Pseudo-first-order kinetics in environmental adsorption: Why are there two distinct equations?

The pseudo-first-order (PFO) kinetic model is conventionally written as ln(q_e − q_t) = ln(q_e) – k₁·t. However, a mathematically distinct equation, 1/q_t = τ/(q_e·t) + 1/q_e, has been repeatedly and erroneously labeled as the PFO model in the literature. This study is the first to systematically examine and clarify this pervasive misidentification. We identify two key factors contributing to the confusion: (1) the equation’s initial designation as the “generalized first-order kinetic equation,” which was later conflated with the authentic PFO model due to scholarly oversight, and (2) its superficial resemblance to the PFO expression under specific conditions, leading to the mistaken assumption of full equivalence. We show that this equation, originally introduced in the 1960s, constitutes a separate kinetic model with no mathematical relationship to the PFO model. To maintain methodological rigor and terminological accuracy, we urge the environmental adsorption community to discontinue its mislabeling and to correctly recognize this equation as a distinct kinetic formulation.