SOLUTIONS OF THE EQUILIBRIUM AND SEMIEQUILIBRIUM MODELS OF CHROMATOGRAPHY

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 = N Lf [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. © 1990.