Analysis of the Langmuir rate equation in its differential and integrated form for adsorption processes and a comparison with the pseudo first and pseudo second order models

We present a new factorized form of the differential Langmuir rate equation and exhaustively examine the conditions under which the Langmuir kinetics reduces to the widely used pseudo-first order (PFO) and pseudo-second order (PSO) models. A graphical method is proposed and tested with experimental data from the literature for assessing the PFO and PSO models as substitutes for the Langmuir equation and estimating the adsorption parameters. We show that the integrated form of the differential Langmuir rate equation, originally derived by Marczewski (Langmuir 26(19):15229-15238, 2010), can be conveniently expressed in terms of five physically interpretable parameters commonly used in adsorption studies, namely the initial solute concentration, the adsorbent dosage, the microscopic adsorption rate constant, the adsorption amount at the equilibrium and the maximum adsorption capacity of the adsorbent. The potential applicability of the integrated Langmuir equation for the determination of the kinetic parameters of the Michaelis-Menten rate equation is also briefly discussed.