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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
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model to account for the elution of a single-component band by a binary eluent? This paper deals mainly with the issues belonging to this third set of questions. Obviously, it would be a great simplification if we could use single-compound adsorption isotherms rather than the competitive binary isotherms of the sorbate and the strong solvent in order to calculate the profile of a large concentration band eluted by a complex mobile phase. The tedious work required for the collection of the data that are necessary for the application of our results would be considerably reduced. Furthermore, knowing under which set of conditions we can use a single-compound model and a single-compound isotherm to predict the profile of a large concentration band will orient and simplify the work required for the extension of the present results to the solution of the two-component separation problem. With a binary eluent, the rigorous solution of this problem would require the use of a three-compound model, and thus, the determination of ternary competitive isotherms, an extremely complex proposition. Under conditions where we can neglect the effect of the strong solvent on the profile of the solute bands, however, the problem reduces to a two-compound problem, an attractive simplification. Thus, this paper contains first a brief description of the two-component model of chromatography, of its properties, and of the calculation methods used. Then, we have compared the band profiles predicted by the one-component model with those derived from the two-component model for different ratios of the adsorption strength of the solute and the strong solvent. Finally, a comparison is made between the experimental elution band profiles recorded with a binary solvent and those calculated from single-component isotherm data.
Equation 1 is thus replaced by the following simpler one:
THEORY I. Semiideal Model of Chromatography for a Single Component. For one compound the mass balance in a column section is written (1) as follows:
dz = H dt = 2H/u,
where z is the abscissa along the column and t is the time; C, and C, are the concentrations of the compound studied, in the mobile and the stationary phases, respectively; F is a geometrical constant, equal to the ratio of the volumes occupied by the stationary and the mobile phases (i-e.,the phase ratio); u is the mobile phase velocity; and D is the axial diffusion coefficient of the compound (the product of its molecular diffusion coefficient and the column packing tortuosity and the contribution of eddy diffusion). The integration of this system has been discussed by many previous authors (1-3, 9-21). It requires an additional equation that gives the concentration in the stationary phase. There is no analytical solution in the general case. The procedure that at present seems the best involves the computer calculation of numerical solutions (2,19,20). The derivation of the proper algorithm for this calculation is somewhat indirect. First, we consider the ideal model of chromatography,which assumes the two phases of the chromatographic system to be constantly at equilibrium and the axial diffusion to be negligible (11-19). Since the two phases are at equilibrium, the equilibrium isotherm provides a relationship between the solute concentrations in the mobile and the stationary phases. The ideal model assumes an infinite column efficiency. This is not very realistic, even if the column efficiencies achieved in most modes of high-performance liquid chromatography (HPLC) are rather high. The numerical integration of the simplified partial differential equation will permit, however, the exact simulation of the effects of a fiiite column efficiency.
with
k’ = dC,/dC,
= f’(C,)
(3)
The second equation is derived by differentiation of the equilibrium isotherm, i.e., C, = f(C,). The classical boundary conditions for elution chromatography correspond to an instantaneous pulse injection (3). At time t = 0, the concentration C, is equal to 0 everywhere in the column, except at the origin, where it is equal to C, in a volume Sdz (S is the column cross section area) such that C$dz is equal to the sample size. At all other times, the solute concentration is equal to 0 at the column inlet (z = 0). It has been impossible so far to derive an exact solution of the system of equations (2) and (3) in the general case, but numerical solutions can be calculated, using a finite difference method. The continuous plane ( z , t ) on which the solute concentration, C,, is defied is replaced by a grid (z, t ) of mesh dz and dt. The concentration is then calculated at each point of the grid, starting from the origin (0,O) and the points (0, dt) and (dz, 0), where the concentration is given by the boundary conditions (3). By the very principle of the method, a systematic error is introduced at each step (20). A study of this error shows that it is equivalent to the replacement of the right-hand side of eq 2 (i.e., 0) by a term equal to D,d2C/dz2, where D, is an apparent diffusion coefficient, which must remain constant. If we chose the following values of the time and space increments (20) (4)
(5)
we have
D,
H u / ~= H L / 2 t ,
(6) where H is the column height equivalent to a theoretical plate (HETP), corresponding to a zero sample size; u, is the velocity of a vanishingly small concentration of solute (u,= u / ( l + k,,’);and tois the retention time of a nonretained compound. With these values, the numerical solution is stable and converges toward a solution of eq 2, as required. More importantly, the procedure provides exact numerical solutions of eq 1. Haarhof and Van der Linde (21) have shown that, if mass transfer between the two phases is fast, but not infinitely fast, it is possible to replace the system of equations combining eq 1 and the relevant kinetic equation by the system made of eq 3 and the combination of eq 1and 6, which is the system for which our numerical algorithm calculates solutions. In other words, we may calculate the exact elution band profile, knowing only the equilibrium isotherm and the column efficiency for a very small sample size, provided the kinetics of mass transfer, i.e., the diffusion coefficient, does not vary with increasing solute concentration. Within the concentration range used in liquid chromatography, this assumption is usually valid for small molecules, not for biopolymers (22). The assumptions made in the model just described have been reviewed and discussed recently (2). We take the convention that the weak solvent is not sorbed (23). We assume that (i) the mobile phase is noncompressible, (ii) the diffusion coefficient of the compound studied does not vary with its concentration, and (iii) the partial molar volumes of the compound considered are the same in the two phases and do not change with Concentration (i.e., the sorption effect is negligible). These assumptions are valid in most cases where preparative chromatography is used. AccordingIy, excellent
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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 eluted by a pure solvent. Such an agreement has been reported previously (1). 11. Binary Solvent Used as Mobile Phase. If the solvent is a binary mixture, we have to take into account the adsorption of the strong solvent on the stationary phase surface and the competition between the molecules of the strong solvent and those of the solute for the adsorption sites on the adsorbent surface. This can be done easily by using the same two-compound model we have already used for the study of the separation of the bands of a pair of compounds eluted by a pure eluent (19). The model contains now two mass balance equations and two isotherm equations, one set of equations for the solute, 1:
ac, a w , ) +=O az
(1 + Fk
