AMI. Chm. 1991, 63,1147-1154
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Comparison of an Experimental Competitive Isotherm and the Levan-Vermeulen Model and Prediction of Band Profiles in a Case of Selectivity Reversal Sadroddin Golshan-Shirazi,Jun-Xiong Huang,l and Georges A. Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6120
The recond-ordar Levan-Vemukn Isotherm pennits a correct account of the compaitlve kotherm data measured for the .quwakm ol trans- and &-androsterone between acetadtrlk/dlchkromethane solutlonr and a rillca gel modlfled wtth a pH = 6.8 phosphate Mer. I t kofspecial bnporlance to roped that the Levan-Vermeukn Isotherm accounts well for the wlectlvlty reversal observed at hlgh concentrations. The ck homer h more strongly retained at low concentratlorw, but Its column saturation capachy k lower than for the trans isomer. The lndlvldual elution band proflies for large a ” t s o f d x h # e s ol bans-and Cdr-andmterone ol relative concentrations 113, 111, and 311 are also accurately predicted.
INTRODUCTION The ability to predict the individual elution profiles of the Components of a mixture in liquid chromatography, under nonlinear conditions, is the key to a fundamental study of the optimization of the experimental conditions of preparative applications. Considerable efforts have been made toward this goal, and various procedures for the calculation of these band profiles by numerical integration of the differential mass balance equations have been reported (1-3). The comparison between experimental and calculated band profiles shows the great importance of an accurate determination of the equilibrium isotherms and of a suitable representation of these data ( 4 , 5 ) . For example, in the separation of enantiomeric N-benzoyl-D and L-alanine on immobilized bovine serum albumin, it was shown that the equilibrium data are well accounted for by the sum of two Langmuir isotherms (4). The first of these two terms corresponds to non-chiral-selective interactions, which are identical for both isomers. The second term corresponds to the chiral recognition mechanism, and althoug the numerical values of the coefficients are quite different for the two enantiomers, the column saturation capacity is the same for both, within the precision of the measurements. In this case, the competitive Langmuir isotherm model could be applied separately to each term of the isotherm. It provided an accurate description of the competitive adsorption behavior of the two enantiomers. The calculated elution bands are in excellent agreement with those obtained experimentally (4). In other cases, for example with 2-phenylethanol and 3phenylpropanol, the agreement between experimental and calculated chromatographic band profdes was less satisfactory (5). It was shown that the discrepancy arises from model errors in the representation of the competitive adsorption *Author to whom correspondence should be addressed, at the University of Tennessee. Present address: Institute of Eco-Environment, Academy of Sciences, Beijing, China. 0003-2700/91/0363-1147$02.50/0
behavior. The Langmuir competitive isotherm is not satisfactory in this case. The Langmuir isotherm has been originally derived on the basis of simple kinetic considerations to represent the adsorption behavior of gases and vapors (6). It can also be obtained by thermodynamic considerations, provided we assume the ideal behavior of the gas phase, an ideal localized adsorption monolayer, and the lack of adsorbate-adsorbate interactions. It is an experimental observation that the Langmuir isotherm accounts well for many gas*lid equilibria and for most liquid-solid equilibria, notably for those of interest in gas and liquid chromatography, as long as single components are involved (7). This justifies the use of the Langmuir equation to account for most single-component experimental data (8). The extension of the Langmuir isotherm to describe the competitive adsorption behavior of two gases is straightforward (9). However, the competitive Langmuir isotherm obtained is not in agreement with the Gibbs-Duhem relationship if the saturation capacities are different for the two components involved (10). For molecules of different sizes, the assumption of equal saturation capacities is unrealistic. Furthermore, the Langmuir isotherm predicts a constant separation factor, independent of the concentrations of either component. This is in contradiction with the experimental results when the saturation capacities are different and especially when the saturation capacity of the lesser retained component is the larger. Thus, it should be expected that the use of the competitive Langmuir model to account for the behavior of the components of a binary mixture cannot permit the calculation of band profiles in satisfactory agreement with those recorded experimentally. Because of the unsatisfactory results obtained with the competitive Langmuir model and the need of equations to account for the competitive adsorption behavior in different areas of separation science, several empirical equations have been suggested. Some use exponential expressions similar to the Freundlich isotherm (11-13). Ruthven and Goddard (14) and Lin et al. (15) have derived isotherms based on simple statistical models and that are the ratio of polynomials. The major inconvenience of these models is the number of their coefficienta that have to be determined, requiring the measurement of numerous, highly accurate data points. The Ideal Adsorbed Solution (IAS) theory was developed by Myers and Prausnitz (16) to permit the prediction of multicomponent isotherms using only data obtained from single-component isotherms. The theory is based on the concept of ideal behavior in the fluid and the adsorbed phase and uses classical surface thermodynamics and the Gibbs adsorption equation. It accounts for the difference in saturation capacity of the two components. Henson and Kabel (17) have shown that the IAS theory predicts competitive adsorption isotherms that are accurate at low coverages but deviate systematically from experimental data at high coverages. Later, Radke and Prausnitz (18) have extended the Q 1991 Amerlcan Chemlcal Society
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IAS theory to competitive liquid-solid equilibria in the case of dilute solutions. Using the IAS theory, Levan and Vermeulen (19) have derived competitive binary isotherms in the form of a rapidly converging series expansion, provided the single-componentisotherms are either Langmuir or Freundlich isotherms. The properties of this isotherm could explain qualitatively the experimental results observed in preparative liquid chromatography. When the saturation capacity is higher for the more retained component of a binary mixture, the displacement effect observed (20) is stronger than predicted on the basis of competitive Langmuir isotherm (21). Conversely, when the saturation capacity of the lesser retained component is the higher, it is very difficult, sometimes even impossible, to perform their separation. The aim of this paper is to compare experimental data regarding the competitive equilibrium isotherms of two closely related isomers with the adsorption data calculated by using the Levan-Vermeulen isotherm. Isotherm data regarding the adsorption of trans- and cis-androsterone on a modified silica surface were acquired and published previously (22). Chromatographic band profiles of large size samples were determined at the same time but remained unpublished as we had not found an equation that could account accurately for these data.
isotherms, provided these isotherms are accounted for by a simple Langmuir isotherm. When the column saturation capacities of the two components are equal, eqs 1 and 2 simplify to
THEORY Competitive Isotherms. Provided that the single-component isotherms can be accounted for by the Langmuir model, the two-term expansion series of the Levan-Vermeulen (19) isotherm for the liquid-solid competitive equilibrium of the two components of a binary mixture can be written as
where z is the column length, t , the time, ug,the mobile-phase velocity, and D,, the apparent dispersion coefficient, which accounts for the band broadening due to the axial dispersion and to the finite kinetics of mass transfer in the column (24). D, is equal to Huo/2, with H being the height equivalent to a theoretical plate of the chromatographic column. A finite difference algorithm developed in the laboratory and previously discussed was used for this calculation (25,26). In the numerical calculations of the individual band profdes, we have taken the isotherm parameters determined later and assumed a value of the column efficiency equal to 5OOO theoretical plates (i.e., an axial Peclet number equal to lOO00).
and
where qi and Ci are the stationary- and mobile-phase concentrations of component i a t equilibrium and ai and bi are numerical coefficients characteristic of component i.
(3) with q,,i = ai/bi and
(4)
The IAS theory and the procedure developed by Levan and Vermeulen (19) give the isotherms as an expansion series. Equations 1 and 2 are the two-term expansion. In most practical cases, convergence is rapid and there is little difference between the results of the second- and third-order expansion. The isotherm should not be used in a concentration range where 8qi/8Ci becomes negative. The isotherm equations (1)and (2) depend only on four parameters, the two column saturation capacities, q j , and the two bi coefficients, which are respectively given by
where F = V J V , is the phase ratio and k :,o is the retention factor under linear (i.e., analytical) conditions. These four parameters can be derived simply from the single-component
which is the classical Langmuir competitive isotherm. When the column saturation capacity for the first eluted component is smaller than that for the second component (diverging isotherms), the amount of first component adsorbed at equilibrium is lower than predicted by the simple Langmuir competitive model and the amount of second component adsorbed at equilibrium is larger than predicted by this model. The converse is true when the column saturation capacity for the first component is the larger (isotherm intersection). These results are in agreement with experimental reports (20, 23). Prediction of Band Profiles. The band profiles are calculated by numerical integration of the equilibrium-dispersive model of chromatography:
EXPERIMENTAL SECTION Apparatus. The system used for the determination of competitive isotherms consists of two chromatographs,one used for the determination of the breakthrough profiles (frontal analysis), the other for the on-line analysis of the fraction grabbed at the intermediate plateau occurring during the elution of a binary mixture step. The first chromatograph was assembled by using a Waters (Milford, MA) gradient system, two Model 510 pumps, an automatic gradient controller, a 10-port Valco (Houston, TX) pneumatic sampling valve fitted with two 5-mL loops, a Rheodyne (Berkeley, CA) Model 7010 sampling valve fitted with a 20-pL loop, the 5-cm-long adsorption column, and a Kratos (ABI, Ramsey, NJ) Spectroflow 757 UV spectrophotometricdetector. The column and the sample loops are immersed in a Haake A81 (Saddle Brook, NJ) water bath at constant temperature. The second chromatograph consisted of a Model 510 Waters pump, the Rheodyne valve, the 25-cm-long analytical column, and another Kratos Spectroflow 757 detector. The signals of both chromatographswere recorded with a Waters NEC data station, using the Waters Maxima 860 Dynamic Solutions (Ventura,CA) software. Large size samples of binary mixtures of various compositions were injected on the analytical column fitted on a 1090A Hewlett-Packcud liquid chromatograph (Palo Alto, CA) equipped with a UV diode-array detector, a multisolvent delivery system, and a computer data station. During the elution of the binary mixture, band fractions were collected from the analytical column, stored, and analyzed on the same chromatograph. A Gilson (Middleton, WI) Model 203 fraction collector was used for fraction collection and analysis. Products. trans-Androsterone and cis-androsterone were purchased from Sigma Chemical Co. (St. Louis, MO) and used without further purification. All solvents were of high purity grade
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1. Competitive equilibrium isotherms of b9ns- (t) and ds-anckosterone(c). Comparison between the experimental data (Huang and Quiochon (221, symbols) and the two-termexpanskn of the L e v a n - V m l e n competirive isotherms (eqs 1-4, lines) calculated by using the singlecomponent Langmuir Isotherm coefficients In Table I (22). Plots of q t (0 and solM line) versus C,and of qc (A and dotted line) versus C,. Experimental conditions. I. Column: 50 X 4.6 mm, packed with Partlsil 10-pm silica modified with a pH = 6.8 phosphate buffer (27). 11. Mobile phase: acetonltrile/dlchloromethane, 8515; flow rate, 0.25 mL/min; temperature, 25 OC. 111. Isotherm data: a t = 8.085, b , = 0.01807 mL/mg, a , = 9.735, b , = 0.02838 mLlmg, Q,,, = a,/b,. (a, left) Feed composition: 2/1 (vlv, trans/cls). (b, middle) Feed composition: 111. (c, right) Feed composition: 1/2.
from American Scientific Products (McGaw Park, IL) and were used as received. Reagents were of analytical reagent grade from Mallinkrodt (Paris, KY). Partisil-10, a silica gel with a specific surface area of 350 m2/g and an average pore size of 85 A, was obtained from Whatman (Clifton, NJ)and treated with a phosphate buffer at pH = 6.8, according to the method described by Schwartzenbach (27). The columns were packed in the laboratory, using a conventional slurry packing method. Columns (5 and 25 cm long, 4.6-mm i.d.) were packed with stainless steel 316 tubing (Alltech, Deerfield, IL). The column was operated at 25 O C . Procedures. The singlecomponent isotherms were determined by using the classical frontal analysis technique (28).Competitive isotherms were determined by frontal analysis, using binary solutions of the two isomers of increasing concentrations, but constant relative composition. The method described by Jacobson et al. (29) was followed. Details of the procedure have been reported elsewhere (22). Large amounts of binary mixtures of different compositions were injected on the analytical column. Fractions were collected during the elution of the bands and analyzed, as described previously (4).
RESULTS AND DISCUSSION We compare successively (i) the competitive isotherms determined experimentally with the isotherms calculated by using the Levan-Vermeulen isotherms (19) and (ii) the individual band profiles of the two isomers obtained by analysis of the collected fractions with those calculated by using the equilibrium-dispersive model of chromatography. The experimental results regarding the competitive equilibrium isotherms of the two androsterone isomers have been published previously (22). A chromatogram showing the total elution profile of a large amount of a 1/3 mixture and suggesting an inversion of the elution order at high concentration has also been published (30). In the following, we use the subscripts t and c to represent the symbols corresponding to trans- and cis-androsterone, respectively. At very low concentrations (i.e., under linear conditions), the trans isomer is eluted first. Validity of the Levan-Vermeulen Competitive Isotherm. In the following figures, the experimental data previously published (22) regarding the competitive equilibrium
Table I. Parameters Used for the Calculation of Isotherms trans-androsterone cis-androsterone Reference 22,Table I a
b, mL/mg %Img/mL *
8.085 0.018 449
9.735 0.0284 330
This Work a
b, mL/mg 0.; = a ; / b ; .ma/mL
8.977 0.0187 480
9.482 0.0237 400
isotherms of trans- and cis-androsterone on phosphatemodified silica gel are represented by symbols. The isotherms calculated by using eqs 1-4 are shown in solid lines for trans-androsterone and in dotted lines for cis-androsterone. The binary competitive isotherms are two surfaces of equation qi = fi(C,, C,). These surfaces can be represented most conveniently by their section by vertical planes of trace C, = kCt on the (Ct, C,) coordinate plane. Data have been acquired with mixtures whose composition corresponds to the values 2, 1, and 0.5 for k. Figure 1 compares experimental data and the LevanVermeulen competitive isotherms calculated by using the single-component Langmuir isotherm coefficients ai and bj reported by Huang and Guiochon (22) (see Table I), at three different relative compositions of the binary mixture. These figures show good agreement between the experimental data and the Levan-Vermeulen isotherm for cis-androsterone (dotted lines). The agreement is poor, however, for transandrosterone. Systematic deviations are observed and the reversal of the separation factor for the 1/1 mixture is not accounted for (Figure lb). This discrepancy can be explained by the conjugation of three factors. We were not aware of the last one when the data were obtained. Firstly, the properties of the phosphate-modified silica drift in time. The single-component isotherms were measured f i t , long before the competitive isotherms were determined. An approximate value of this drift can be derived from the data in Table I11 of the publication by Huang and Guiochon (22) (see Table I), which contains data on the influence of tem-
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Flgure 2. Competitive equilibrium isotherms of trans- (1) and cis-androsterone (c). Comparison between the experimental data (symbols) and
the two-term expansion of the Levan-Vermeulen compewiveisothemrs (eqs1-4) calculated using the coefRdemsobtained by fHUng the expdmntal competithre Isotherm data by Huang and Gulochon (22)on these Isotherms (lines). Same plots and experimental conditions as for Figure 1. (a, left) Feed compositlon: 211. (b, middle) Feed compositlon: 111. (c, right) Feed composition: 112.
perature and mobile-phase composition. These data were obtained near the end of the study. Secondly, the slope of the isotherms, i.e., the values of the coefficient ai determined by nonlinear regression of the experimental data, does not agree with those derived from the retention times of very small concentration pulses, under linear conditions. We have often observed a systematic difference between the value of the coefficient ai derived from eq 5 and the value calculated by the computer, using a nonlinear least-squares fitting program (31). The computer is trying to force the data on the Langmuir isotherm and uses the 2 degrees of freedom to the higher extent possible. There is no reason to give in this calculation an exceptionally heavy weight to the data point obtained at very low concentration. However, the value of a = a z / a l is too large and overestimates the actual ease of separation. Finally, and more importantly, we know that the silica surface is not completely homogeneous. Usually, the adsorption energy on a small fraction of the surface is higher than on the rest of the surface (32). The value of the coefficient ai corresponding to these sites is high while the saturation capacity is very low. The measurement made at very small sample sizes corresponds to that ai value for the active sites, but when isotherm data are determined, they fit well with an isotherm corresponding to a smaller value of ai. The combination of all these sources of errors means that the Langmuir isotherm may not represent perfectly well the single-component adsorption data. Therefore, the LevanVermeulen isotherm, derived from moderately accurate single-component isotherms, cannot account very accurately for the competitive isotherms of the mixed components. The deviation remains small, however, so an alternative fitting procedure was used to account for these sources of systematic errors. We have taken the 23 sets of competitive isotherm data (i.e., qt and qe for 23 couples of values, C,,C,)determined by binary mixture frontal analysis (22),and we have fitted them on the Levan-Vermeulen isotherm equations. A system nonlinear regression program permits the calculation of the best values of the four parameters. The values obtained for these parameters are listed in Table I. Figure 2 compares the experimental data to the best Levan-Vermeulen isotherms so obtained. The agreement is now excellent for both isomers, with 2/1 (Figure 2a) and 1/2 (Figure 2c) relative concentrations. With the 1/1 relative concentration mixture, the
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Flgure 3. Influence of the mobile-phase composition on the competitive equilibrium Isotherms of trans- (t) and cisandrosterone (c). Comparison between the experimental data (symbols) and the tweterm expansion of the Levan-Vermeulen competitive Isotherms (eqs 1-4) calculated using the coefflcients obtained by fitting the experimental data by Huang and Guiochon (22). Same plots and experimental conditions as for Figure 2b, except moblle-phase compositlon, as indicated. Mobile-phase composition: curves 1. acetonitrile, 90%, dichloromethane, 10% ; curves 2, acetonltrile, 85 % , dichkromethane, 15 % ; curves 3, acetonitrile, 80%, dichloromethane, 20%.
agreement is excellent for trans-androsterone. It is somewhat less good but still satisfactory for cis-androsterone. The calculated isotherm is slightly above the experimental data, and the separation factor reversal is predicted at about twice too large a mobile-phase concentration (Figure 2b). Competitive isotherms had also been determined experimentally at different mobile-phasecompositions (90/10,85/15, and 80/10)and at different temperatures (22). Comparisons between the experimental results and the Levan-Vermeulen isotherms calculated by nonlinear regression of the corre-
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Flgue 4. InRuence of the temperature on the competitive equlllbrium Lsatherms of trans- (t) and ds-androstwone (c). Comparison between the experlmental data (symbols) and the two-term expansion of the Levan-Vermeulen competitive isotherms (eqs 1-4) calculated using the coafflclents obtained by fMhg the experknental data by Huang and Gukchon (22). Same plots and experlmental conditions as for Flgure 2b, except temperature, as indicated. Column temperature: curves 1, 5 OC; curves 2, 15 OC; curves 3, 25 OC.
sponding data in the publication by Huang and Guiochon (22) are shown in Figures 3 (composition) and 4 (temperature). There is again an excellent agreement between the experimental data and the values predicted by the Levan-Vermeulen equation (19). Calculation of Individual Elution Band Profiles. The previous section has demonstrated the ability of the LevanVermeulen isotherm equations to permit an accurate interpolation of the experimental competitive adsorption data. A demanding test of a competitive isotherm model is its ability a t permitting the prediction of the individual band profiles for large size samples of binary mixtures (4). We compare in Figures 5-8 the experimentalband profdes for three samples of different composition with the calculated profiles. The experimental elution curves (Figures 5a, 6a, and 7a) were determined by fraction collection and analysis. The theoretical profiles (Figures 5b, 6b, 7b, and 8a) were calculated by numerical integration of the system of mass balance equations (eq 7) coupled by the Levan-Vermeulen competitive isotherm using the parameters in Table I. Figure 8b was calculated by using the competitive Langmuir isotherm. In Figure 5, we report the elution profiles of the two androsterone isomers obtained upon the injection of a 7-mg sample of a 1/1mixture. The number of fractions collected (one every 10 s) is sufficient to give an accurate representation of each individual profile. These profiles are consistent with the total chromatogram derived from the detector response. There is good agreement between the experimental results and the calculated profiles. The retention time predicted for the band front is slightly longer than observed (about 40 s), which can be explained by a small error on the flow rate or the phase ratio of the column packing. The ratio of the retention times of the two bands fronts or of the two maxima is the same for the experimental and calculated results. The calculated band ends earlier and sharper than observed, which is usual and may be explained by the fact that we have ne-
, Figure 5. Overloaded elution of a binary mixture of trans- and cisandrosterone. Comparison between the band profiles calculated by using the twc-term expansion Levan-Vermeulen lsothem (eqs 1-4) and the experimental profiles. (a, top) Experimental profiles. Conditions. I. Column: Same as for F w e 1, except length 25 cm and efficiency 5000 theoretical plates, column phase ratio f = 0.357; flow rate 1 mllmin. 11. Sample: relative feed composition, 111; sample size, 20 gL of a solution containing 176.3 mg/mL of each isomer. (b, bottom) Band profiles calculated using the values of the experimental parameters of Figure 5a and the isotherm as in Figure 2.
glected the contribution of the active sites to the retention and to the isotherm, as explained in the previous section. The slight hump on the rear of the second component profile (Figure5b) is due to the tag-along effect (2,211. The frequency of the sample collection is too low and the definition of the experimental profile (Figure 5a) is too coarse to demonstrate this effect. In Figure 6, we compare the band profiles of the two androsterone isomers for the elution of a 3.88-mg sample of a 3/1 mixture (2.91 mg of trans- and 0.97 mg of cis-androsterone). The elution time of the band front is slightly longer than in the previous Chromatogram,which is explained by the lower concentration of the components. The experimental and calculated profiles are in good agreement. The difference between the elution times of the calculated and recorded bands is the same as in Figure 5, consistent with the explanation given. Again, the band of &-androsterone tails slightly longer than predicted. The tag-along effect on the rem of the second component band, clearly seen in the theoretical profile (Figure 6b), is slightly visible in the experimental profile (Figure 6a).
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Flgurr 6. Overload elution of a binary mixture of trans- and clsandrosterone. Comparlson between the band profiles calculated by using the two-term expansion Levan-Vermeulen isotherm (eqs 1-4) and the experimental proflles. (a, top) Experimental profiles. Same Conditions as for Figure 5, except relathre feed composition 311 and sample size 20 pL of a solutlon containing 145.5 mg/mL of transandrosterone and 48.6 mg/mL of c/s-androsterone. (b, bottom) Band proflles calculated using the values of the experimental parameters of Figure 6a.
Figure 7 compares the experimental and calculated profiles of the isomers for a 3.63-mg sample of a 1/3 mixture (0.94 mg of trans- and 2.69 mg of cis-androsterone). In this case, there is a significant difference between the experimental and calculated isotherms, especially the elution profile of cisandrosterone. The experimental profiles exhibit a reversal of the elution order of the two isomers, compared to the chromatograms in Figures 5 and 6 and to the chromatogram obtained under linear (i.e., analytical) conditions. The total chromatogram given by the detector signal is quite similar to a chromatogram published by Gonzalez et al. (30),who suggested a reversal of the elution order of trans- and cisandrosterone at high concentration but could not demonstrate it. However, there is no reversal order of the elution bands in the calculated chromatogram in Figure 7b. Also, the two calculated band heights are nearly equal, while the recorded cis-androsterone band is much higher than the trans-androsterone band. The experimental band of the cis isomer begins before and ends after the band of trans isomer, which is quite unusual.
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Figure 7. Overloaded elution of a blnary mixture of ffans- and cisandrosterone. Comparison between the band profiles calculated by using the two-term expansion Levan-Vermeulen Isotherm (eqs 1-4) and the experimental profiles. (a, top) Experimental profiles. Same conditions as for Figure 5, except relative feed composition 1/3 and sample size 20 pL of a solution Containing 47.1 mg/mL of trans-androsterone and 134.5 mg/mL of cis-androsterone. (b, bottom) Band profiles calculated using the values of the experimental parameters of Figure 7a.
These differences may be explained in part by the fact that the equilibrium isotherms and the overload chromatograms were obtained with two different columns. The isotherms were measured first, with a 5-cm-long column, while the overloaded chromatograms were recorded by using a 25-cm-long column, packed with the same material but at a later time. Although the parameters of the adsorption isotherms should ideally be the same for both columns, the column-to-column reproducibility of these data has not been fully demonstrated (31). Furthermore, the efficiency of the second column may have been insufficient for an accurate estimate of the integral mass balances of the two components, which is needed in the frontal analysis method of competitive isotherm determination (22, 29). The band profile calculations performed by using the Levan-Vermeulen isotherms cannot show elution order reversal because the column saturation capacities for the two isomers are too close. Another set of experimental data has been collected by Gonzalez et al. (30) on the very same system. These isotherms have been measured by frontal analysis on the same column used for overloaded elution. The individual band profiles of
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Flguo 8. Overload elution of a binary mixture of trans- and &-androsterone. Comparison between calculated and experlmental band profiles. (a, left) Band profwes calculated by ushg the tweterm expanskn Levan-Vermeulen isotherm (eqs 1-4) and the singleGomponent Langmut isotherm parameters of Gonzaler et al. (30). Solid lines, trans-androsterone. Dotted lines, cis-androsterone, Dashed line, total concentration proflle of the hbh concentration chromatogram. I. Sample. Small sample: 0.155 mg of trans- and 0.026 mg of clsandrosterone. Large sample: 1.8 mg of &am-and 5.2 mg of &-androsterme. 11. Column: 250 X 4.6 mm, packed wlth Partisil lGpm particles, modified wlth a pH = 6.8 phosphate buffer (27),phase ratlo 0.17. 111. Moblle phase: acetonlbileldlchlthane, 9O:lO; flow rate, 0.98 mL/min. (b, tight)Calculated chromatograms using the Langmuir competitive isotherms. Same simulation conditions as in a.
the two isotherms have not been determined, however. In order to check the usefulness of the Levan-Vermeulen isotherm, we calculated the band profiles under the experimental conditions of Gonzalez et al. (30),using their single-component isotherm data in a Levan-Vermeulen isotherm. The two calculated chromatograms, corresponding to a small and a large sample, respectively, are shown in Figure 8a. The experimental chromatograms are shown in the figure insert. There is excellent agreement between the calculated and experimental chromatograms, and the elution order reversal is correctly predided. When comparing the experimental and calculated chromatograms a t high concentrations, it should be remembered that the calculated chromatogram is a concentration profile while the recorded one is an optical density profile, that the UV detector used is not linear in the concentration range, and that the response factor a t the wavelength selected is higher for trans- than for cis-androsterone. The reversal in elution order in the calculated chromatogram takes place because the column saturation capacity is more than twice larger for trans- than for cis-androsterone. Finally, the chromatogram in Figure 8b has been calculated by using the conventional competitive Langmuir isotherm, but with the same numerical coefficients as used for Figure 8a. Obviously, the result is not satisfactory.
CONCLUSION The Levan-Vermeulen isotherm (19)accounts well for the competitive adsorption data collected experimentally. It predicts the intersections of the competitive isotherms that are observed experimentally (22). This isotherm model permits the calculation of individual band profiles in chromatography, which are in very good agreement with experiment results, most of the time. It predicts a significant variation of the separation factor with the relative composition of the feed. Accordingly, it predicts a competitive behavior that is in much better agreement with
experimental results than the simple competitive Langmuir isotherm. The displacement effect is markedly increased if the saturation capacity of the component that is leaser retained a t low concentrations is the smaller. On the contrary, if the saturation capacity of the component that is lesser retained under linear conditions is the higher, as is the case in the present work, the displacement effect is considerably reduced (20) and a reversal of the elution order may occur (30).The model accounts well for these observations. In spite of these advantages, the Levan-Vermeulen isotherm, like the Langmuir isotherm, has the drawback of assuming that the mobile and the stationary phase have an ideal behavior, that molecular interactions between the mixture components in either phase are negligible, and that the activity coefficients of solutes are constant. These assumptions limit the validity of the model to a moderate range of concentrations, which does not include entirely those achieved in preparative chromatography. Registry No. trans-Androsterone,481-29-8; cis-androsterone, 53-41-8.
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RECEIVED for review November 30,1990. Accepted February 21, 1991. This work has been supported in part by Grant CHE-8901382 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We acknowledge the support of our computational effort by the University of Tennessee Computing Center.
Internal Standardization Technique for Capillary Zone Electrophoresis Eric V. Dose and Gorges A. Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Analytical Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120
A new method of Internal standardization for caplllary zone electrophoresis (CZE) k founded on the Ilnear relation between each bn’s effective vdume Injected and lts inherent “Iy.Tho use oftwointmd standards dkwrthe analyst to WaMkh thk relation quantltatlvely for each sample and to correcl the analyte concentratlons for variations In each Inl.ctkn. Tho computatbnal method given k rlmple, and we d ” t r a t e that lt can give reproduclMlltIe8 for manual CZE hydrodynamlc and electrostatic InJectlons of under 1% relative standard devlatlon.
INTRODUCTION Capillary zone electrophoresis (CZE) will not find the same kind of acceptance that chromatography has found in the broad analytical realms of quality control, purity assessment, and trace analysis until much more attention is paid to its quantitative aspects. Detector, instrument and column reliability, and quality and speed of separations have benefited greatly from the last decade’s numerous studies. We feel it is time to address CZE quantitation as well. Electrokinetic (EK) injection is popular among CZE users largely because it requires little or no instrumentation other than that required to effect the separation itself. In contrast, hydrodynamic (HD) injection requires that one apply a pressure differential between the column ends during injection. This requires in turn either the application of vacuum, pressure, or vertical sample displacement. This last approach requires that the column be moved, increasing the size of the instrumental and complicatingtemperature equilibration, for example. Since EK injection merely applies a potential difference between the column ends, injection automation requires only that sample be moved to the buffer cell and that potential pulses be timed precisely. Thus, EK injection equipment should be simpler and more reliable than HD 0003-2700/81/0363-1154$02.50/0
injection equipment. However, since EK injection draws each analyte into the column at a rate proportional to its electromigration velocity, faster ions are overrepresented in the electrophoregram. This velocity discrimination leads directly to apparent bias in detector peak areas. The expected effects of sample solution conductivty and ion mobility differences on the quantitative results obtained using EK injection were described recently (1,2). In the short time since those works were published, bias has been mentioned in several reviews (3-5) and other articles (6-13). Though HD injection seems to be preferred by many workers (4-6,9,13-15), there is little evidence that it offers greater precision than does EK injection. Two problems arise in using EK injection for quantitative analysis. First is the mobility bias problem (Figure l),which can be corrected for if the ion mobilities are known or can be calculated. Second, run-to-run variations in the injection voltage, injection time, and sample conductivity also generally exist. These variations could be corrected for by using internal standards in the same manner as done in chromatography if the relative contributions of electroosmotic flow velocity and electromotive migration velocities were constant. However, variations in the electroosmotic and electromotive injection contributions are not perfectly correlated. To correct for both sources of error, we have developed a new and easily performed internal standardization technique, using two internal standards added to each sample. In this article, we describe this technique and give initial data supporting the technique’s ability to correct for common sources of run-to-runvariations, especially where electrokinetic injection is used.
EXPERIMENTAL SECTION The electrophoretic system consisted of a Hipotronics 25-kV high-voltagepower supply, untreated silica capillary tubing (length 65 cm, working length 42 cm, and inside diameter 75 Mm, Polymicro Technologies, Phoenix, AZ), and an on-column UV detector (JASCOModel UV-100-111) operated at 230-nm wavelength. A Data Translation Model 2801 board digitized the analog detector 0 1991 Amerlcan Chemlcal Soclety
