An Empirical Isotherm Model for Cationic Dye Sorption onto Organosilica: A Focus on Adsorption-Partitioning Behavior

This study presents a novel three-parameter empirical isotherm model, the Shin model, developed to improve the prediction of cationic dye sorption. It integrates the Freundlich isotherm at low concentrations (adsorption) and transitions to a linear isotherm at high concentrations (partitioning). The model was evaluated using the sorption of crystal violet (CV), thioflavin T (TFT), and acriflavine neutral (AFN) onto hexadecyltrimethylammonium-modified silica (HDTMA–silica). Sorption affinity and capacity followed the order CV > TFT > AFN. The mechanism involves ion exchange with surface cations and two-dimensional adsorption onto a pseudo-organic layer formed by HDTMA’s hydrocarbon chains. Comparative analysis showed that three-parameter models—including Sips, dual-mode, Khan, modified Song, and Shin—provided better fits than two-parameter models like Freundlich and Langmuir, due to enhanced flexibility. Among these, the Shin model demonstrated superior performance, especially in capturing the asymptotic behavior of sorption at varying concentrations. Its ability to represent both adsorption and partitioning makes it particularly suitable for modeling cationic dye sorption on HDTMA–silica surfaces. This work highlights the Shin model’s effectiveness in describing complex sorption dynamics and supports its application in environmental remediation studies.