## 2 Materials and Methods
### Adsorption Modelling
#### 2.5.2 Adsorption Isotherm  
Adsorption isotherms use concentration-dependent data to describe the distribution of the adsorbate between the solid and liquid phases at equilibrium. The adsorption isotherm can give useful information about the adsorbate, the adsorbent, and the adsorption process.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Contaminant**} & \multicolumn{6}{c}{**Parameters**} \\ \cline{2-6}  & & \multirow{2}{*}{pH} & \multirow{2}{*}{C\({}_{0}\) (mg/L)} & \multicolumn{2}{c}{\(\mathbf{C_{ads}}\)} & \multirow{2}{*}{t (Min)} & \multirow{2}{*}{T (\({}^{\circ}\)C)} \\  & & & & & & \\ \hline \multirow{4}{*}{CV} & pH effect & 2–12 & 100 & 0.1 & 45 & 30 \\  & C\({}_{\mathrm{ads}}\) effect & 9 & 100 & 0–0.5 & 45 & 30 \\  & Effect of t & 9 & 100 & 0.1 & 0–120 & 30 \\  & Effect of T & 9 & 100 & 0.1 & 45 & 25–50 \\ \hline \multirow{4}{*}{Cr (VI)} & Isotherm & 9 & 50–500 & 0.1 & 60 & 30 \\  & pH effect & 2–10 & 100 & 0.15 & 70 & 30 \\  & C\({}_{\mathrm{ads}}\) effect & 6 & 100 & 0–0.5 & 70 & 30 \\  & Effect of t & 6 & 100 & 0.15 & 0–120 & 30 \\  & Effect of T & 6 & 100 & 0.15 & 70 & 25–50 \\ \hline \multicolumn{6}{c}{} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Experimental conditions of the different adsorption experiments.  
adsorption isotherms were studied using the Langmuir Equation (7) [53], the Freundlich Equation (9) [54], and the Temkin Equation (10) [55] isotherm models:  
\[\mathrm{q_{e}}=\frac{\mathrm{q_{m}K_{L}C_{e}}}{1+\mathrm{K_{L}C_{e}}} \tag{7}\]  
where \(\mathrm{K_{L}}\) is the equilibrium adsorption constant or Langmuir constant (L/mg); \(\mathrm{q_{e}}\) is the quantity adsorbed at equilibrium per gram of adsorbent (mg/g); \(\mathrm{C_{e}}\) is the concentration of the adsorbate in the liquid phase at equilibrium (mg/l); \(\mathrm{q_{m}}\) is the maximum adsorption capacity for a monolayer (mg/g).  
A dimensionless constant commonly known as the separation factor (\(\mathrm{R_{L}}\)) can be represented as follows (Equation (8)) [56]:  
\[\mathrm{R_{L}}=\frac{1}{1+\mathrm{K_{L}C_{0}}} \tag{8}\]  
where \(C_{0}\) is the initial concentration of the adsorbate in (mg/L) and \(\mathrm{K_{L}}\) is the Langmuir constant in (mg/g).  
This separation factor is used to indicate the nature of the adsorption, which can be either linear (\(\mathrm{R_{L}}=1\)), irreversible (\(\mathrm{R_{L}}=0\)), unfavourable (\(\mathrm{R_{L}}>1\)), or favourable (\(0<\mathrm{R_{L}}<1\)).  
\[\mathrm{q_{e}}=\mathrm{K_{F}C_{e}^{1/n}} \tag{9}\]  
where \(\mathrm{K_{F}}\) is the Freundlich constant characterising the adsorption capacity of the solid in (mg/g) (L/mg)\({}^{1/n}\); \(1/n\) is the intensity of adsorption or surface heterogeneity, indicating the relative energy distribution and heterogeneity of the adsorption sites. When \(1/n\) is between 0 and 1 (\(0<1/n<1\)), adsorption is favourable. When \(1/n\) is greater than 1, adsorption is unfavourable. It is irreversible when \(1/n=1\). The irreversibility of the isotherm can be attributed to the fact that the concentration must extremely decrease to a low value before desorption of the adsorbate molecules from the surface [57].  
\[\mathrm{q_{e}}=\frac{\mathrm{RT}}{\mathrm{b}}\ln(\mathrm{AC_{e}}) \tag{10}\]  
where A is the equilibrium binding constant of the Temkin isotherm (L/mg); b is Temkin isotherm constant; R is the universal gas constant (8.314 \(\mathrm{J/mol/K}\)); T is the absolute temperature (K).