## 2 Materials and Methods
### Adsorption Isotherm Analysis  
To describe the adsorption process and the physiochemical interaction between the adsorbents and the dyes, adsorption isotherm models including Langmuir, Freundlich, Temkin, and BET isotherm models were used in this study. The isotherm tests were conducted by adding a fixed dosage of 0.04 g/L of adsorbents into 125 mL flasks containing different initial concentrations of dyes (4, 10, 15, 25, 50, and 100 mg/L). Samples were agitated on a shaker at 50 rpm at a constant temperature until equilibrium was reached, which was determined during the kinetic experiments. The final equilibrium concentrations were then measured using a UV-Vis spectrophotometer.  
The Langmuir model assumes a monolayer adsorption behaviour with uniform energy for all available sites on the surface of the adsorbents to adsorb the dyes. The constants of the Langmuir model are calculated as follows [18] (Equation (7)):  
\[\mathrm{q_{e}}=\frac{\mathrm{K_{L}q_{m}C_{e}}}{1+\mathrm{K_{L}C_{e}}} \tag{7}\]where \(q_{m}\) is the monolayer adsorption capacity predicted by the Langmuir model (mg/g), \(K_{L}\) is one of the Langmuir constants that is related to the energy of adsorption for monolayer coverage (L/mg). \(q_{m}\) and \(K_{L}\) were obtained from the slope and the intercept of plots of \(\frac{C_{e}}{q_{e}}\) vs. \(C_{e}\).  
The Freundlich isotherm is an empirical model that describes multilayer adsorption behaviour by assuming different energy of adsorption for the available empty sites on the surface of the adsorbents [18] (Equation (8)).  
\[q_{e}=K_{f}C_{e}^{1/n} \tag{8}\]  
where \(K_{f}\) is the Freundlich isotherm adsorption constant, and \(n\) indicates adsorption intensity. Both parameters were obtained from the slope and the intercept of the \(\log q_{e}\) vs. \(\log C_{e}\) plot. \(1/n\) reveals adsorption intensity [18].  
The Temkin isotherm model assumes interactions between adsorbate and adsorbent can linearly reduce the heat of adsorption in the different layers. The linearized equation for this isotherm is outlined as follows [21] (Equations (9) and (10)):  
\[q_{e}=B_{T}lnK_{T}+B_{T}lnC_{e} \tag{9}\]  
\[B_{T}=\frac{RT}{b_{T}} \tag{10}\]  
where temperature is T (K), Temkin constant or heat of adsorption \(b_{T}\) (J/mol) is obtained after the calculation of the \(B_{T}\) value according to Equation (9), using Equation (10), where R is the universal gas constant (8.314 J/(mol.K)). \(b_{T}\) has a direct correlation to the interaction between the adsorbate and the adsorbent; the lower \(b_{T}\) value indicates weak interaction between the two. The maximum binding energy constant is \(K_{T}\) (L/mg) [21].  
The BET isotherm model assumes the interaction of adsorbate and adsorbent can form multilayer coverage [21] (Equation (11)).  
\[\frac{C_{e}}{(C_{s}-C_{e})d_{e}}=\frac{1}{K_{b}q_{m}}+\frac{C_{e}(K_{b}-1)}{K _{b}q_{m}C_{s}} \tag{11}\]  
where \(K_{b}\) is BET isotherm constant, and \(C_{s}\) is the saturation concentration of dyes (mg/L). \(C_{s}\) values were found using the solubility data in water for MB (40 g/L) and RhB (10 g/L) at 20 \({}^{\circ}\)C [21]. \(C_{s}\) values were calculated based on the working volume (125 mL) in this study 5000 mg/L and 1250 mg/L for MB and RhB, respectively. The BET isotherm constants \(K_{b}\) and \(q_{m}\) were found from the intercept and slope of a plot of \(\frac{C_{e}}{(C_{s}-C_{e})d_{e}}\) versus \(C_{e}\)[21].