## 4 Kinetic Modelling
It is also worth observing that, because Cr(VI) sorption rate varies with operative conditions such as the initial solute concentration and the sorbent dosage, it is usually not possible to compare the values of \(k_{PFO}\) or \(k_{PSO}\) from different reports. To partially overcome this problem, one could, for example, compare the values of initial sorption rate (\(v_{0}\)) as predicted by the PFO and the PSO model, using Equations (33) and (34), respectively:  
\[v_{0(PFO)}=k_{PFO}q_{e} \tag{33}\]  
\[v_{0(PSO)}=k_{PSO}q_{e}{}^{2} \tag{34}\]  
Values of \(v_{0(PFO)}\) and \(v_{0(PSO)}\) from recent works are reported in \[q=q_{e}\Big{(}1-e^{-k_{PFO}t}\Big{)} \tag{28}\]0 With very few exceptions [32; 53], most of the studies report higher values of \(v_{0(PSO)}\) compared to \(v_{0(PFO)}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Protocol**} & \multicolumn{6}{c}{**Protocol**} & \multicolumn{6}{c}{**Protocol**} \\ \hline
**Sorbent Type** & **G** **c** **mound L-1.7** & **Sorbent Duggy (g-1.5)** & **d-mound G-1.9** & **k-core D-1.9** & **k-core D-1.9** & **d-mound G-1.9** & **d-mound G-1.9** & **k-core D-1.9** & **k-core D-1.9** \\ \hline Magnetic ion exchange ratio & 1.00 & 1 & 0.77 & 5.6 & 7.702 & 6.88 & 29.4 & 78.29 & 25 & 4 & [3] \\ \hline Zero-isothermal inner-neighbourly distributed & 1.85 & 0.75 & 1.648 & 0.78 & 1.285 & 1.49 & 0.652 & 2.81 & 25 & 56 & [4] \\ \hline Zienemann multi-objective heats & 0.83 & 2.5 & 0.0490 & 0.024 & 2.57\(\cdot 10^{5}\) & 0.0288 & 3.536 & 4753.1\(\cdot 10^{5}\) & 25 & 5 & [4] \\ \hline Mg-M hybridization & 3.88 & 2. & 1.671 &  
The Weber-Morris equation [94] is a semi-empirical model often used in Cr(VI) sorption studies for evaluating whether the rate of the process is affected by solute diffusion within sorbent pores (intra-particle diffusion):  
\[q=q_{e}\Big{(}1-e^{-k_{PFO}t}\Big{)} \tag{28}\]1  
Here \(k_{D}\) is a kinetic diffusion parameter and \(\kappa\) is a constant proportional to the thickness of the diffusion boundary layer. A linear correlation between \(q\) and the root square of time suggests that the rate of Cr(VI) sorption is controlled by diffusion [19; 32; 73]. Some authors argued that the presence of multiple linear portions in the \(q\) vs. \(\sqrt{t}\) plot indicates that the sorption process proceeds through two or more consecutive diffusive steps [75]. Another model used for modelling the sorption kinetic data is the Elovich equation [95], which envisages an exponential decrease of the sorption rate with an increase of bound sorbate \(q\), without reaching equilibrium (for \(t\rightarrow\infty\), \(q\rightarrow\infty\)):  
\[q=q_{e}\Big{(}1-e^{-k_{PFO}t}\Big{)} \tag{28}\]2  
where \(\alpha\) is the initial (\(t=0\)) sorption rate and \(\beta\) indicates the extent to which the rate varies with \(q\). It is agreed that the applicability of the Elovich model to experimental data provides evidence of heterogeneity of sorption sites [75]. Equation (36) can be integrated for the boundary conditions \(t=0\) to \(t=t\) and \(q=0\) to \(q=q\), yielding:  
\[q=q_{e}\Big{(}1-e^{-k_{PFO}t}\Big{)} \tag{28}\]3  
Owing to historical difficulties (lack of computational power) associated with the application of non-linear curve fitting [96], an _approximated_-linearized form of Equation (37) was introduced by [PERSON] and [PERSON] [97] for testing the applicability of the Elovich model. The approximated form of the integrated Elovich equation was derived on the assumption that for sufficiently high values of \(t\), the term \(\alpha\)-\(\beta\)-\(t\) becomes much greater than 1 so that Equation (37) reduces to:  
\[q=q_{e}\Big{(}1-e^{-k_{PFO}t}\Big{)} \tag{28}\]4  
Thus, the validity of Equation (38) is verified by the linear plot of \(q\) vs. \(\ln t\). Although non-linear curve fitting is now easy to perform, linearization is still the first-choice procedure for testing the Elovich model, which may lead to incorrect interpretation of data.