## 4 Results and Discussions
### Adsorption Equilibrium Studies of Methylene Blue (MB)  
In order to evaluate the equilibrium characteristics of adsorption, three models were used to analyze the adsorption data; Langmuir; Freundlich and Frumkin models. The non-linearized forms of Langmuir, Freundlich and Frumkin models can be described by Equations (1)-(3), respectively. \[q_{e}=\frac{q_{\max}K_{L}C_{e}}{1+K_{L}C_{e}} \tag{1}\]  
\[q_{e}=K_{f}C_{e}\frac{1}{\hbar} \tag{2}\]  
Figure 10: Pore size distribution curves of the raw material (**S\({}_{0}\)**) and activated clays (**S\({}_{1}\)**, **S\({}_{2}\)**, **S\({}_{3}\)**) with \(\pm\)5% error bars. \[q_{e}=\frac{q_{\max}K_{Fr}C_{e}}{\left[e\big{(}\frac{2a_{\rm{rg}}}{\max}\big{)}+K_ {Fr}C_{e}\right]} \tag{3}\]  
where \(K_{L}\)(L/mg) is the Langmuir adsorption constant related to the energy of adsorption, \(q_{\max}\) and \(q_{\rm{e}}\) (mg/g) are the maximum and equilibrium adsorption capacity, respectively. \(K_{f}\) and \(n\) are the Freundlich constants corresponding to adsorption capacity and intensity, respectively. For Frumkin model, the constant \(K_{Fr}\) is related to equilibrium adsorption and \"\(a\)\" is the interaction parameter. A positive value of \"\(a\)\" corresponds to repulsive interactions among adsorbed molecules, whereas a negative value means attraction between the adsorbed molecules. When \(a=0\) the adsorbed molecules do not interact each other and the Frumkin isotherm reduces to Langmuir isotherm. The adsorbed quantity of MB dye at the equilibrium (\(q_{e}\)), versus residual concentration (\(C_{e}\)) is presented in Figure 11.  
The adsorption isotherms exhibited the classical type-I behaviors of adsorbents. The Langmuir model provided the best fit to the experimental data in all range of concentrations (Figure 11). The Feundlich and Frumkin isotherms failed to correlate the results and showed large deviation (not presented). The linearization of Feundlich model (In \(q_{e}\) vs. In \(C_{e}\)) shows a correlation factor (\(R^{2}\)) of 0.87, 0.89, 0.84 and 0.97 for sample isotherms S\({}_{0}\), S\({}_{1}\), S\({}_{2}\) and S\({}_{3}\), respectively. However the plot of equation \(\ln\Big{(}\frac{\theta}{(1-\theta)}\frac{1}{C_{e}}\Big{)}\) versus \(\theta\) for Frumkin isotherm shows linear relationship with correlation factor, \(R^{2}\) which did not reach 0.8 in all sample isotherms; S\({}_{0}\), S\({}_{1}\), S\({}_{2}\) and S\({}_{3}\) (\(\theta\) = \(\frac{q_{e}}{q_{\max}}\) represent the degree of surface coverage by the adsorbent). Figure 11: Adsorption isotherms of MB on raw material (**S\({}_{0}\)**) and activated clay samples (**S\({}_{1}\)**, **S\({}_{2}\)**, **S\({}_{3}\)**) with \(\pm\)5% error bars in uptake: The lines represent the fit of [PERSON] model, (T = 20 \({}^{\circ}\)C, pH = 7, m\({}_{\rm{ads}}\) = 200 mg). The adsorption data were compared with the adsorption isotherm model of [PERSON] and the parameters were estimated with regression by minimizing the sum of squared residuals. The fitting function \(\delta q\) to be minimized is established in the following equation:  
\[\delta q=\sum_{i=1}^{N}\biggl{(}\frac{q_{\text{exp}}-q_{cal}}{q_{\text{exp}}} \biggr{)}^{2} \tag{4}\]  
\(q_{\text{exp}}\) and \(q_{cal}\) are the experimental and the calculated adsorbed quantities at concentration \(C_{e}\), respectively. \(\delta q\) is an objective function and has a relative value that depends on the fitting model and the experimental data. The deviations between the fitting model and the experimental data were determined by Equation (4), \(N\) is the number of experimental points. \[\Delta q(\%)=\frac{100}{N}\sum_{i=1}^{N}\biggl{(}\frac{q_{\text{exp}}-q_{cal} }{q_{\text{exp}}}\biggr{)} \tag{5}\]  
For Langmuir model, the overall mean deviation calculated by Equation (4) was <2%.