## 3 Results and Discussion
### Removal of Lead from Aqueous Solution
#### 3.4.3 Adsorption Isotherm
The model is given by the following linear equation (Equation (4)):  
\[\mathrm{q_{e}}=\mathrm{\ B\ln A}+\mathrm{\ B\ln C_{e}} \tag{4}\]  
where A is the Temkin isotherm equilibrium binding constant (L/g) and B is a constant related to the heat of adsorption and is given by Equation (5) as follows:  
\[\mathrm{B}=\frac{\mathrm{RT}}{\mathrm{L}} \tag{5}\]  
L is the Temkin isotherm constant (J/mol), T is the absolute temperature and R is the general gas constant (8.31 J/mol K). Furthermore, the Dubinin-Radushkevich (D-R) isotherm model was used to correlate the adsorption data in the present work, where the model generally expresses the adsorption mechanism with a Gaussian energy distribution onto a heterogeneous surface [52; 53; 54; 55; 56; 57]. The model has often successfully fitted the intermediate range of concentration data well and high solute activities, where its linearized equation form (Equation (6)) can be written as follows:  
\[\mathrm{ln_{e}}=\mathrm{\ ln\,q_{s}}-\mathrm{\ \beta\,\varepsilon^{2}} \tag{6}\]  
where \(\mathrm{q_{e}}\) is the amount of adsorbate in the adsorbent at equilibrium (mg/g); \(\mathrm{q_{s}}\) is the theoretical isotherm saturation capacity (mg/g); \(\mathrm{\beta\,(mol^{2}/KJ^{2})}\) is the activity coefficient, which is useful in obtaining the mean sorption energy E (KJ/mol), which can be expressed by Equation (7). \[\mathrm{E}=\frac{1}{\sqrt{2\beta}} \tag{7}\]  
where \(\varepsilon\) is the Dubinin-Radushkevich isotherm polanyi potential constant and it is given by Equation (8) as a function of the equilibrium concentration of adsorbate in solution (\(\mathrm{C_{e}}\)) (mg/L) [41; 42; 43; 44; 45; 46]. \[\varepsilon=\mathrm{RTln}\Big{[}1+\frac{1}{\mathrm{C_{e}}}\Big{]} \tag{8}\]  
Table 4 reports the various parameters calculated from the fitting of Pb(II) adsorption data to the four models. As seen from Table 4 and Figure 11a-e, the Langmuir isotherm best describes the uptake of Pb (II) ions onto the C1, C2, C3 and C4 clays (\(r^{2}\) ca. 0.99) (Figure 11a), which suggests that Pb (II) adsorbed onto clay-uniform sites. The maximum monolayer (\(\mathrm{q_{max}}\)) adsorption capacity for the uptake of Pb (II) ions was found to be 74.07 mg/g for C1 clay, which can be attributed to the presence of high amounts of palygorskite in this sample and its high surface area [40]. On the other hand, the multilayer adsorption model ([PERSON], Figure 11b) and adsorbent-adsorbate interaction model (Temkin, Figure 11c) may also play an important role in the lead uptake onto C1, C3 and C4 clays (\(r^{2}\) for Freundlich is ca. 0.95 and for Temkin is ca. 0.98). However, in the case of C5, the Temkin adsorption isotherm best described the uptake of Pb (II) ions (\(r^{2}\) value of 0.990) (Figure 11c). The good regression coefficient of the Temkin model for C5 suggests a strong affinity for the sorption of Pb (II) onto calcite, which is a dominant mineral in this clay [42]. Again, our results are consistent with the findings of [PERSON] et al. [42], who reported that the adsorption mechanism of Pb (II) onto limestone solid with 41.0% calcite can be better described by the Temkin model. Moreover, we cannot neglect the fact that the D-R model also fitted the lead adsorption data of C5 clay (\(r^{2}\) of 0.90) better than other clays (Figure 11d and Table 4). In addition, the E value calculated from D-R model of C5 clay was higher than 8 KJ/mol, indicating that the adsorption of Pb (II) ions onto C5 clay is through chemisorption in nature [58].