the true elution order. One has to be more careful, however, in making such interpretations. For example, the tabulated u(u) values imply that both Q ~ S and T USTAT for the fourth vs. fifth entry, and for the seventh vs. eighth entry, are really not significantly different. For eluent combinations (Y)and (Z),one therefore has insufficient grounds to decide whether the lowest u or the second lowest u should be chosen to represent the best set of elution orders. It is therefore fortuitous that the correct elution orders were also predicted by eluent combinations (Y) and (2).A clearer choice results from eluent combination ( X ) ,where we find that the first and second entries are separated by about one u(u) for uINST and even better for STAT. We can thus pick the elution order corresponding to the lowest uINST with 76% confidence, as discussed above. In examining the third lowest (T for each eluent combination,one finds that the values are all separated from the second lowest u by several u(u)'s. One can then reject the third and higher choices in each case as being correct with
'I 10
6
0
0
6
6
10
10
16
15
e0
T I M E , MIN
Figure 1. Reconstructed chromatograms showing the separation of the four components in mixture I1 in the three eluents: (1) toluene; (2) n-butyl chloride; (3) 60:40 H,O:CH,CN.
at least 99.99% confidence (see supplementary material). As expected, the top two choices in each eluent combination
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
Table 111. Correlation Results and Calculated Parameters predicted elution ordera analyte
eluent 1
chloroform benzaldehyde 3-pentanone tetrahydrofuran
1 2 3 4
eluent 2 Mixture I
eluent 3
calcd vol, pL
vol, pL
1
4 3 2
0.25 0.24
0.25 0.25
0.24
0.25
1
0.22
0.25
4 3 2
0.20 0.10 0.40 0.26
0.20
2 3 4
true
calcd RI 1.4437 1.5412 1.3854 1.3916
Mixture I1 chloroform benzaldehyde 3-pentanone tetrahydrofuran a
1
1
2 3 4
2 3 4
These are also the correct elution orders.
1
0.10
0.40 0.30
1.4438 1.5367 1.3861 1.3918
Assuming n, = 1.4936 and n, = 1.4009.
correspond to interchanging the assigned elution order for components 3 and 4, which have nearly identical RI’s and identical concentrations. We note that for the sixth entry, the two a’s did not produce the same ranking. In general, if experimental us values are available, one should rely on aINsT. USTAT is used only if single chromatograms are obtained or if there is reason to believe the uncertainties are proportional to the magnitudes of S. The bottom line is that we should use only the first entry in Table I, based on urnsT and u(umsT), for a correlation of the elution orders at a 76% confidence level. Table I1 provides a much more difficult case for testing the correlation scheme, because of the proximity of the areas in one chromatogram. For eluent combination (X), the two smallest u ~ S Tare separated by about two times a(a). We can thus choose this particular elution order with 92% confidence. If we examine the other two eluent combinations, it is not even possible to distinguish among the top three choices in each case based on the ).(a values. In fact, many other choices (not tabulated) beyond the first three are also less than one a(a) away from the minimum value. Therefore, the only valid prediction is offered by the eluent combination (X), which also corresponds to the correct elution order as determined independently. All of the tabulated choices in Table I1 for eluent combinations (Y)and (2)were incorrect elution orders. The point of interest is that the overall smallest u is not necessarily the one corresponding to the best choice. One must be able to establish a confidence level for the choice, which may mean trying each combination of the two “predicting” eluents until a reasonable confidence level is achieved in one of them. A qualitative explanation can be given as to why in this particular sample the eluent combination of (X)used for prediction seems to outperform the others in the correlation scheme. An examination of the raw data shows that the data obtained in eluent 3 for either mixture are overall more precise than those in the other eluents. In calculation of the residuals in eq 7 or 8, it is better to have a precise, and thus presumably more accurate, value for one of the terms, i.e., S”,rather than to have the precision (and accuracy) of this data set diluted when it is incorporated into the Sr‘ term. The value of R in eq 12 is not a true correlation coefficient but does give an indication of the “accountability” of the errors. The value is always positive, and can be larger than unity. Tables I and I1 show that R alone cannot be used to determine the elution order. However, the better fits all show a reasonable value for R of the order of unity. The value of r in eq 11 accounts for all the areas observed in the third chromatogram. Tables I and I1 show that for the best fits, the areas are properly accounted for, i.e., one did not “lose” area by having some material not eluting in the third chro-
matographic system and did not “gain” area by eluting some extra components off the column. After the elution orders are determined, the concentration of each component can be calculated from eq 4, and its RI can be calculated from eq 9. These are tabulated in Table 111. The “true” volumes are also listed there and the literature RI values were given earlier in this section. One should note that the RI’s here are measured at a wavelength of 520 nm because of the particular instrument. The comparison with literature values is only appropriate if one assumes that the RI’s of the analytes and of the eluents can be linearly extrapolated from their Na D line values. The tetrahydrofuran volumes are somewhat low and may be related to nonideal solutions, as evidenced by heat produced when the mixtures are prepared. Good agreement is indicated therefore in all cases. We have repeated this procedure for a third mixture consisting of benzaldehyde,methyl acetate, methyl ethyl ketone, and cyclohexanone in the same three chromatographic systems. In normal-phase LC, this was the elution order. In reversed-phaseLC, benzaldehyde became the last component to elute while the others maintained their order of elution. Our correlation scheme works even in this case with some scrambling of the peaks. The other findings are about the same as in mixture I and 11. There, the precision of the three chromatograms was about the same, and the level of confidence in the least-squares choice of elution orders turns out t o be about the same regardless of which eluent pair is used for the predictions. We can now discuss the limitations of this correlation procedure. First, if the values of the peak areas in each chromatographic system are separated by more than the individual precisions, as,and if precision implies accuracy, one can expect to arrive at a reliable correlation at a high confidence level. The more different the peak areas are from each other, the higher is the confidence level of the correlation. We note that similar areas will be produced by substances with both similar concentrations and similar RI’s, e.g., the last two entries in Table I1 in the supplementary material. However, similar areas can also be produced by fortuitious combinations of substances with different concentrationsand different RI’s, e.g., the last four entries in Table 111 in the supplementary material. Secondly, to obtain the initial correlation in three eluents, their RI’s must be quite different to allow eq 6 to become significant. Once the initial correlation is obtained, however, any two of these chromatograms can be used to predict the areas, and thus correlate the elution order, in an arbitrary fourth eluent, regardless of its RI. This is because one no longer needs to go through N! combinations. The problem reduces to one for the direct application of eq 6. Thirdly, to obtain the best quantitative information after the
Anal. Chem. 1984, 56, 1457-1460
elution orders are determined, one needs only two of the chromatograms (5). The best choice will be either the pair that is most different in RI (for the best detection limit) or the pair that has the most precisely determined peak areas (for the best precision) while providing a moderate RI difference. Fourthly, when some of the peaks in the chromatograms can be correlated independently, e.g., identified analytes, one can eliminate these (mathematically) before using the correlation scheme. This reduces the number N that must be correlated without identification. This is in fact closer to a “real” situation. For the difficult unknown cases, e.g., the fourth vs. fifth entries in Table I, one has actually limited the choices to 2!. A more sensitive test for consistency then results, particularly if the experiments can be repeated to produce more precise results for these two analytes in the three eluents. In summary, we have developed a completely objective scheme for correlating elution orders in arbitrary eluents without analyte identification. This should aid in the development of the optimum chromatographic conditions for separating the components in an unknown sample. It also extends the usefulness of quantitiation schemes that do not require analyte identification (5)to allow for changes in elution orders in different chromatographic systems. Supplementary Material Available: Derivations of key equations and three tables of experimental data (8 pages). Photocopies of the supplementary material from this paper or microfiche (105 X 148 mm, 24X reduction, negatives) may be
1457
obtained from Microforms Office, American Chemical Society, 1155 16th Street, N.W., Washington, D.C. 20036. Orders must state whether for photocopy or microfiche and give complete title of article, names of authors,journal issue date, and page numbers. Prepayment, check or money order for $13.50 for photocopy ($15.50 foreign) or $6.00 for microfiche ($7.00 foreign), is required and prices are subject to change. LITERATURE CITED Hamming, M. C.; Foster, N. G. ”Interpretation of Mass Spectra of Organic Compounds“; Academlc: New York, 1972. Gross, M., L., Ed. “High Performance Mass Spectrometry: Chemical Applications”; American Chemical Society: Washington, DC, 1978. Miiier, R. G. J., Stace, B. C., Eds. “Laboratory Methods in Infrared Spectroscopy”, 2nd ed.; Heyden: London, 1972. Jlnno, K.; Fujimoto, C.; Ishii, D. J. Chromatogr. 1982, 239, 625-632. Synovec, R. E.; Yeung, E. S. Anal. Chem. 1983, 54, 1599-1603. Synovec, R. E.; Yeung, E. S. J. Chromatogr. 1984, 283, 183-190. Wilson, S. A,; Yeung, E. S. Anal, Chlm Acta 1984, 157, 53-83. Wilson, S. A.; Yeung, E. S.; Bobbitt, D. R. Anal. Chem. 1984, 56, 1457-1460. Bevington, P. R. “Data Reduction and Error Analysis for the Physical Sciences”; McGraw-Hili: New York, 1969; Chapter 6. Younger, M. S. “Handbook for Linear Regression”, 1st ed.; Duxbury Press: North Scltnate, MA, 1979; pp 234-244.
RECEIVED for review October 28, 1983. Accepted March 28, 1984. The Ames Laboratory is operated by the U.S. Department of Energy by Iowa State University under Contract
No. W-7405-eng-82. This work was supported by the Office of Basic Energy Sciences. R.E.S. thanks the Dow Chemical Company for a research fellowship.
Quantitative Ion Chromatography without Standards by Conductivity Detection Steven A. Wilson, Edward S. Yeung,* and Donald R. Bobbitt
Department of Chemistry and Ames Laboratory, Iowa State University, Ames, Iowa 50011
Nonsuppressed Ion chromatography with a conductlvlty detector Is used to study solutlons of ions. A quantitative method is demonstrated whlch Is based on the detector response using first an eluent with a hlgh equivalent Ionic conductance and then an eluent with a low equivalent Ionic conductance. One can thus predlct the number of equlvalents and the equlvalent Ionic conductance of unidentlfled Ions. The Indlvidual charges of the ions are deduced from a study of the adlusted retentlon tlmes, 80 that the molar concentrationscan be determined. The method does not require any prlor knowledge of the Identity of the sample Ion or any of Its physlcal propertles.
Ion chromatography has developed into a very useful analytical technique in the past few years (1-3). Research in the development of low capacity columns and in detection methods has reached the point where suppression of the eluent conductance is no longer necessary (4-6). Methods of ion detection other than by conductivityhave been developed and some give slightly better overall detectabilities. However, the conductivity detector has some advantages and remains the most common detector for use in ion chromatography. Recently, we showed that absorbance due to samples of ions eluted successively by a strongly absorbing ion and by a weakly 0003-2700/84/0356-1457$01.50/0
absorbing ion can be used to determine the concentrations and the molar absorptivities of the sample ions (7). In what follows, we shall show that the conductivity detector can also be used as a detector in this quantitative scheme. Although the conductivity detector may show a slightly poorer overall detectability, it allows better characterization of most inorganic ions. This is because most inorganic ions do not absorb in the visible or near-UV spectral regions (8,9).So, even though our scheme allows the determination of the molar absorptivities (7), these are not very useful for characterizing the ions. With the use of the conductivity detector, this problem is solved due to the wide range of equivalent ionic conductances that inorganic ions display. The conductance of a solution of ions is related to their equivalent ionic conductances and their concentrations. The conductance of a solution of ions consisting of one anion and one cation, as is the case for a self-buffered eluent, is
C is the concentration (normality) of the ions, G is the conductance (mhos), and K is the cell constant (cm-’). X + O and Loare the limiting equivalent conductances of the cation and anion, respectively. These closely approximate the actual equivalent conductances of the ions in dilute solutions (IOw3 to N), such as those used in single column ion chroma@ 1984 American Chemical Society
