Theoretical Analysis of Langmuir Adsorption Kinetics for Defining Initial Concentration-Based Transition Boundaries Between Pseudo-First and Pseudo-Second Order Models: A Mathematical Approach

Despite the widespread application of pseudo-first-order (PFO) and pseudo-second-order (PSO) kinetic models in adsorption processes, the selection of these models often relies on an empirical approach, lacking explicit thermodynamic justification and well-defined operational boundaries. To address this phenomenological ambiguity, a theoretical and dimensionless framework derived from Langmuir adsorption kinetics is developed in this study. By formulating a kinetic phase diagram, analytical boundary equations were derived within a dimensionless space defined by the initial concentration and the capacity parameter. This theoretical phase diagram was validated by mapping the experimental kinetic trajectories for the adsorption of penicillin and cephalosporin antibiotics. Employing a sensitive relative goodness-of-fit criterion based on error variance, the concentration-dependent transition from the PFO regime to HOTZ was quantitatively confirmed, demonstrating excellent agreement with the theoretical boundaries. Ultimately, by decoupling the intrinsic adsorption kinetics from macroscopic operating conditions, this reference diagram can provide chemical, environmental, and textile engineers with a robust tool for predictive reactor design and rational scale-up. Particularly in textile engineering, the application of this framework for a more precise analysis of fiber dyeing kinetics, as well as the optimal design of adsorbents for colored wastewater treatment, can minimize the need for costly trial-and-error experiments.