## 3 Results and Discussion
### Adsorption Kinetics  
Figure 5a elucidates the fitting of the pseudo-second-order and three-stage kinetic models with the experimental kinetics data for the adsorption of Co(II) onto the b-HdA. The correlation coefficients (\(R^{2}\)) in Table 5 show a better fit to the pseudo-second-order than the pseudo-first-order model, revealing that the pseudo-second-order model was more suitable for describing the kinetic mechanisms of Co(II) adsorption onto the b-HdA. Hence, the Co(II) adsorbed onto the adsorbent might have been controlled by a chemisorption step. Furthermore, the adsorption behavior of Co(II) on the exterior and interior adsorptive sites of the HdA pellets was revealed through the three-stage kinetic model. The three-stage model considers three different adsorption stages, including the first (sharp) portion reflecting external mass transfer, the second (gradual) portion representing the internal surface adsorption (intraparticle diffusion), and the third (constant) portion denoting the equilibrium stage [9; 32]. It can be seen from Figure 5a that the \(C_{t}/C_{o}\) values achieved from the calculations of the three-stage model were very close to the experimental data. In addition, a significantly lower instantaneous adsorption portion (\(\xi_{1}\)) compared to the interior adsorption portion (\(\xi_{2}\)) indicates that a major part of the adsorption occurred on the internal adsorptive sites of the b-HdA, as presented in Table 5 [32]. Consequently, the mechanism controlling the process might have been intraparticle diffusion [32; 43].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Temperature** & \(K_{0}\) & \(\Delta G^{0}\) & \(\Delta S^{0}\) & \(\Delta H^{0}\) \\
**(K)** & **(kJ/mol)** & **(kJ/mol\(\cdot\)K)** & **(kJ/mol)** \\ \hline
293 & 1.56 & \(-1.08\) & & \\
303 & 2.24 & \(-2.03\) & 0.20 & 58.72 \\
313 & 7.33 & \(-5.18\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Thermodynamic parameters of Co(II) during the adsorption onto b-HdA.  
Figure 4: The fitting results of thermodynamic analysis: (**a**) plots of ln(\(C_{s}\)/\(C_{e}\)) versus \(C_{s}\) at different experimental temperatures and (**b**) ln(\(K_{0}\)) versus \(1/T\).  
Furthermore, the value of \(\xi_{1}\) was very small, suggesting that the external surface adsorption of Co(II) ions occurred rapidly and that the adsorption process quickly progressed to the intraparticle diffusion stage [9].  
Figure 5b shows the graph of \(q_{t}\) versus \(t^{0.5}\), which fits the experimental data with the intraparticle diffusion model. The plot presents a linearized section (i.e., red dotted line) during the second (gradual) portion of the adsorption process, which indicates that intraparticle diffusion took place in the rate-limiting step. This result is consistent with the data obtained from the three-stage kinetic model (discussed above). However, the plot did not pass through the origin (\(I\
eq 0\)); thus, intraparticle diffusion was not the sole rate-controlling step of the adsorption process [30]. It is noteworthy that two linearity fractions with different slopes corresponding to different stages in the adsorption process were observed in Figure 5b, implying that the multi-kinetic stage might have governed the adsorption of Co(II) onto the b-HdA.