## 4 Kinetic Modelling  
The study of sorption kinetics is of practical significance because it provides information on the time required for effective removal of a solute from the aqueous phase. Among the models proposed for describing the kinetics of Cr(VI) sorption, the most popular ones are the pseudo-first order (PFO) and the pseudo-second order (PSO) model; other frequently used models are the Weber-Morris and the Elovich model. The PFO model [91] assumes that the observed rate of sorption (\(dq/dt\)) is proportional to the distance from equilibrium expressed as the difference between the amount of Cr(VI) sorbed at equilibrium (\(q_{e}\)) and at any time (\(q\)):  
\[\frac{dq}{dt}=k_{PFO}(q_{e}-q) \tag{27}\]  
where \(k_{PFO}\) is the PFO kinetic rate constant. By integrating Equation (27) for the boundary conditions \(t\) = 0 to \(t\) = \(t\) and \(q\) = 0 to \(q\) = \(q\), we obtain:  
\[q=q_{e}\Big{(}1-e^{-k_{PFO}t}\Big{)} \tag{28}\]  
The PSO model [92] differs from the PFO model because the sorption rate is assumed to be proportional to the square of the distance from equilibrium:  
\[\frac{dq}{dt}=k_{PSO}(q_{e}-q)^{2} \tag{29}\]  
Here \(k_{PSO}\) is the PSO kinetic rate constant. Equation (29) can be integrated, for the same boundary conditions as for Equation (27), leading to:  
\[q=\frac{q_{e}{}^{2}k_{PSO}t}{1+q_{e}k_{PSO}t} \tag{30}\]It is widely maintained that the application of the PSO model to Cr(VI) sorption kinetic data gives better fitting results than the PFO model. Examples include studies carried out using as sorbent bacterial cellulose/attapulgetic composites [83], palm oil kernel shell [80], cross linked-chitosan-grafted-polyaniline composite [64], teff straw [70] and material derived from harmful algal bloom biomass [27]. In contrast, [PERSON] et al. [63] found that the PFO model was best suited to describe the kinetic data of Cr(VI) sorption onto chitosan/poly(vinyl amine cryogel. Better applicability of the PFO model appears to be associated with weak Cr(VI)-sorbent interaction (physisorption) [63], whereas better applicability of the PSO model is indicative of strong interaction (chemisorption) [41]. Noticeably, better performance of the PSO model may also be an artifact from improper analysis of the data. In most cases this results from the application of Equation (30) in the following linearized form [17; 93]:  
\[\frac{t}{q}=\frac{1}{k_{PSO}q_{e}}^{2}+\frac{t}{q_{e}} \tag{31}\]  
According to Equation (31), if the data obey the pseudo-second order model, a plot of \(t\)/\(q\) vs. \(t\) should produce a straight line. The main drawback of this method is that for data points at (or very close to) equilibrium, the plot of \(t\)/\(q\) vs. \(t\) becomes linear independently of sorption kinetics [93]. The goodness of the results obtained by the plot of \(t\)/\(q\) vs. \(t\) may be verified by using a linearized form of the PSO model [17]:  
\[\frac{q}{t}=k_{PSO}q_{e}^{2}-k_{PSO}q_{e}q \tag{32}\]  
If the PSO model is appropriate for the data, the plot of \(q\)/\(t\) against \(q\) will be linear. As an example, the comparison between \(t\)/\(q\) vs. \(t\) and \(q\)/\(t\) vs. \(q\) for the sorption of Cr(VI) onto MnO\({}_{2}\)[24] is reported in Figure 5. The authors claimed that the sorption kinetics was adequately described by the PSO model because the linear curve-fitting of \(t\)/\(q\) vs. \(t\) data gave excellent results (R\({}^{2}\) = 0.999, see Figure 5). However, as can be seen from Figure 5, this conclusion is incorrect because the same data, rearranged in the form \(q\)/\(t\) vs. \(q\), significantly deviate from linearity. Figure 5: Comparison between the plots of \(q\) vs. \(t\) and \(q\)/\(t\) vs. \(q\) for the sorption of Cr(VI) onto MnO\({}_{2}\); initial Cr(VI) aqueous concentration = 0.02 mM, sorbent dosage = 2 g L\({}^{-1}\). Adapted from [PERSON] et al. [24]. As a general rule, the non-linear curve fitting is recommended over the linear curve fitting to prevent possible misinterpretation of the data.