# Cationic and Anionic Dye Adsorption on a Natural Clayey Composite
## 2 Materials and Methods
### Adsorption Kinetic Modeling  
The adsorption kinetic data is vital to understanding the adsorption rate, the reaction pathways and adsorption mechanisms, and predicting the rate-controlling step [23]. The adsorption kinetic models express a mathematical relation between the absorate adsorbed and time. There are different adsorption mechanisms; therefore, various phenomenological models describing adsorption kinetics, consisting of different model parameters. We studied the adsorption kinetic data by analyzing the pseudo-first-order, pseudo-second-order, and the intra-particle diffusion model's feasibility. The criterion for choosing the more feasible model was that it should be a good fit between the model and the experimental data. The best-fit model was determined by carrying out linear regressions of the linearized kinetic equation. The values analysis allowed the models' evaluation; below a short overview of the feasible models describing the adsorption kinetic data. [PERSON] proposed the empirical pseudo-first-order model (PFOM) [28]. The PFOM assumes that the absorption could be described by Equation (9). Solving Equation (9) with the boundary condition \(q_{t=0}=0\), one obtains Equation (10). Equations (10) and (11) describe the nonlinear and linear forms of the PFOM. \[\frac{dq_{t}}{dt}=k_{1}(q_{e}-q_{t}) \tag{9}\]  
\[q_{t}=q_{e}\Bigl{(}1-e^{-k_{1}t}\Bigr{)}\text{ when }q_{t=0}=0 \tag{10}\]  
\[\ln(q_{e}-q_{t})=\ln q_{e}-k_{1}t\text{ when }q_{e}>q_{t} \tag{11}\]  
The values of \(q_{e}\) and \(k_{1}\) parameters are usually determined by applying the commonly accepted linear regression procedure, based on Equation (11). Where \(q_{e}\)(mg/g) and \(q_{t}\)(mg/g) are the amounts of adsorbate adsorbed at equilibrium and at any \(t\)(min), respectively. The \(k_{1}\)\(\left(\min^{-1}\right)\) represents the rate constant of the PFOM. Near to the equilibrium \(q_{t}\to q_{e}\), the validation of the PFOM by Equation (11) is significative, affected by random data uncertainty [23; 29] Alternatively, the PFOM validation could be performed with its nonlinear forms in Equation (10). The pseudo-second-order model (PSOM) proposed by [PERSON] et al. [30] assumes that the adsorption follows second-order chemisorption (Equation (12)) [31]. PSOM has been used to describe chemisorption and ion exchange adsorption mechanisms [32]. Equation (13) describes the nonlinear and linear forms of the PSOM. By plotting \(q_{t}/t\)vs. \(q_{t}\), both \(k_{2}\) and \(q_{e}\) constant can be obtained. Furthermore, the initial adsorption rate \((h_{0};\ \text{mg}/\text{g}\times\text{min})\) can be determined using the relation \(h_{o}=k_{2}q_{e}^{2}\). \[\frac{dq_{t}}{dt}=k_{2}(q_{e}-q_{t})^{2} \tag{12}\]  
\[q_{t}=\frac{k_{2}q_{e}^{2}t}{1+k_{2}q_{e}t}\ \text{and}\ \frac{q_{t}}{t}=-(k_{2}q_{e}) q_{t}+k_{2}q_{e}^{2} \tag{13}\]  
where \(q_{e}(\text{mg}/\text{g})\) and \(q_{t}(\text{mg}/\text{g})\) are the amounts of adsorbate adsorbed at equilibrium and at any \(t(\text{min})\), respectively. The \(k_{2}(\text{g}/\text{mg}\times\text{min})\) represents the rate constant of the PSOM. The \(t(\text{min})\) is the label for the contact time between the adsorbate and the adsorbent. Although the PSOM can adequately describe adsorption kinetic experimental data, this model does not help to reveal the adsorption mechanism. The intra-particle diffusion model (IPDM) can help identify the reaction pathways and adsorption mechanisms and predicting the rate-controlling step [23]. Equation (14) shows the linearized transformation of the IPDM proposed by [PERSON] and [PERSON] [33]. In Equation (14) \(K_{ipd}\left(\text{mg}/\text{g}\ \text{min}^{\frac{1}{2}}\right)\) represents the rate constant of the IPDM and \(C(\text{mg}/\text{g})\) is a constant associated with the boundary layer thickness. A higher value of \(C\) corresponds to a more significant effect on the limiting boundary layer. By plotting \(q_{t}\) vs. \(\sqrt{t}\), both \(K_{ipd}\) and \(C\) constants can be obtained.