## 3 Results and Discussion
### Kinetic Characteristics  
Different kinetic models have been employed to assess the adsorption reaction mechanism and control steps of the potential rate, such as pore diffusion, mass transport, and chemical reaction process. [PERSON]'s pseudo-first-order, pseudo-second-order, Elovich, and intraparticle diffusion are the kinetic models used for the experimental results to expect the sorption process of the yttrium ions on 3-amino-5-hydroxypyrazole impregnated treated bleaching clay (AHIBC). The pseudo-first-order model is calculated by the linear equation as the following:  
\[\log(\mathrm{q_{e}}\ -\ \mathrm{q_{t}})\ =\ \log\mathrm{q_{e}}\ -\ \biggl{(}\frac{ \mathrm{k_{1}}}{2.303}\biggr{)}, \tag{5}\]  
where \(\mathrm{q_{e}}\) (mg\(\cdot\)g\({}^{-1}\)) is the equilibrium adsorption capacity of Y(III), and \(\mathrm{q_{t}}\) (mg\(\cdot\)g\({}^{-1}\)) is the amount of ions adsorbed after time t (min), while \(\mathrm{k_{1}}\) (min\({}^{-1}\)) is the constant rate of pseudo-first-order. The plot of \(\log(\mathrm{q_{e}}-\mathrm{q_{t}})\) versus t proposed the applicability of the kinetic model of yttrium ions on AHIBC. The value of \(\mathrm{k_{1}}\) was truly calculated from the slope, while \(\mathrm{q_{e}}\) was calculated from the intercept of the graph. The obtained result in Figure 2a and Table 2 established that the correlation coefficient \(\mathrm{R^{2}}\), as well as \(\mathrm{q_{e}}\)(cal) value for adsorption mechanism, did not match with a pseudo-first-order kinetic model. With the aim of this result, the pseudo-first-order model is unrelated to the adsorption of Y(III) onto AHIBC. Figure 2: Kinetic models for Y(III) adsorbed on the AHIBC adsorbent (**a**) pseudo-first-order, (**b**) pseudo-second-order, (**c**) Elovich kinetic, and (**d**) intraparticle diffusion. The pseudo-second-order kinetic model was performed and is presented in the subsequent equation [37]:  
\[\frac{t}{q_{t}}\ =\ \frac{1}{k_{2}q_{e}^{2}}+\left(\frac{1}{q_{e}}\right)t, \tag{6}\]  
where \(k_{2}\) is a constant rate of pseudo-second-order (g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\)). The model was more potential to foretell the kinetic performance of adsorption, with the chemical adsorption process being the rate-controlling step. If pseudo-second-order kinetic model is viable, the chart of t/q\({}_{t}\) against t should give a straight line. \(q_{e}\) could be premeditated from the slope, and \(k_{2}\) was calculated from the intercept of the diagram. From the data in Figure 2b and Table 2, the value of the correlation coefficients is about 0.99, almost close to unity. Additionally, both the values of calculated (q\({}_{e}\)) and experimental (q\({}_{exp}\)) adsorption capacity are close. The data recommend that the adsorption process of Y(III) on the AHIBC adsorbent was consistent with the pseudo-second-order model. The Elovich kinetic model designated the chemical nature of the adsorption process of yttrium ions and was generally presented as a non-linear equation [38; 39]. The integrated form is presented in the next equation as:  
\[q_{t}\ =\ \frac{1}{\beta}\ln(\alpha\beta)\ +\ \frac{1}{\beta}\ln(t), \tag{7}\]  
where \(\alpha\) (mg\(\cdot\)g\({}^{-1}\cdot\)min\({}^{-1}\)) refers the rate of initial adsorption, and \(\beta\) (g\(\cdot\)mg\({}^{-1}\)) is the Elovich constant relevant to coverage of the adsorbent surface, respectively. Figure 2c shows a graph of q\({}_{t}\) versus ln(t) with a linear relation. (1/\(\beta\)) is the slope, and the (1/\(\beta\))ln(\(\alpha\beta\)) represents the intercept. The Elovich kinetic parameters (\(\alpha\), \(\beta\) are correlation coefficients) were calculated and are listed in Table 2. The coefficient of \(\alpha\) is in relation to chemisorption rate, while \(\beta\) is related to surface coverage. The value of the determination coefficient (R\({}^{2}\)) is 0.92. These data state that the adsorption process of Y(III) was irrelevant to the Elovich model.