* Corresponding author. Department of Chemistry, The University of Tennessee, 611 Buehler Hall, Knoxville, TN 37996-1600, USA. Tel.: #1-423-974-0733; fax: #1-423-974-2667. Chemical Engineering Science 54 (1999) 1677—1696 Determination of binary competitive equilibrium isotherms from the individual chromatographic band profiles Franiois James, Mauricio Sepu´lveda, Fre´de´ric Charton, Igor Quin  ones, Georges Guiochon  * MAPMO, UMR CNRS 6628, Universite ´ d+Orle ´ ans, F-45067 Orle ´ ans, France Departamento de Ingenerı & a Matema ´ tica, Universidad de Concepcio ´ n, Casilla 4009, Concepcio ´ n, Chile Department of Chemistry, The University of Tennessee, 414 Buehler Hall, Knoxville, TN 37996-1600, USA Division of Chemical and Analytical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6120, USA Received 11 December 1997; received in revised form 2 December 1998; accepted 10 December 1998 Abstract A numerical solution of the inverse problem of nonlinear chromatography is described and validated. This method allows the determination of best numerical estimates of the coefficients of an isotherm model from the individual elution profiles of the two components of a binary mixture. The sample size must be large enough for the two bands to interfere strongly and for their maximum concentrations to exceed the range within which the isotherm equation is needed. In two cases, excellent agreement was observed between the equilibrium isotherm equations obtained by this new method and those determined by the classical combination of elution by characteristic points and binary frontal analysis. In the first case, the adsorption of the ketoprofen enantiomers on a cellulose-based chiral phase is accounted for by a competitive Bilangmuir isotherm. In the second case, the adsorption of benzyl alcohol and 2-phenylethanol on C18 silica is accounted for by a competitive Langmuir model. The importance of using the proper boundary conditions (i.e., a realistic injection profile) is stressed. The new method seems especially well suited for the rapid determination of the isotherms of enantiomers needed for the computer-assisted optimization of the separation of mixtures of these compounds, e.g., in simulated moving-bed applications.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Isotherm; Chromatography; Band profiles; Enantiomers 1. Introduction Adsorption equilibrium isotherms relate the excess amount of adsorbates close to a surface and the chemical potentials of these adsorbates in the bulk fluid and in the vicinity of the surface. Isotherm plots are not linear but deviate from linear behavior for a variety of reasons, mainly related to the saturation and/or the heterogeneity of the surface or to adsorbate—adsorbate interactions in the bulk fluid and/or close to the surface. Conversely, the experimental determination of equilibrium isotherms allows the test of assumptions made regarding the extent and nature of surface heterogeneity and of adsorbate— adsorbate interactions exhibited by an adsorbent—adsor- bate system (Ruthven, 1984; Jaroniec and Madey, 1988; Rudzinski and Everett, 1992). Isotherms are for- mally described in terms of excess amounts close to the surface and are related to the chemical potentials of each species. It is much easier, however, and it has become conventional in applied fields to substitute adsorbate and fluid phase concentrations for the more correct physicochemical parameters. This practice is commonly accepted in chemical engineering and we will follow it here. Isotherms, whether those of pure compounds or those of the different components of mixtures are of great importance in separation sciences. They allow calcu- lations of unit performance, the comparison of the poten- tial of different separation schemes and, to some extent, the prediction of the production rates and the extent of purification which will probably be achieved. At least, they tell clearly what would be the best possible separ- ation and allow simple comparisons between different 0009-2509/99/$ — see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 5 3 9 - 9