## 3 Results and Discussion
### Equilibrium Isotherms  
adsorption isotherms provide critical information that describes the interaction between the adsorbate and the adsorbent [48]. In the present research, the characteristics of the adsorption were investigated by fitting the experimental data to the Langmuir, Freundlich and Temkin isotherms. The Langmuir isotherm, as one of the commonly used isotherms, is valid for monolayer adsorption, which takes place onto a homogenous surface containing a finite number of adsorption sites [51]. The linear form of Langmuir isotherm is as follows:  
\[\frac{C_{e}}{q_{e}}=\frac{1}{q_{m}K_{L}}+\frac{C_{e}}{q_{m}} \tag{12}\]  
\[R_{L}=\frac{1}{1+K_{L}c_{0}} \tag{13}\]  
where \(c_{0}\), \(c_{e}\) and \(q_{e}\) represent the initial concentration of MG (mg/L), the equilibrium concentration of the MG solution (mg/L) and the amount of adsorbate adsorbed per unit mass at equilibrium (mg/g), respectively, \(q_{m}\) is the maximum adsorption capacity of the adsorbent (mg/g), \(K_{L}\) is the Langmuir constant (L/mg), and \(R_{L}\) is a dimensionless constant. The value of \(R_{L}\) suggests the adsorption to be irreversible (\(R_{L}=0\)), favorable (\(0<R_{L}<1\)), linear (\(R_{L}=1\)), or unfavorable (\(R_{L}>1\)) [52]. The Freundlich isotherm, as an empirical equation, assumes that the adsorption takes place on heterogeneous surfaces and the sorption is not restricted to the formation of the monolayer [53]. The linear form of Freundlich isotherm is expressed as follows:  
\[\log q_{e}=\log k_{f}+\frac{1}{n}\log c_{e} \tag{14}\]  
where \(k_{f}\) is the Freundlich constant and \(1/n\) represents the heterogeneity factor. A close to zero value of \(1/n\) manifests the rise of heterogeneity on the surface [54], and the value of \(1/n\) between \(0\) and \(1\) shows a favorable adsorption process, while \(1/n\) above \(1\) demonstrates bi-mechanism and cooperative adsorption [55]. The Temkin isotherm assumes that the adsorption energy would decrease linearly with the surface coverage due to the adsorbent-adsorbate interactions [56]. The linear form of Temkin isotherm is given as follows:  
\[q_{e}=B\ln A+B\ln c_{e} \tag{15}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**RSM**} & \multicolumn{2}{c}{**ANN-PSO**} & \multicolumn{2}{c}{**ANN-GA**} \\ \cline{2-7}
**Process Variables** & **Predicted** & **Experimental** & **Predicted** & **Experimental** & **Predicted** & **Experimental** \\  & **Parameters** & **Parameters** & **Parameters** & **Parameters** & **Parameters** & **Parameters** \\ \hline Initial pH & 4.16 & 4.20 & 4.00 & 4.00 & 5.70 & 5.70 \\ Temperature (\({}^{\circ}\)C) & 29.16 & 29.20 & 30.00 & 30.00 & 27.19 & 27.20 \\ Contact time (min) & 114.54 & 115.00 & 120.00 & 120.00 & 110.62 & 111.00 \\ Initial concentration (mg/L) & 619.61 & 620.00 & 600.00 & 600.00 & 607.03 & 600.00 \\ Percentage of removal (\%) & 89.50 & 77.37 & 92.55 & 84.87 & 94.12 & 90.72 \\ Absolute errors (\%) & & 12.13 & & 7.68 & & 3.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the confirmatory results with predicted results. RSM: Response Surface Methodology; ANN-PSO: Artificial neural network coupled with particle swarm optimization; ANN-GA: Artificial neural network coupled with genetic algorithm. where \(B\) is the Temkin isotherm energy constant (L/mol) and \(A\) is the Temkin isotherm equilibrium binding constant (L/g). The values of parameters obtained from the three isotherm models are summarized in Table 7. For the adsorption isotherms, the fitting results manifest that the Langmuir isotherm with \(R^{2}\) of 0.9988 is suitable for describing the adsorption of MG onto nZVZ. Furthermore, its calculated adsorption capacity is in a satisfactory agreement with the experimental adsorption capacity, which further proves the suitability of the Langmuir isotherm model (Figure 17). According to this model, the maximum adsorption capacity was calculated to be 1000.00 mg/g, which was significantly higher than those of the other materials (Table 8).