## 3 Results and Discussion
### Theoretical Models
#### 3.3.4 Intraparticle Diffusion Model  
The other kinetic equation is the intraparticle diffusion model, given by:  
\[q_{t}=\ k_{l}\sqrt{t}+C_{i} \tag{7}\]  
where \(q_{t}\) describes the amount per unit mass of nanoparticles adsorbed by the zeolite at time \(t\), \(k_{i}\) (mg g\({}^{-1}\cdot\)min\({}^{-0.5}\)) is the rate constant of intraparticle diffusion, and \(C_{i}\) is the intercept [31,32]. Figure 5a shows the Lagergren kinetics adsorption model of AgNPs into clinoptiolite. In this case, a significant correlation between the experimental results and the \(R^{2}\) factor is observed (Table 2). The highest correlation values for the first-order Lagergren model are presented in the concentrations of 3 and 4 mg\(\cdot\)L\({}^{-1}\). This result supports the experimental values shown in Figure 4 where the adsorption efficiency values are similar in both cases and their behavior is linear. Of the employed theoretical models, the pseudo-second-order model (Figure 5b) presents the highest correlation values which are of the order of 0.99. Therefore, the second-order model describes the adsorption process with greater accuracy. This represents chemisorption or chemical adsorption due to the formation of chemical bonds between adsorbent and adsorbate in a monolayer on the surface [26,33]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Kinetictic** **Model** \\ **Model** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Lagergren** \\ **First-Order** \\ **(K1) min\({}^{-1}\)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Pseudo-Second-Order \(\mathbf{K_{2}}\)** \\ **(g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\))** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Elovich \(\mathbf{a}\) (mg\(\cdot\)g\({}^{-1}\cdot\)min)** \\ **(g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\))** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Intraparticle Diffusion** \\ **(g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\))** \\ \end{tabular}  } \\ \cline{1-1}  & & & & \(\mathbf{K_{2}}\) & \(1.88\times 10^{2}\) & \(\mathbf{a}\) & 73,690.88 & \(\mathbf{C_{i}}\) & \(4.93\times 10^{-4}\) \\ \cline{1-1}  & & & \(\mathbf{q_{e}}\) & \(8.38\times 10^{-4}\) & \(\mathbf{\beta}\) & 0.000115 & \(\mathbf{K_{i}}\) & \(2.76\times 10^{-5}\) \\ \cline{1-1} \cline{2-7}  & & \(\mathbf{R^{2}}\) & 0.762 & \(\mathbf{R^{2}}\) & 0.99108 & \(\mathbf{R^{2}}\) & 0.89589 & \(\mathbf{R^{2}}\) & 0.91193 \\ \cline{1-1} \cline{2-7}  & & \(\mathbf{K_{1}}\) & 0.032 & \(\mathbf{K_{2}}\) & 73.4 & \(\mathbf{a}\) & \(1.98\times 10^{4}\) & \(\mathbf{C_{i}}\) & 0.00102 \\ \cline{1-1} \cline{2-7}  & & \(\mathbf{K^{2}}\) & 0.857 & \(\mathbf{R^{2}}\) & 0.98635 & \(\mathbf{R^{2}}\) & 0.88164 & \(\mathbf{R^{2}}\) & 0.92399 \\ \hline \multirow{2}{*}{ \begin{tabular}{c} 3 mg\(\cdot\)L\({}^{-1}\) \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **\(\mathbf{K_{1}}\)** \\ **(K1) min\({}^{-1}\)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} 0.031 \\ **(g\(\cdot\)m g\({}^{-1}\cdot\)min\({}^{-0.5}\))** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **\(\mathbf{K_{2}}\)** \\ **(g\(\cdot\)m g\({}^{-1}\cdot\)min\({}^{-0.5}\))** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} 37.8 \\ **(g\(\cdot\)m g\({}^{-1}\cdot\)min\({}^{-0.5}\))** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **\(\mathbf{a}\)** \\ **(g\(\cdot\)m g\({}^{-1}\cdot\)min\({}^{-0.5}\))** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} 2.33 \(\times\)\(10^{3}\) \\ **(g\(\cdot\)m g\({}^{-1}\cdot\)min\({}^{-0.5}\))** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **\(\mathbf{C_{i}}\)** \\ **(g\(\cdot\)m g\({}^{-1}\cdot\)min\({}^{-0.5}\))** \\ \end{tabular}