## 2 Materials and Methods
### Adsorption Isotherm Models  
The adsorption capacity of MB on ternary materials was calculated using the following expression:  
\[\text{q}_{\text{e}}=\frac{(\text{C}_{0}-\text{C}_{\text{t}})\times\text{V}}{ \text{m}} \tag{1}\]  
where \(\text{q}_{\text{e}}\) (mg/g) is the amount adsorbed at time t (minute), and \(\text{C}_{0}\) and \(\text{C}_{\text{t}}\) (mg/L) are the initial and at time t dye concentrations, respectively. V is the volume of solution (L), and m is the number of adsorbent materials (g).  
The equilibrium relationship between dyes adsorbed and many equations described un-adsorbed amounts. [PERSON] and [PERSON]'s models were tested to provide the best description of dye adsorption [53; 54; 55; 56; 57].  
The Langmuir model predicts whether the adsorption is favourable or unfavourable; the model describes monolayer adsorption to a surface very well. The Langmuir expression and the linearised form are presented in Table 1. A plot of \(\text{C}_{\text{e}}\)/\(\text{q}_{\text{e}}\) versus \(\text{C}_{\text{e}}\) should indicate a straight line of slope 1/\(\text{q}_{\text{max}}\) and intercept of 1/\(\text{K}_{\text{L}}\times\text{q}_{\text{max}}\). A dimensionless constant separation factor (\(\text{S}_{\text{F}}\)) is the characteristic of Langmuir isotherm, expressed by Equation (2). The value of \(\text{S}_{\text{F}}\) represents the nature of the adsorption process, which is unfavourable if \(\text{S}_{\text{F}}>1\), linear if \(\text{S}_{\text{F}}=1\), favourable if 0\(<\text{S}_{\text{F}}<1\), and irreversible if \(\text{S}_{\text{F}}=0\)[58; 59].  
\[\text{S}_{\text{F}}=\frac{1}{1+\text{K}_{\text{L}}\times\text{C}_{0}} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Isotherm Models** & **Math Expression** & **The Linearised Form** & **Plots** & **Parameters** \\ \hline \multirow{6}{*}{Langmuir} & & & & \(\text{q}_{\text{max}}\) = maximum amount of \\  & & & dye per gram of catalyst \\  & & & (mg.g-1) \\  & & & \(\text{K}_{\text{L}}\) = the Langmuir \\  & & & adsorption constant \\  & & & (L-mg\({}^{-1}\)) \\ \hline \multirow{2}{*}{Freundlich} & \(\text{q}_{\text{e}}=\text{K}_{\text{F}}\times\text{C}_{\text{e}}{}^{1/n}\) & \(\text{ln}\text{q}_{\text{e}}=\text{ln}\text{K}_{\text{F}}+\frac{1}{n}\times \text{ln}\text{C}_{\text{e}}\) & \(\text{ln}\text{q}_{\text{e}}\) vs. ln \(\text{C}_{\text{e}}\) & \(\text{K}_{\text{F}}\) = the Freundlich \\  & & & \(\text{A}_{\text{e}}\) = the adsorbed dye quantity per gram of adsorbent (mg.g-1). \\ \hline \hline \end{tabular}  
\(\text{C}_{\text{e}}\) = dye concentration at equilibrium (mg.L-1), \(\text{q}_{\text{e}}\) = the adsorbed dye quantity per gram of adsorbent (mg.g-1).  
\end{table}
Table 1: The adsorption isotherm models.  
The Freundlich model corresponds to a heterogeneous adsorbent surface. The empirical and linearised form of the Freundlich equation can be written as presented in Table 1[60], where \(\mathrm{K_{F}}\) and n are indicators of adsorption capacity and adsorption intensity, respectively. A plot of \(\mathrm{ln(q_{e})}\) versus \(\mathrm{ln(C_{e})}\) should indicate a straight line of slope 1/n and intercept \(\mathrm{ln}\)\(\mathrm{K_{F}}\).