Physical insights into kinetic models of adsorption

The ratio of the two roots of the Langmuir kinetic rate equation at equilibrium is a critical factor for assessing the possibility of reducing the Langmuir kinetics to the pseudo first- (PFO) and second-order (PSO) models of batch adsorption. This critical factor solely depends on the ratio of available adsorbate to adsorbent capacity and the product of the Langmuir equilibrium constant and the adsorbent capacity. [Display omitted] •Derivation of physical meaning of two roots of the Langmuir rate equation.•Derivation of applicability criteria for the pseudo first- and second-order models.•The criteria, the ratio of the two roots, are solely determined by two factors.•Determination of correct pseudo-first- and second-order rate constant.•Detailed analyses on the errors introduced if the criteria are not satisfied. The pseudo first- (PFO) and second-order (PSO) kinetic models have been extensively applied to study the kinetics of interfacial processes, for example, separation or removal of toxic metal ions from aqueous solution by adsorption. However, the two models were used without any specialization of operating conditions and their theoretical origin is not well understood. The aim of the present study is to provide clear physical understanding on the relationships of the Langmuir kinetics with the two models. The conditions under which the Langmuir kinetics can be reduced to the simplified models and their explicit dependence on operating conditions and isotherm parameters are provided. We adopt the square root method to obtain two roots of the differential Langmuir rate equation at equilibrium, then prove that one root is the surface coverage fraction and the other is the reciprocal of adsorption efficiency at equilibrium, and show that the root ratio is a critical factor in assessing the applicability of the PFO and PSO models. This critical factor solely depends on the ratio of available adsorbate to adsorbent capacity and the product of the Langmuir equilibrium constant and adsorbent capacity. The reduction of the Langmuir kinetics to the PFO and PSO models can be valid only under extreme conditions. The system with too high or too low available adsorbate relative to the maximum adsorbent capacity or with too small adsorption rate constant multiplied by the maximum adsorbent capacity relative to the desorption rate constant might be approximately described by the PFO model. On the other extreme that available adsorbent capacity matches available ads