## 3 Results
### TC Adsorption Isotherms  
To describe the relationship between the adsorbent and adsorbate at equilibrium for varying initial concentrations, Langmuir, Freundlich, Sips, and Temkin isotherm models were applied, as illustrated in Figure 11. The parameters derived from these models provide insight into the adsorption mechanism and the specific surface properties of the materials. The Langmuir model assumes monolayer adsorption on a surface with a finite number of identical sites, where the adsorption energy remains constant. The parameter q\({}_{\text{m}}\) represents the maximum adsorption capacity, and for BA1, a q\({}_{\text{m}}\) of 110 mg\(\cdot\)g\({}^{-1}\) was reached in 30 min at a C\({}_{\text{e}}\) of 43.46 mg\(\cdot\)L\({}^{-1}\). This suggests that BA1 possesses a high density of active sites, enhancing its adsorption potential. The Langmuir constant (K\({}_{\text{L}}\)) reflects the affinity between the adsorbate and adsorbent, with a higher K\({}_{\text{L}}\) value indicating stronger interactions on the material's surface. For BA1, a K\({}_{\text{L}}\) value of 0.23 was obtained, signifying stronger chemical interactions with tetracycline (TC) compared to BN. The enhanced interaction is due to the acid treatment that leaches surface impurities, exposing more active groups for interaction, thus favoring chemisorption as the dominant mechanism [30; 31]. For BN, the equilibrium data were analyzed at both low and high concentrations (Figure 11A,B). At low concentrations (Figure 11A), the Langmuir and Freundlich models provided good fits to the experimental data, suggesting that both monolayer adsorption and multilayer adsorption phenomena occur. However, at higher concentrations (Figure 11B), the Langmuir model exhibited a superior fit, indicating that monolayer adsorption becomes the predominant mechanism as the available active sites become more occupied. The discrepancy observed for the Temkin model at lower concentrations suggests that the assumption of a uniform decrease in adsorption energy may not hold under these conditions. Figure 11: Equilibrium isotherms for BN at low concentrations (**A**) and high concentrations (**B**) and BA1 (**C**). The Freundlich model, which describes adsorption on heterogeneous surfaces and the formation of multilayers, was also considered. The parameter K\({}_{\rm F}\) in this context represents the adsorption capacity, while the exponent \(n\) provides insight into adsorption intensity. Experimental data show that BA1 exhibited a Freundlich exponent (\(1<n<10\)), confirming a favorable adsorption process. Additionally, the relatively high K\({}_{\rm F}\) value obtained for BA1 further supports a strong affinity for TC, reinforcing the presence of heterogeneous adsorption sites and the suitability of the adsorbent for tetracycline uptake [32; 33]. In Figure 11A,B, the Freundlich model suggests that BN also exhibits surface heterogeneity, though its fit varies depending on concentration. The better fit at lower concentrations supports the presence of multiple types of adsorption sites, while the deviation at higher concentrations suggests a shift toward monolayer saturation. The Sips (Simplified Modified Langmuir-Freundlich) model, which combines elements of both the Langmuir and Freundlich models, accounts for surface heterogeneity while still considering the possibility of monolayer adsorption at higher adsorbate concentrations. The excellent fit of the Sips model to the data (R\({}^{2}\) = 0.9767 for BA1 and for BN, the model did not converge) indicates that both materials exhibit moderate surface heterogeneity and substantial interaction with the adsorbate. The equilibrium constant K\({}_{\rm S}\) from the Sips model quantifies the affinity between the adsorbate and adsorbent. A higher K\({}_{\rm S}\) value for BA1 suggests a more optimized balance between surface heterogeneity and adsorption capacity, implying that this material possesses better overall adsorption characteristics than BN.