## 2 Materials and Methods
### Methods
#### 2.2.4 Mathematical Models
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Linearized Equation** & **Plot** & **Model Parameters** \\ \hline \multirow{3}{*}{Langmuir [29; 30; 31]} & \(\frac{\mathrm{C}_{\mathrm{e}}}{\mathrm{k}_{\mathrm{e}}}=\frac{\mathrm{C}_{ \mathrm{e}}}{\mathrm{q}_{\mathrm{m}}}+\frac{1}{\mathrm{q}_{\mathrm{m}}\mathrm{ K}_{\mathrm{c}}}\) & \multirow{3}{*}{\(\mathrm{C}_{\mathrm{e}}/\mathrm{q}_{\mathrm{e}}\) against \(\mathrm{C}_{\mathrm{e}}\)} & \(\mathrm{q}_{\mathrm{m}}\): maximum monolayer adsorption capacity (mg/g) \\  & \(\mathrm{R}_{\mathrm{L}}=\frac{1}{\mathrm{T}+\mathrm{R}_{\mathrm{L}}+\mathrm{C}_{ 0}}\) & & \(\mathrm{K}_{\mathrm{L}}\): Langmuir adsorption constant (L/mg) \\ \hline Freundlich [32; 33; 34] & \(\mathrm{ln}\mathrm{q}_{\mathrm{e}}=\frac{1}{n}\cdot\mathrm{ln}\mathrm{C}_{ \mathrm{e}}+\mathrm{ln}\mathrm{K}_{\mathrm{F}}\) & \(\mathrm{ln}\mathrm{q}_{\mathrm{e}}\) against \(\mathrm{In}\mathrm{C}_{\mathrm{e}}\) & \(\mathrm{K}_{\mathrm{F}}\): constant related to the adsorption capability (mg/g)-(L/mg)\({}^{1/n}\) \\  & & & \(\mathrm{n}\): constant related to adsorption intensity \\ \hline \multirow{3}{*}{Elovich [35]} & \(\ln\frac{\mathrm{q}_{\mathrm{e}}}{\mathrm{C}_{\mathrm{e}}}=\ln(\mathrm{K}_{ \mathrm{E}}\cdot\mathrm{q}_{\mathrm{m}\mathrm{E}})-\frac{\mathrm{q}_{\mathrm{e} }}{\mathrm{q}_{\mathrm{m}\mathrm{E}}}\) & \(\mathrm{ln}(\mathrm{q}_{\mathrm{e}}/\mathrm{C}_{\mathrm{e}})\) against \(\mathrm{q}_{\mathrm{e}}\) & \(\mathrm{q}_{\mathrm{m}\mathrm{E}}\): maximum Elovich adsorption capacity (mg/g) \\  & & \(\mathrm{(mg/g)}\) \\  & & \(\mathrm{K}_{\mathrm{E}}\): Elovich equilibrium constant (L/mg) \\ \hline Temkin [36; 37; 38] & \(\mathrm{q}_{\mathrm{e}}=\mathrm{B}\cdot\mathrm{ln}\mathrm{A}+\mathrm{B}\cdot \mathrm{ln}\mathrm{C}_{\mathrm{e}}\) & \(\mathrm{q}_{\mathrm{e}}\) against \(\mathrm{ln}\mathrm{C}_{\mathrm{e}}\) &
\begin{tabular}{l} A: adsorption equilibrium constant (L/mg) \\ B: the Temkin constant (mg/g) \\ \end{tabular} \\ \hline [PERSON] [39; 40] & \(\mathrm{ln}\mathrm{q}_{\mathrm{e}}=\mathrm{ln}\mathrm{q}_{\mathrm{m}}-\mathrm{K}_{ \mathrm{J}}\cdot\mathrm{C}_{\mathrm{e}}\) & \(\mathrm{ln}\mathrm{q}_{\mathrm{e}}\) against \(\mathrm{C}_{\mathrm{e}}\) & \(\mathrm{q}_{\mathrm{m}}\): maximum adsorption capacity (mg/g) \\  & & \(\mathrm{K}_{\mathrm{J}}\) is the Jovanovic constant (L/mg) \\ \hline \multirow{3}{*}{Dubinin–Radushkevich [41; 42]} & \(\mathrm{ln}\mathrm{q}_{\mathrm{e}}=\mathrm{ln}\mathrm{q}_{\mathrm{m}}-\mathrm{K} \cdot\mathrm{\epsilon}^{2}\) & \multirow{3}{*}{\(\mathrm{ln}\mathrm{q}_{\mathrm{m}}\): maximum adsorption capacity (mg/g)} \\  & & \(\mathrm{K}\): constant related to the sorption energy \\  & & \(\mathrm{(mol^{2}/\mathrm{P}^{2})}\) \\  & & \(\mathrm{\epsilon}\): adsorption potential (J/mol) \\  & & & \(\mathrm{E}\): mean free energy of adsorption \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Adsorption equilibrium mathematical models. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Linearized Equation** & **Plot** & **Model Parameters** \\ \hline \multirow{3}{*}{Pseudo-first order [43; 44]} & \(\ln(\mathrm{q}_{\mathrm{e}}-\mathrm{q}_{\mathrm{t}})=\mathrm{ln}\mathrm{q}_{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrmmathrm{\mathrm{\mathrmmathrmmathrm{\mathrm{                                                           \! \! \!\!\!\!\The adsorbate molecules diffuse through the interior of the adsorbent particles (intra-particle diffusion). Any one of them, or a combination of both [51], can control the rate of the adsorption process. The Weber-Morris intraparticle diffusion model and the Boyd model are used to identify the rate-controlling step of the adsorption process (Table 3). When pure intraparticle diffusion exists, the Weber-Morris representation shows a straight line passing through the origin.