Characterization of sigmoidal adsorption isotherms for pharmaceutical contaminants

The Frumkin–Fowler–Guggenheim (FFG) and Hill–de Boer (HdB) models are frequently used to describe sigmoidal equilibrium isotherms for aqueous-phase adsorption of contaminants, including pharmaceutical compounds. Although their implicit nature requires the adsorbed-phase concentration to be determined numerically, these models are often fitted using linearized forms. This approach is fundamentally flawed because it relies on arbitrary assumptions, such as fixing the saturation capacity at the experimentally observed plateau. Moreover, experimental datasets often do not exhibit linearity over their full range, prompting subjective selection of data subsets to obtain apparently satisfactory linear fits. More importantly, an apparently good linear fit can yield lateral interaction parameters exceeding the critical values of 4 for the FFG model and 6.75 for the HdB model, producing multivalued isotherms that contradict the observed monotonic behavior. These critical values originate from lattice-gas theory and are associated with two-dimensional condensation, a phenomenon not typically observed in conventional aqueous adsorption systems. Using literature data for niflumic acid and ciprofloxacin, we demonstrate that direct nonlinear fitting avoids these limitations by utilizing complete datasets and enforcing physically meaningful parameter constraints. The implicit nature of the models can be readily accommodated by embedding a numerical root-finding procedure within each iteration of the nonlinear parameter estimation. Consequently, nonlinear fitting provides a more rigorous and reproducible framework for applying the FFG and HdB models to sigmoidal adsorption data.