## 3 Results and Discussion
### Removal of Lead from Aqueous Solution
#### 3.4.4 Adsorption Kinetics  
Three different kinetic models have been used to investigate the adsorption kinetics of Pb(II) on different clays--namely, Lagergren pseudo-first-order [59], pseudo-second-order [60] and intraparticle diffusion [61]. The linear forms of the three different kinetic model can be written as Equations (9)-(11), respectively:  
\[\ln\bigl{(}\mathrm{q_{e}}-\mathrm{q_{t}}\bigr{)}=\ \ln\mathrm{q_{e}}-\ \mathrm{k_{t}} \tag{9}\]  
\[\frac{\mathrm{t}}{\mathrm{q_{t}}}=\frac{1}{\mathrm{k_{2}}\,\mathrm{q_{e}^{2}} }+\frac{\mathrm{t}}{\mathrm{q_{e}}} \tag{10}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**adsorption-Model** & **Constants** & **C1** & **C2** & **C3** & **C4** & **C5** \\ \hline Langmuir & q\({}_{\mathrm{max}}\) (mg/g) & 74.07 & 40.0 & 68.03 & 53.48 & 99.01 \({}^{\mathrm{a}}\) \\  & b (L mg\({}^{-1}\)) & 0.008 & 0.027 & 0.007 & 0.0060 & 0.001 \({}^{\mathrm{a}}\) \\  & r\({}^{2}\) & 0.997 & 0.999 & 0.996 & 0.993 & 0.607 \\ Freundlich & K\({}_{\mathrm{f}}\) & 1.69 & 2.66 & 1.35 & 0.80 & 7.35 \\  & n & 1.68 & 2.19 & 1.64 & 1.56 & 0.93 \\  & r\({}^{2}\) & 0.946 & 0.892 & 0.9488 & 0.953 & 0.916 \\ Temkin & B & 14.61 & 6.45 & 13.48 & 10.59 & 13.74 \({}^{\mathrm{b}}\) \\  & A (L/g) & 0.116 & 1.088 & 0.102 & 0.079 & 0.038 \({}^{\mathrm{b}}\) \\  & L (J/mol) & 169.5 & 384.0 & 183.7 & 233.9 & 186.02 \({}^{\mathrm{b}}\) \\  & r\({}^{2}\) & 0.985 & 0.903 & 0.983 & 0.981 & 0.990 \\ D-R & \(\beta\) (mol\({}^{2}\)/KJ\({}^{2}\)) & 0.0004 & 0.00001 & 0.00004 & 0.00007 & 0.0005 \\  & E (KJ/mol) & 111.80 & 223.60 & 111.80 & 84.51 & 31.62 \\  & q\({}_{\mathrm{s}}\) (mg/g) & 42.2 & 30.22 & 38.12 & 28.56 & 32.56 \\  & r\({}^{2}\) & 0.758 & 0.769 & 0.756 & 0.753 & 0.908 \\ \hline \hline \end{tabular} \({}^{\mathrm{a}}\) unreliable value and \({}^{\mathrm{b}}\) reliable value for C5 clay.  
\end{table}
Table 4: List of estimated parameters from Langmuir, Freundlich, Temkin and Dubinin–Radushkevich (D-R) adsorption models; \(\mathrm{r^{2}}\) is the regression coefficient.  
Figure 11: Statistical fitting of data from adsorption isotherms for lead onto different clays using different models: (**a**) Langmuir, (**b**) Freundlich, (**c**) Temkin and (**d**) Dubinin–Radushkevich (D-R).  
\[\mathrm{q_{t}}=\ \mathrm{k_{i}t^{0.5}}+\ C \tag{11}\]  
where \(\mathrm{q_{t}}\) (mg/g) is the adsorption capacity at time t and \(\mathrm{k_{1}}\) (min\({}^{-1}\)) and \(\mathrm{k_{2}}\) (g/mg min), are the rate constants of the pseudo-first-order and pseudo-second-order adsorptions, respectively. \(\mathrm{k_{i}}\) (mg/g min\({}^{0.5}\)) is the intraparticle diffusion rate constant and C (mg/g) is a constant.  
As shown in Table 5 and Figure 12, the pseudo-second-order rate equation is the best fit model for adsorption of Pb (II) onto different clays in the study, where a good regression coefficient (r\({}^{2}\)) was obtained for the pseudo-second-order rate kinetic model compared with the pseudo-first-order and intraparticle diffusion kinetics models. It suggests that chemical adsorption is the rate controlling step of adsorption of Pb (II) onto different clays in the study, even though these clays contain different type of minerals with different weight percentages [62; 63]. In addition, the values of intercept C of all clays (Table 5), as estimated by Equation (11), were not zero, which indicates that intraparticle diffusion is not the controlling rate determining step in adsorption of Pb (II) onto different clays [64].