Kinetics of Solute Adsorption at Solid/Solution Interfaces:  A Theoretical Development of the Empirical Pseudo-First and Pseudo-Second Order Kinetic Rate Equations, Based on Applying the Statistical Rate Theory of Interfacial Transport

Abstract

For practical applications of solid/solution adsorption processes, the kinetics of these processes is at least as much essential as their features at equilibrium. Meanwhile, the general understanding of this kinetics and its corresponding theoretical description are far behind the understanding and the level of theoretical interpretation of adsorption equilibria in these systems. The Lagergren empirical equation proposed at the end of 19th century to describe the kinetics of solute sorption at the solid/solution interfaces has been the most widely used kinetic equation until now. This equation has also been called the pseudo-first order kinetic equation because it was intuitively associated with the model of one-site occupancy adsorption kinetics governed by the rate of surface reaction. More recently, its generalization for the two-sites-occupancy adsorption was proposed and called the pseudo-second-order kinetic equation. However, the general use and the wide applicability of these empirical equations during more than one century have not resulted in a corresponding fundamental search for their theoretical origin. Here the first theoretical development of these equations is proposed, based on applying the new fundamental approach to kinetics of interfacial transport called the Statistical Rate Theory. It is shown that these empirical equations are simplified forms of a more general equation developed here, for the case when the adsorption kinetics is governed by the rate of surface reactions. The features of that general equation are shown by presenting exhaustive model investigations, and the applicability of that equation is tested by presenting a quantitative analysis of some experimental data reported in the literature.

A New Approach for Analysis of Adsorption from Liquid Phase: A Critical Review

“…Equations (8) and (10) are named pseudo-first-and pseudo-second-order equations, respectively. These equations are used frequently for analysis of data of adsorption kinetics experiments and some authors have tried to validate them theoretically [10,11].…”

“…Other points observed regarding these two equations are as follows: (1) as referred in the KASRA model and ideal-second-order and KASRA equations [2], effect of occupied sites on the behavior of free surface sites results in non-ideality observed in their interactions with adsorbate molecules and thus there are two regions (and also two parts in the second region) before plateau region in the adsorption breakthrough curve. Therefore, it is not possible to use these equations in the whole time range of experiment, (Figures 3a and 3b), (2) Due to the shape of adsorption kinetics graphs in the first and second regions, when data of the whole time range of experiments are used in these two equations, most data of the first region (compared to the second region) fit to the pseudo-first-order equation while more data of the second region fit to the pseudo-second-order equation [2,12,13], (3) Data of plateau satisfy the pseudo-second-order equation where no adsorption process occurs in this region, (Figures 4a and 4b) [2,14,15], (4) Only by using data of the whole time range, q e,1 and q e,2 values are similar to experimental q e ,(5) Authors of theoretical works use the whole time range as an integrated region for their calculations to calculate a pseudo-first-or pseudo-second-order rate constant for adsorption process that is not a right assumption [7][8][9][10]16] and finally (6) The pseudo-first-and pseudo-second-order rate constants are mixed constants [17] and include c t variable values and usually do not change orderly with temperature and we can't use them in the Arrhenius equation to calculate activation energy of adsorption process [1,18,19].…”

“…But, some authors [57,58] incorrectly attribute these sequential adsorption kinetic curves to one kind of adsorption site, (Figure 10), (7) Data of plateau region can't be used in kinetic and thermodynamic calculations. Nevertheless, during study of rapid adsorption process, some authors [59] reported data of plateau region rather than data of regions before it, ( Figure 11) and (8) Under alkaline conditions, metal ions participate as metal hydroxide. But, in some cases it was wrongly reported [60] that adsorption capacity …”

Splitting methods for relaxation two-phase flow models

“…The probability of each event is calculated using a first-order perturbation analysis of the Schrödinger equation, together with the Boltzmann definition of entropy. Since its introduction by Ward et al (1982), SRT has been used to model a number of different transport processes, including crystal growth (Dejmek & Ward, 1998), solution/solid adsorption (Azizian et al, 2008;Rudzinski & Plazinski, 2006), gas/solid adsorption (Elliott & Ward, 1997a;, temperature programmed desorption (Elliott & Ward, 1997b), ion permeation across lipid membranes (Bordi et al, 2000), chemical reactions (Harding et al, 2000), and evaporation and condensation (Ward & Fang, 1999;Kapoor & Elliott, 2008;Ward & Stanga, 2001). Lund and Aursand (2012) developed an explicit expression for the phase transfer source term Γ in Eqs.…”

A New Adsorption Rate Equation in Batch Systems

“…Activation energy is the energy which must be available to a chemical system with potential reactants to result in a reaction. Activation energy is obtained by Arrhenius equation and the rate constants of different temperature.Activation energy can't be evaluated by the many of previous adsorption kinetic models, such as the pseudo-order models [1][2][3][4][12][13][14][15] and its changed types [5][6][7][8][20][21][22][23] . These models involve the adsorption amount q e (No.…”