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Anal. Chem. 1988, 6 0 , 2630-2634
Comparison between Experimental and Theoretical Band Profiles in Nonlinear Liquid Chromatography with a Pure Mobile Phase Sadroddin Golshan-Shirazi, Samir Ghodbane, and Georges Guiochon*
Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37966-1600,and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Experlmentalproflles have been obtalned for large size samples of phenol In normal phase liquid chromatography on slllca, using dlchloromethane as moblle phase. These profiles are compared to the profiles predlcted by the numerlcal lntegratlon of the equatlons of the theory of nonllnear chromatography. This requlres the prlor determlnatlon of accurate sorption Isotherms of phenol between dlchloromethane and slllca. The agreement between the experimental and the theoretlcal proflles Is quantttatlve In all cases.
In a previous paper we have presented a model of nonlinear chromatography based on the mass balance equation of the component studied and a method of calculation of the numerical solutions of this equation (1). This model makes practically no s i m p l i i g assumption and should give an exact prediction of the elution profiies of large concentration bands, provided the correct equilibrium isotherm is available. The results obtained by computer simulation using this model under a wide variety of experimental conditions have also been discussed (1).Since this model is demonstrably correct and takes into account all the phenomena that seem to have a significant effect on the band profile (2))it is important to compare as accurately as possible the results of its predictions to a series of experimental results. Some minor effects have been neglected (influence of the solute concentration on the viscosity, adsorption enthalpy, etc.). It would be useful to make sure that they are negligible. This comparison is the aim of the present paper. It is not very difficult to find out experimental conditions under which a simple, one-compound model of nonlinear chromatography applies and where, consequently, there should be exact agreement between theoretical and experimental results. These experimental conditions are not very common, however, nor highly representative of those under which most applications (e.g., preparative separations) are carried out, nor very practical. Strict adherence to the model demands the use of a very simple chromatographic system, where a pure eluent is used. As we know on the other hand, most chromatographic separations, including most preparative applications, are carried out using solvent mixtures, with at least a binary mobile phase (3).In ion-pair chromatography or in the separation of peptide mixtures, for example, still more complex eluents are often used. It is known that, under such conditions, in analytical chromatography, the injection of the sample generates a number of system peaks (4-6). The injection of the large sample used in nonlinear chromatography may result in considerable band distortion if some conditions are not met (7). The behavior of a chromatographic system using a mixed mobile phase, under nonlinear conditions, is studied in other papers (7-9).
* Author to whom correspondence
should be sent.
Among the previous results most pertinent to the present work, the strong dependence of the exact band shape on the curvature of the isotherm at the origin has been stressed (1). Obviously, a change in sign of the curvature (i.e., from convex to concave isotherm) would result in a dramatic modification of the profile. But minor changes in the numerical value of one of the coefficients of an equilibrium isotherm almost always lead to important changes in the band profile (I). Consequently, any attempt a t a meaningful comparison between theoretical and experimental band profiles generated for the same set of experimental conditions requires a high precision in the determination of the adsorption equilibrium isotherm. Thus, this paper contains a brief description of the model of chromatography and of the calculation methods used, a description of the different methods of measurement of the equilibrium adsorption isotherms employed and a comparison between the results they provide, and a comparison between elution profiles of bands recorded and calculated from these isotherm data.
THEORY I. Semiideal Model of Chromatography. For one compound the mass balance in a column section is written (I,2) dC,
dC,
dt
dt
- + F-
due, d2C, +=D dz dz2
(1)
where C, and C, are the concentration of the compound in the mobile and stationary phases, respectively, z is the abscissa along the column (of length L)and t the time, u is the mobile phase velocity, F is a geometrical constant, equal to the ratio of the volumes occupied by the stationary and the mobile phases, and D is the compound axial diffusion coefficient (molecular diffusion coefficient modified by the tortuosity of the packing and eddy diffusion). Integration of eq 1 requires first a relationship between the concentrations in the mobile and stationary phases, or rather between their first derivatives, and second a set of boundary conditions. This latter question is simple and will be dealt with later. The former raises the issue of the kinetics of mass transfer between phases. Even in the linear case, it has never been solved satisfactorily, as shown by the impossibility to predict the exact efficiency of a column (10,11). Another approach is required. We have adopted the semiideal model (1,2), an improvement over the ideal model which assumes that the kinetics of the various mass transfer steps between the mobile and the stationary phases do not depend on the solute concentration, except at the liquid-solid interface, where adsorption takes place. This is justified by the facts that the solutions used in preparative liquid chromatography are dilute, never more concentrated than ca. 8% (w/w), i.e., 0.1-0.8 M depending on the molecular weight, and that the diffusion coefficients do not vary significantly in this range (12),at least for com-
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988
pounds with low molecular weight. For polymers, and notably for proteins, the situation is different and further investigation is required. The ideal model assumes that the column efficiency is infinite (13-19). Accordingly, the mobile and the stationary phases are constantly in equilibrium and the relationship needed between C, and C, is the therodynamic isotherm. The following simplified system of equations is obtained, in the case of a single compound eluted with a pure solvent as mobile phase dCm dt
(1 + Fk?-
+ F-=duC, dz
0
with
(3) The second equation is derived by differentiation of the equilibrium isotherm, i.e., C, = f(C,). The classical boundary conditions correspond to an instantaneous pulse injection (2). At time t = 0, the concentration C is equal to 0 everywhere in the column, except a t the origin, where it is equal to C, in a volume S dz (S is the column cross-section area) such that C,Sdz is equal to the sample size, m. At all other times, the solute concentration is equal to 0 at the column inlet ( z = 0). It is not possible to integrate the partial differential equation ( 2 ) in the general case. In order to obtain solutions of the system of eq 2 and 3, we need to calculate them numerically. This is a relatively easy task for a properly programmed computer. In order to write such a program, we have chosen the finite difference method and adopted the Godunov algorithm (2,20),which has an extremely interesting property in the case of our system. The errors resulting from the use of finite values for the space and time increments in the numerical integration are exactly equivalent to the diffusion term of eq 1,neglected in eq 2 (21),provided that we chose the following values of the time and space increments:
dz = H dt = 2H/u,
(4)
(5)
where H is the column height equivalent to a theoretical plate (HETP), corresponding to a zero sample size, and u, is the velocity of the solute band a t infinite dilution (u/(l k d ) . With these values of the space and time increments, the numerical solution is stable and converges toward a weak solution of eq 2, as required (2). However, the numerical solution thus obtained is an exact solution of eq 1, with
+
L
LCO
where to is the retention time of a nonretained compound, D, is the apparent diffusion coefficient of the chromatographic system, and L is the column length. 11. Summary of the Assumptions Made. The assumptions made in the model just described are the following: The solvent is not sorbed. Rather than an assumption, it is a convention regarding the definition of the adsorption (22). The mobile phase is a noncompressible fluid. Its viscosity and the molecular diffusion coefficients do not change with the local pressure. The pressure variations of the mobile phase density, viscosity, and diffusion coefficients are, at worst, very small (23).Because of the experimental errors, a slight variation of these parameters along the column would probably have no significant effect on the elution band profile, anyway. The diffusion coefficient of the compound studied does not vary with its concentration in the range considered in prep-
2831
arative liquid chromatography (12). This is very reasonable for small molecules (molecular weight below 500-1000), but it is not so with polymers. If the diffusion coefficient decreases with increasing solute concentration, the band will grow wider around its top and consequently will be shorter than predicted by the model. The partial molar volumes of the compound considered are the same in the two phases and do not change with concentration. The sorption effect is negligible. These assumptions have a negligible effect on the representativity of the model. Accordingly, excellent agreement between elution band profiles recorded for large size samples and simulated profiles using our model is to be anticipated in the case of a single, pure compound of low molecular weight eluted by a pure solvent. EXPERIMENTAL SECTION I. Equipment. An HP 1090 liquid chromatograph (Hewlett-Packard, Palo Alto,CA) equipped with a multisolvent delivery system, a diode-array W detedor, an automatic sampling system, and a computer data acquisition system has been used to acquire both the data necessary to determine the equilibrium isotherm of phenol and large sample elution profiles on the same column, which maximizes the accuracy. 11. Materials. Stationary Phases. We have used a 25 cm long, 4.6 mm i.d. column packed with 15-25 hm porous silica particles, Nucleosil 100 (Alltech, Deerfield, IL). Mobile Phase and Solute. We have used dichloromethane to elute phenol on the silica column. The phenol and dichloromethane are from Aldrich (Milwaukee,WI) and from J. T. Baker (Phillipsburg, NJ), respectively. 111. Determination of Adsorption Isotherms. We have used the three following methods for the determination of the adsorption isotherm of phenol on silica, from a dichloromethane solution: frontal analysis (FA), frontal analysis by characteristic point (FACP) and elution by characteristic point (ECP). We briefly discuss here the principle of these methods and compare their advantages and drawbacks. Frontal Analysis (FA). Frontal analysis was introduced by James and Phillips (24) and by Schay and Szekely (25). A stream of solution of the compound studied in the mobile phase, at concentration Ci, is equilibrated with the column, until the detector base line is stable. Then the stream is abruptly replaced by another one, at the same flow rate, with a higher concentration, c b , for concave (i.e., Langmuir type) isotherms. The resulting breakthrough curve is a self-sharpening front. An integral mass balance equation can be written, to relate the amount sorbed by the stationary phase to the concentration in the new stream (7)
where Qband Qi are the concentrations of solute in the stationary phase, at equilibrium with the solution of solute in the mobile phase, at concentrations cb and ci,respectively, vRis the retention volume of the breakthrough front, V, is the column void volume, and V, is the volume of adsorbent in the column. The isotherm is determined by measuring a number of successive points, corresponding to as many concentration steps. Frontal Analysis by Characteristic Point (FACP). In FACP, introduced by Glueckauf (16),a complete isotherm is determined from the diffuse profile obtained by replacing a stream of solute solution in the mobile phase by a stream of pure mobile phase (in the case of concave, Le., Langmuir-type isotherms) or, conversely, in the case of convex, i.e., anti-Langmuir isotherms, a stream of pure mobile phase by a stream of a solute solution at the proper concentration. Each point of the isotherm at a concentration C is derived from the retention volume, V, of the characteristic point of the diffuse profile, at the same concentration, C in the mobile phase. A differential mass balance yields
v - v,
dC
The main drawback of the FACP method is that the profile
ANALYTICAL CHEMISTRY, VOL. 80,NO. 23, DECEMBER 1, 1988
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LC
R 285.4
550.100
ot
GPIFR D
300
000
700:
E
500
4 200
I
$
I
Figure 1. Detector recordings obtained during frontal analysis chromatography with solutions of phenol in dlchloromethane on a silica column. The step heights correspond to concentrations of phenol of 0, 0.0053,0.0106,0.0212, 0.0318,0.0424,0.053, 0.0836,and 0.0848,respectively. The diffuse front obtained by desorption of the total step is used for the FACP method. See details in text. Flow rate was 1 mL/min. UV detector wavelength used was 285 nm.
obtained, which represents the elution curve of the concentration step, is in fact the variation of the detector signal with time. It must be transformed into a profile of solute concentration versus eluent volume. This requires conversion of the detector signal into concentration units, which in turn demands proper, accurate calibration prior to any series of isotherm determination. Elution by Characteristic Point (ECP). The ECP method is a very simple experimental technique, introduced by Cremer and Huber (26). A large sample pulse is injected and its elution profile is recorded. Application of eq 8 to the diffuse side of this overloaded band profile permits the determination of the point of the isotherm, with coordinates C and Q,. By use of this procedure for a series of points of the diffuse part of the profile, it is possible to determine the arc of the isotherm in the concentration range between zero and clase to the maxjmum concentration of the band. Like with FACP, a single experiment permits the determination of an entire arc of the isotherm. Since a single sample pulse is injected, the amount of product required to determine the isotherm is kept much smaller with this method than with FA or even FACP. This is a major advantage in the case of expensive compounds. Like with FACP also, the detector must be calibrated. IV. Procedures. Injection of Large Samples. The use of a 250-~Lsample loop permits the easy injection of samples large enough to overload the column to the extent desired, whether the acquisition of overloaded elution bands is desired for comparison with predicted profiles (see below) or for determination of the isotherm by the ECP method. Injection of Steps. The solvent delivery system incorporates a provision for the use of three different solvents. We have found the following procedure extremely practical to carry out either FA or FACP isotherm determinations. The pure mobile phase is stored in one of the solvent flasks; a concentrated solution of the studied compound in the mobile phase is stored in another flask. At the appropriate time, the "no-injection" of a sample is programmed, followed by a change in solvent composition,which is executed by the pump as a step gradient. The change in solvent pumped to the column is very steep, there is very little apparent dead volume, and the step introduced is as close as possible to the theoretical vertical step of the FA method. Data are acquired until the base-line stabilizes and a plateau is recorded for a short period. Then the procedure is repeated automatically, as often as needed, by the properly programmed computer, each time with a new concentration step. This procedure permits the determination of an arc or isotherm in the range that is useful for nonlinear chromatography, within less than a few hours. A similar procedure is used to carry out determinations with the FACP method. This time a stream of pure solvent replaces the last solution pumped in the column. Determination of the Isotherms. Figure 1 shows a typical record obtained with the chromatograph for a series of FA steps and for the FACP method. Figure 2 shows a typical record used for the determination of the isotherms by using the ECP method. The data obtained through the use of the different procedures described above (concentration in the mobile phase and amount sorbed at equilibrium, reported as concentration per unit volume
Flgure 2. Band profile used for the determination of the isotherm of phenol on silica, with dichbromethane as solvent (wavelength, 285 nm): flow rate, 1 mL/min; sample size, 250 WLof a 1 M solution of phenol in CH,CI,. L9
0
Lo
0
t
0
r)
.=0
.0
2 0.00
0.01
0.02
0.03
0.04
0.05
I 36
C Flgure 3. Equilibrium isotherms of phenol between silica and dichloromethane, experimental points and best Langmuir curves. There are three data point sets, obtained respectively by FA (0). FACP (A), and ECP (0) methods. The curves are the Langmuir isotherms (0 = aC,/(l 4- bC,) best representing the different sets. Best values of a are as follows: 14.55(ECP), 14.45 (FACP), and 14.25 (FA). Best values of 6 are as follows: 8.05 (ECP), 8.1 (FACP), and 7.8 (FA).
of packing material used) are fitted on a Langmuir isotherm equation. The best values of the coefficients obtained by a least-squares fit procedure are reported in the caption of Figure 3.
Determination of the HETP. The column HETP at the flow velocity used during these experiments, which is constant and equal to 1mL/min, is determined by injecting a very small sample and measuring the bandwidth at mid-height. The sample size used is selected to be well within the range of validity of Henry's law (i.e., linear equilibrium isotherm) for the compound studied, which is made easy by the availability of the entire equilibrium isotherm. The column efficiency is approximately 5000 plates at the flow rate where it is operated. Calibration of the Detector. The detector is calibrated by pumping solutions of known concentration of the solute studied in the mobile phase used. The solvent delivery system is used for that purpose. The conventional solvent tanks are replaced
ANALYTICAL CHEMISTRY, VOL. 60, NO. 23, DECEMBER 1, 1988 0.09 0.08
0.07
r---
1 4
7
I
A\
i'
0
11
13
15
17
TIUE(rnl")
Flgure 4. Comparison between predicted and recorded band profiles of phenol on silica: mobile phase, dichloromethane; flow rate, 1 mL/min; plate height, ca.50 Mm; sample sizes, (1) 0.07 mmol, (2) 0.1 mmol, (3) 0.15 mmol, (4) 0.25 mmol; predicted profile (continuous lines) and experimental profile (data points). The experimental points are corrected for the nonlinearity of the UV detector response (see text).
by small flasks filed with the standard solutions. Each standard solution is pumped until the detector signal reaches a plateau, which is taken as giving the detector response for the corresponding solute concentration. The column is replaced by a narrow, empty restrictor. The plateaus of FA may also be used for detector calibration (see Figure 1). The detector response has been found not to be linear in the range of concentrations used in our experiments, because the wavelength selected for the determination cannot be at the maximum of the UV spectrum of the compound studied where light absorption is too intense. The detector response at the selected wavelength was fitted on a fifth degree polynomial and the response function obtained was used to correct the data obtained by FACP or ECP. When experimental band profiles are compared to the profiles predicted by the numerical solution of the model, selected points of the chromatogram (i.e., of the computer-stored file, which is a plot of the optical density of the column eluent versus time) are converted into actual solute concentrations, using the response function. Measurement of the Dead Volume. The dead volume was derived from the retention time of ethylbenzene in normal phase chromatography. A small error is made by assuming this compound to be unretained (27),but at this stage of our work this error is of limited consequences for our present purpose, the study of the agreement between experimental and predicted profiles of overloaded elution bands.
RESULTS AND DISCUSSION I. Determination of the Isotherms. We have determined the adsorption isotherm of phenol on silica, using pure dichloromethane as solvent. Figure 3 shows the results obtained by the three methods described above, FA, FACP, and ECP. The parameters of the Langmuir isotherms that account for the experimental results are given in the caption. There is an excellent agreement between the seta of parameters derived from the three series of measurements (FA, FACP, and ECP) and reported in the caption of Figure 3. Because of the compensation between the values of the parameters a and b, the differences between the isotherms are actually very small, well within the range of the experimental errors. The results obtained through the use of the three procedures described above are practically identical. We have found the ECP method to be much more attractive than the other two. It uses a very small sample and is very economical, especially when applied to the study of expensive compounds, such as are encountered frequently in the life sciences. Furthermore, only one chromatographic run is necessary. All the parameters are derived from a single band profile. The use of the ECP method does require, however, the calibration of the detector, which, in turn, with the procedure adopted
2833
here, necessitates a significant amount of product, albeit easily recoverable. The band dispersion due to the limited efficiency of the column affects the results of the ECP and the FACP methods, not those of the FA method. The comparison made proves that the consequences are not serious, although the columns used here do not have a very large efficiency (ca.5000 theoretical plates). Finally, the results of the ECP and FACP methods are valid only if the kinetics of mass transfer between phases is fast. FA is much more time-consuming. It does not require detector calibration, which is a major advantage, but a large amount of the compound(s) under investigation is necessary, unless narrow bore columns are used. FACP requires a much larger amount of product than ECP, although it is otherwise a very similar method.
11. Comparison between Experimental and Predicted Band Profiles. Figure 4 shows a series of four-band profiles recorded with the same column and under the same experimental conditions as previously described (see Experimental Section), for increasing sample sizes (points), and the profiies predicted by the computer program we have written, based on the numerical analysis summarized above and described in preceding papers (I,2). The general agreement between each predicted profile and the one recorded under the same experimental conditions is strikingly good. Given the nature of the compound used for this f i t test of the theory, this kind of agreement was certainly expected. The program "engine" we use to relate overloaded elution band profiles and equilibrium isotherms is rigorous: it has only one approximation which is susceptible to break under some conceivable set of experimental conditions, it is the assumption that the solute diffusion coefficient in the mobile phase does not depend on its concentration. For compounds such as phenol, which has a molecular weight and a viscosity very close to those of dichloromethane, this is an excellent approximation. For the polymers that are found in the life sciences, such as long-strand DNA or proteins, a correction will certainly be necessary to account for a concentration-dependent rate of diffusion. Finally, we want to dispel the erroneous idea that the comparison between overloaded elution band profiles predicted from isotherm data obtained on a given column, using a chromatographic method, and the corresponding profiles recorded for large concentration bands eluted under the same experimental conditions would be somehow tautological. Admittedly, the relationships between the equilibrium isotherm and the concentration signals obtained at column outlets in the FA, FACP, and ECP, methods are contained in the set of equations we use in this work (eq 1to 3), but the converse is not true. Especially, simple theoretical constructions derived from eq 7 or 8 cannot account for the progressive dilution of the band profile, nor for the interaction between the band front and rear, nor for the slope of the steep front (27). Secondly, it has been demonstrated without reasonable doubt that in the case of adsorbents used for highperformance chromatography, the equilibrium adsorption data derived from FA or ECP are in excellent agreement with those derived by using the classical, static methods of adsorption studies (26,B).Only for nonhomogenous surfaces could some disagreement arise, due to the different time scale upon which static methods and chromatography operate. ACKNOWLEDGMENT We are grateful to the Hewlett-Packard Corp. for the gift of a Model 1090A liquid chromatograph with its data system and to ADA, Inc., for the generous gift of a powerful software package that permitted the transfer of the data fiies recorded by the liquid chromatograph to the VAX 8700 computer of the university of Tennessee Computer Center. Registry No. Phenol, 108-95-2.
2634
Anal. Chem. 1988, 60,2634-2641
LITERATURE CITED (1) Guiochon. G.: Golshan-Shirazi, S.; Jauimes, A. Anal. Chem. 1988, 60, 1856. (2) Rouchon, P.; Schonauer, M.; Vaientin. P.; Guiochon G. Sep. Sci. Techno/. 1087, 22, 1793. (3) Guiochon, G.; Katti, A. Chromatographla 1888, 2 4 , 165. (4) Soims, D. J.; Smuts, T. W.; Pretorius, V. J. Chromatogr. Sci. 1971, 9 , 600. ( 5 ) Levin, S.; Grushka, E. Anal. Chem. 1086, 58, 1602. (6) Levin, S.; Grushka, E. Anel. Chem. 1087, 59, 1157. (7) Goishan-Shirazi, S.; Gulochon, G. Anal. Chem.. following paper in this issue. (8) Goishan-Shlrazi, S.; Guiochon, G. J. Chromafogr., in press. (9) Golshan-Shlrazi, S.; Guiochon. 0. J. Chromatogr., in press. (10) Huber, J. F. K. Ber. Bunsen-Ges. fhys. Chem. 1073, 77, 179. (11) Horvath, C.; Lin, H. J. J. Chromatogr. 1078, 149. 43. (12) Bird, R. B.; Stewart, W. E.; Llghtfoot, E. N. I n Transport Phenomena ; Wiiey: New York, 1960. (13) Wilson, J. N. J . Am. Chem. Soc. 1840, 62, 1583. (14) DeVauR, D. J. Am. Chem. SOC. 1043, 65, 532. (15) Giueckauf, E. R o c . R. SOC.London, A 1946, 186, 35. (16) Glueckauf, E. Trans. Faraday Soc. 1955, 51, 1540. (17) Schay, G. ceundlegen der Chromatographle;Akademie Verlag: Berlin, DDR, 1962. (18) Jacob, L.; Guiocon, G. Chromatogr. Rev. 1971, 14, 77.
(19) Rhee, H. K.; Aris, R.;Amundson, N. phlbs. Trans. R. SOC.London, A 1970, 267, 419. (20) Godunov, S. K. Math. Sb. V 1859, 47, 271. (21) Lin, 8.; Guiochon, G. Sep. Sci. Techno/., in press. (22) Kovats, E. sz I n The ScEence of Chromatography; Bruner, F.. Ed.; Journal of Chromatography Library 32; Elsevler: Amsterdam, 1985; p 205. (23) Biu, G.; Martin, M.;Eon, C.; Guiochon. G. J. Chromatogr. Sci. 1973, 1 1 , 641. (24) James, D. H.; Phllllps. C. S. G. J. Chem. Soc. 1954, 1066. (25) Schay, G.; Szekeiy, G. Acta CMm. Hung. 1954, 5 , 167. (26) Cremer, E.; Huber, J. R. K. Angew. Chem. 1061, 73, 461. (27) Poppe, H.; Kraak, J. C. J . Chromafogr. 1983, 255, 395. (28) Kiselev, A. V.; Yashin, Ya. I. I n Gas Adsorption Chromatography; Plenum: New York, 1971.
RECEIVED for review April 19,1988. Accepted July 21,1988. This work has been supported in part by Granta CHE-8519789 and CHE-8715211of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory.
Comparison between Experimental and Theoretical Band Profiles in Nonlinear Liquid Chromatography with a Binary Mobile Phase Sadroddin Golshan-Shirazi and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37966-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
The experimental proflles obtained for large size samples in normal- and reversed-phase liquid chromatography are compared to the profiles predicted by the theory of nonilnear chromatography. When the sorption issthetms of the analytes are determhed accurately, the agreement between the experimental and the theoretlcal profiles Is quantltatlve In ail cases. When the solvent Is a mlxture of a strong and a weak solvent, and If a nonselective detector Is used, both the competitive adsorption Isothermsof the analyte and the strong solvent should be taken Into account. Then a two-component model of nonilnear chromatography must be used, and the system peaks are predicted. If a nonselective detector Is used, however, whlch gives no response for the solvent (Le., no system peak Is detected), and the strong solvent is less strongly adsorbed than the solute studied, the eiutlon profile of a large concentratlon band can be predkted wlth a onecompound model, knowlng the adsorption Isotherm of the solute from the solution. The conditions under whkh a slngie-component model can be used to predict with good or excellent agreement the profiles of large concentration bands of analyte eluted with a binary mobile phase are Investlgated.
In a previous paper we have discussed the band profiles of large size samples of phenol eluted by dichloromethane on a silica column ( I ) . We have reported the adsorption isotherm of phenol in this system. Finally, we have compared the
* Author to whom correspondence should be addressed at the University of Tennessee.
experimental profiles to those derived from a theoretical model of nonlinear chromatography published earlier ( 2 , 3 ) . Near exact agreement between the experimentally recorded band profiles and the theoretically predicted profiles was observed (1). The experimental conditions under which this former study was done (one single, pure solute, a pure solvent as mobile phase), however, are not representative of those under which most applications of nonlinear chromatography (e.g., preparative separations) are carried out. Strict adherence to our original single-component model demands the use of a very simple chromatographic system, with a pure eluent (1,2). But most preparative separations are carried out by using solvent mixtures that are often complex and almost always contain a mixture of two solvents (4). In ion-pair or in hydrophobic interaction chromatography, for example, complex mobile phases containing several modifiers, counterions, or additives are often used. It is known that, when a mixture is used as the mobile phase in analytical chromatography, the injection of the sample generates a number of system peaks, (5-7). Thus, if we want to compare experimental band profiles to the prediction of nonlinear chromatography, three sets of problems must be solved. First, what kind of perturbations are we going to see in preparative liquid chromatography when complex eluents are used (8)? Secondly, can we predict these perturbations with the classical models of nonlinear chromatography (a two-compound model is needed to account for the elution of a single-compound band with a binary mobile phase, the strong solvent being the second compound involved in the model)? Finally, under which sets of conditions and while accepting what kind of error can we use a single-component
0003-2700/88/0360-2634$01.50/00 1988 American Chemical Society
