From Langmuir kinetics to first- and second-order rate equations for adsorption

So far, the first- and second-order kinetic equations have been most frequently employed to interpret adsorption data obtained under various conditions, whereas the theoretical origins of these two equations still remain unknown. Using the Langmuir kinetics as a theoretical basis, this study showed that the Langmuir kinetics can be transformed to a polynomial expression of dtheta t /d t = k 1(theta e - theta t ) + k 2(theta e - theta t ) (2), a varying-order rate equation. The sufficient and necessary conditions for simplification of the Langmuir kinetics to the first- and second-order rate equations were put forward, which suggested that the relative magnitude of theta e over k 1/ k 2 governs the simplification of the Langmuir kinetics. In cases where k 1/ k 2 is greater than theta e or k 1/ k 2 is very close to theta e, adsorption kinetics would be reasonably described by the first-order rate equation, whereas the Langmuir kinetics would be reduced to the second-order equation only at k 1/ k 2 << theta e. It was further demonstrated that both theta e and k 1/ k 2 are the function of initial adsorbate concentration ( C 0) at a given dosage of adsorbent, indicating that simplification of the Langmuir kinetics indeed is determined by C 0. Detailed C 0-depedent boundary conditions for simplifying the Langmuir kinetics were also established and were verified by experimental data.