## 3 Results
### TC Adsorption Kinetics  
The evaluation of adsorption kinetics reveals (Figure 10 and Table 3) that BA1 exhibits the highest experimental adsorption capacity (\(q_{e}\) = 41.35 mg\(\cdot\)g\({}^{-1}\)) at 30 min, with a drug concentration (Ce) of 500 mg\(\cdot\)L\({}^{-1}\). Adsorption studies with BA1 (\(q_{e}\) = 40.98 mg\(\cdot\)g\({}^{-1}\)) show a slight increase in the amount of tetracycline adsorbed compared to BN (\(q_{e}\) = 40.60 mg\(\cdot\)g\({}^{-1}\)), highlighting the effectiveness of acid treatment in enhancing tetracycline adsorption. Figure S1 in the Supplementary Material depicts the maximum adsorptive capacity of BN and BA1 adsorbents as a function of time for the adsorption TC, and the experimental data for the adsorption kinetics are summarized in Tables S1 and S2. \begin{table}
\begin{tabular}{c c c} \hline
**Kinetic Parameters** & **BN** & **BA1** \\ \hline \(q_{\text{e,exp}}\) (mg\(\cdot\)g\({}^{-1}\)) & 3.84 & 4.08 \\ Pseudo-first order & & \\ \(k_{1}\) (min\({}^{-1}\)) & \(-0.0008\) & \(-0.0026\) \\ \(q_{\text{e,cal}}\) (mg\(\cdot\)g\({}^{-1}\)) & 0.271 & 0.483 \\ \(\text{R}^{2}\) & 0.9799 & 0.848 \\ Pseudo-second order & & \\ \(k_{2}\) (g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\)) & 0.0071 & 0.1232 \\ \(q_{\text{e,cal}}\) (mg\(\cdot\)g\({}^{-1}\)) & 3.87 & 4.04 \\ \(\text{R}^{2}\) & 0.9998 & 0.9998 \\ Elovich & & \\ \(\alpha\) (mg\(\cdot\)g\({}^{-1}\cdot\)min\({}^{-1}\)) & \(7.30\times 10^{-5}\) & \(1.30\times 10^{-2}\) \\ \(\beta\) (g\(\cdot\)mg\({}^{-1}\)) & 0.309 & 0.906 \\ \(\text{R}^{2}\) & 0.9681 & 0.9679 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters for the fitting to pseudo-first-order, pseudo-second-order, and Elovich models. Figure 9: Thermogravimetric analysis (TG/DTG) of BN, BNF, BA1, and BA1F samples. Among the three models studied, the pseudo-second-order model exhibited the best fit to the data, with a coefficient of determination close to 1 (R\({}^{2}\) = 0.999). This model showed a minimal difference between the experimental (41.35 mg\(\cdot\)g\({}^{-1}\)) and calculated (40.98 mg\(\cdot\)g\({}^{-1}\)) qe values, suggesting that chemisorption is the rate-determining step in the tetracycline adsorption process, involving electron exchange or sharing between the adsorbent and adsorbate [26]. The kinetic parameters derived from the pseudo-second-order model are critical for understanding the adsorption mechanism. The parameter k\({}_{2}\) represents the adsorption rate or the speed at which the adsorbate interacts with the adsorbent. As shown in Table 3, BA1 exhibits a higher adsorption rate than BN, indicating faster and more efficient adsorption compared to the natural clay [5]. The Elovich equation was applied to describe the adsorption kinetics; however, the model showed a poor fit for BA1 (R\({}^{2}\) < 0.9412), indicating that it did not adequately capture the variability of the experimental data. As shown in Figure 10, the adsorption capacity (q\({}_{\rm t}\)) increases with \(ln\)\(t\), but the data for BA1 exhibit a higher initial adsorption capacity compared to BN. The Elovich model is often used to describe chemisorption on heterogeneous surfaces, where the parameter \(\alpha\) represents the initial adsorption rate, and \(\beta\) reflects the extent of surface activation over time. The relatively low correlation observed for this model suggests that the adsorbent surfaces do not exhibit extreme heterogeneity, reinforcing the predominance of controlled chemisorption mechanisms. Additionally, the lower slope observed for BN suggests a slower adsorption process compared to BA1, which exhibited a steeper increase in q\({}_{\rm t}\), indicating a more efficient adsorption performance in the evaluated time range [27]. The pseudo-first-order model showed MSE values of 0.00413 for BN (R\({}^{2}\) = 0.7701) and 0.01391 for BA1 (R\({}^{2}\) = 0.56192). The pseudo-second-order model, which best describes the adsorption process, presented MSE values of 0.001538 for both BN and BA1 (R\({}^{2}\) = 0.9998).