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The applicability of a frequency response (FR) method for determination of single solute adsorption isotherms is investigated. The case study concerns the adsorption of 4-tert-butylphenol and ethyl benzoate as single solutes on octadecyl silica from a methanol and water mixture (60:40, v/v). The method is based on the analysis of the nonlinear frequency response of a chromatographic column to a periodical change of the inlet concentration using the concept of higher-order frequency response functions (FRFs). These functions are used for estimation of the first, second, etc. derivatives of the adsorption isotherm for a chosen steady-state. In this particular case the derivatives up to the third order are considered and the corresponding FRFs are analyzed. Signs of the second and third derivatives of the adsorption isotherm are estimated from the low frequency asymptotes of the phases of the second and third order FRFs, respectively, while absolute values of the derivatives are estimated from the low frequency asymptotes of the first derivatives of the corresponding FRFs [1].
The chosen compounds exhibit significantly different equilibrium behaviour. The adsorption of 4-tert-butylphenol can be described for studied conditions by the Langmuir adsorption isotherm and the adsorption of ethyl benzoate by the BET isotherm. Small injections of the compounds into the not preloaded column, as well as conventional frontal analysis experiments are performed for comparison and rough estimation of the adsorption isotherm coefficients. The experiments were performed using a standard HPLC system which is capable to realize inlet concentration changes in a nearly sine wave form around the chosen steady-state. Experiments were performed for different steady-state concentrations and amplitudes of the inlet concentration change. The concentration changes over time are analyzed in the frequency domain that is obtained by performing the harmonic analysis of the concentration changes, i.e. by applying the fast Fourier transform to selected portions of the inlet and outlet concentrations of the same length. Afterwards, the FRFs are estimated from one or two amplitudes of the inlet concentration change. The coefficients of the adsorption isotherm are obtained by fitting the adsorption isotherm derivatives of the assumed isotherm equation to the isotherm derivatives estimated from the experiments.
[1] Petkovska M., Seidel-Morgenstern A. (2005), Nonlinear frequency response of a chromatographic column. Part I: Application to estimation of adsorption isotherm with inflection point, Chem. Eng. Comm. 192, 1300-1333
