Sadroddin Golshan-Shirazi

Comparison of the various kinetic models of non-linear chromatography

Abstract It is not possible to separate simply the influences of a slow mass transfer kinetics and of a slow kinetics of adsorption-desorption on the elution profile of a component. Four Kinetic models of chromatography are studied: (i) the Thomas or reaction model, which assumes Langmuir Kinetics of adsorption-desorption and no axial dispersion; (ii) the reaction-dispersive model, which uses the same Langmuir kinetics as the Thomas model but assumes a finite axial dispersion; (iii) a transport model, which uses the linear solid film driving force model to account for a slow kinetics of mass transfer and assumes no axial dispersion; and (iv) a transport-dispersive model, using the same mass transfer kinetics as the transport model and assumes finite axial dispersion. The analytical solution of the Thomas model can be fitted, with an accuracy which exceeds the precision of experimental measurements, on bands calculated using either one of the other three kinetic models of chromatography. Thus, the Thomas model, which assumes a slow adsorption-desorption kinetics and infinetely fast mass transfer kinetics, accounts very well for profiles calculated with a model making the reverse assumption. However, the values of the lumped kinetic coefficient obtained by curve fitting depend on the sample amount. Thus, the examination of an elution profile and its fitting to a model do not permit an easy solution of the inverse problem of chromatography.

Solutions of the equilibrium and semi-equilibrium models of chromatography

Abstract In contrast to the kinetic models, the ideal and semi-ideal models of chromatography assume the distribution of the compounds studied to be constantly at equilibrium (ideal model) or very close to equilibrium (semi-ideal model). An exact solution of the ideal model can be obtained under close form for a pure compound with any isotherm and for a binary mixture with competitive Langmuir isotherms. No exact solution of the semi-ideal model can be derived but numerical solutions are available for all isotherms. Approximate analytical solutions for this model can be obtained by assuming that the concentration of the compound studied in the mobile phase is small and, accordingly, that the equilibrium isotherm is parabolic and by neglecting some terms in the derivation. Depending on the assumptions made, the Houghton and the Haarhoff—Van der Linde equations are obtained. These different solutions are compared. It is shown that the Haarhoff—Van der Linde equation is a much better approximation than the Houghton equation and that its range of validity depends essentially on the deviation between the true isotherm and its two-term expansion in the concentration range sampeld by the band during its elution. It is usually valid for loading factors below 0.2% for an ideal column and bCMax ≤ 0.05 for real columns (the loading factor is the ratio of the sample size and the column saturation capacity, b is the second coefficient of a Langmuir isotherm and CMax is the maximum concentration of the band). In practice, however, it can be used for loading factors up to 1% (bCMax ≥ 0.1 for real columns). The ideal model, in contrast, gives a valid presentation of experimental band profiles only at high sample size and column efficiencies. The reduced sample size, m = NLf [k′0/(1 + k′0_]2 (N = column plate number, Lf = loading factor, k′0 = column capacity factor), must be higher than 35. In the intermedia range, only numerical solutions can predict the band profiles accurately. In the case of two components, the exact solution of the ideal model can be obtained under close form with competitive Langmuir isotherms. Numerical solutions can be obtained to simulate real columns. No other analytical solution, even approximate, is available. A correction made to the ideal model to account for the band-broadening effect of a finite efficiency gives good results and permits the investigation of the optimization of the experimental conditions of a separation for maximum production rate.

Modeling of preparative liquid chromatography

Abstract In order to achieve an acceptable production rate at reasonable cost, preparative chromatography must be carried out with phase systems in which the kinetics of mass transfers and adsorption—desorption are fast. Accordingly, band profiles in overloaded chromatographic columns are best understood by considering the ideal model, while the process itself is most suitably modeled using the equilibrium-dispersive model. The former model assumes an infinite column efficiency, while the latter lumps the contributions of axial dispersion and mass transfer resistances into a single apparent dispersion coefficient. The properties and solutions of these models are reviewed. The conditions under which they give satisfactory results are summarized. The excellent agreement between the experimental band profiles of the components of binary mixtures and the individual band profiles calculated with the equilibrium-dispersive model is demonstrated. The degree of agreement is limited only by the accuracy with which the competitive equilibrium isotherms are accounted for.

Study of the representation of competitive isotherms and of the intersection between adsorption isotherms

Abstract Theoretical investigations of the separation of the components of a mixture by preparative liquid chromatography require a knowledge of the competitive equilibrium isotherms of the components of the mixture between the two phases of the chromatographic system. In many instances the set of competitive Langmuir equations provide a satisfactory model. When the competitive isotherms of the two components of a binary mixture intersect each other, however, the Langmuir model becomes unsuitable. This model postulates constant selectivity for the equilibrium between the two phases of a chromatographic system. A modified Langmuir model is proposed, using the ratio between two second-degree polynomials. This model is supported by general results from statistical mechanics. It accounts well for some experimental data that could not be explained in terms of the Langmuir isotherm.

Measurement of the heats of adsorption of chiral isomers on an enantioselective stationary phase

Abstract The adsorption isotherms of the N-benzoyl- d - and l -alanine were measured at different temperatures, and the enthalpy of adsorption and the isosteric heat of adsorption were extracted from the data. These thermodynamic functions provide further evidence that a bimodal retention mechanism is present for the separation of enantiomeric pairs on a bovine serum albumin stationary phase. The first mode of interaction is associated with the chiral selective properties of the column whereas the second is associated with the non-chiral selective properties.

Nonlinear chromatography Recent theoretical and experimental results.

The theory of nonlinear chromatography has been advanced by the incorporation of recent results obtained by the theory of partial differential equations. The system of equations of the ideal model has been solved analytically in the case of a single component for which the equilibrium isotherm between the mobile and the stationary phases is given by a Langmuir equation. A series of computer programs has been written which permits the calculation of numerical solutions of the semi-ideal model. The properties of the solutions obtained are described and discussed for a one-component system (profile of high concentration bands of a pure compound eluted by a pure solvent), several two-component systems (elution of a pure compound band by a binary mobile phase, separation of a binary mixture eluted by a pure mobile phase), and three-component systems (separation of a binary mixture eluted by a binary solvent, displacement and separation of a binary mixture). Experimental results are reported which validate the conclusions derived from the numerical integration of the model. The conclusions of the work apply to all high-performance chromatographic procedures, i.e., to those where the kinetics of mass transfer are fast enough for the mobile and stationary phases always to be near equilibrium. More specifically, the contribution from the kinetics of the retention mechanism to the mass transfer resistance must itself be negligible. This clearly excludes affinity chromatography.

Analytical solution of the ideal model of elution chromatography in the case of a binary mixture with competitive Langmuir isotherms : II. Solution using the h-transform

Abstract Using the results published by Helfferich and Klein, an exact solution of the ideal model of chromatography (infinite column efficiency) is derived, giving the band profiles for the two-component elution problem in the case when the equilibrium isotherms are given by the classical competitive Langmuir equations. The variations of the band profile of each component during elution is analyzed and the interactions between the two profiles are investigated. Two concentration shocks appear, one at the front of each component elution profile. The chromatogram is separated into three zones. The first zone, between the two concentration shocks, contains only the first component. The second zone, immediately after the second shock, contains a mixture of the two components. The third zone, at the rear, contains only the second component. The profiles of the two component in the three zones and their concentrations on both sides of the second shock are given by simple analytical equations. If the sample is injected as a rectangular pulse, it takes some time to erode the corresponding concentration plateaux of each component. On both sides of the mixed zone, a second concentration plateau appears for each component. The heights of these plateaux remain constant as long as they are present. The first component plateau disappears rapidly, but the second component plateau, whose formation explains the “tag-along” effect, remains stable as long as the second zone has not vanished and decreases progressively after the two bands are resolved. Comparison between the profiles obtained as solutions of the ideal model and those calculated using the program of the semi-ideal model, which accounts for the finite efficiency of actual columns, shows very good agreement when the column efficiency exceeds a few hundreds to 1000 plates. The extent of the agreement depends on both the sample size and the column efficiency. The concentration shocks are replaced by shock layers whose thickness is proportional to the column plate height, but depends also on the shock height. The thickness of the second shock, which separates the first and second zones, seems to depend much more than the thickness of the first shock on the actual column efficiency.

Optimization of experimental conditions in preparative liquid chromatography : trade-offs between recovery yield and production rate

Abstract The optimization of the experimental conditions in preparative chromatography under constraints of recovery yield, product purity and maximum available pressure is discussed. It is shown that there are optimum values of the loading factor and the column limit efficiency which permit the achievement of the maximum production rate under specified constraints of recovery yield and product purity. The optimum loading factor is given by a simple equation. The optimum column efficiency is calculated from numerical solutions of the semi-ideal model. The optimum column length for a given packing material and particle size and the optimum mobile phase velocity are then derived. These optimum values depend on the maximum available pressure, and the production rate increases rapidly with increasing pressure. If a given column is available and it is shorter than the optimum length, it should be operated at the optimum loading factor and at the optimum column efficiency. The mobile phase velocity, and hence the production rate, are less than those for the column of optimum length. If the column is longer than the optimum and cannot be cut, it should be operated at the maximum available pressure, at a mobile phase velocity lower than the optimum and with a loading factor larger than the optimum.

Effect of the intersection of the individual isotherms in displacement chromatography

Abstract Neither the existence of an intersection between the two single-component isotherms drawn on the same graph, nor the fact that the column saturation capacity for the more retained component is lower than that of the lesser retained component, have any major consequence on the chromatographic behavior of elution bands or on the formation of the isotachich train, as long as the equilibrium isotherms of the two components are properly described by the competitive Langmuir model. Significant deviation from this model could make impossible the formation of an isotachic train in displacement chromatography, but definitive experimental proof of the existence of this effect is lacking.

A retrospective on the solution of the ideal model of chromatography

In an attempt to clarify the difficulties encountered in finding solutions of the ideal model of chromatography and to justify these solutions, the history of the theoretical work performed on this model is reviewed. The publications of Wilson, DeVault, Glueckauf, Helfferich, Klein, Rhee, Aris and Amundson are analyzed, the efforts and successes of these authors are discussed and the rationale for this endeavor is explained.