## 3 Results and Discussion
### Batch Adsorption Evaluation
\[\text{R}^{\circ}=41.47+1.19\text{x}_{2}+27.53\text{x}_{3} \tag{12}\]  
\[\text{Q}\text{e}\bigg{(}\frac{\text{mg}}{\text{g}}\bigg{)}=3.68+0.0051\text{x }_{2}-2.69\text{x}_{3}+0.95\text{x}_{3}^{2} \tag{13}\]  
where R\({}^{\circ}\) and Q\({}_{e}\) (mg/g) are, respectively, the removal and adsorption capacity, x\({}_{2}\) is time (min), and x\({}_{3}\) is solid mass (g). The experimental results obtained were R\({}^{\circ}\) = 98.1, while the model predictions indicated R\({}^{\circ}\) = 167.93 and Q\({}_{e}\) = 2.406 mg/g. Although the determination coefficients (R\({}^{2}\)) for both equations show that the model performs well overall, the comparison between the experimental and predicted values suggests that the model does not adequately capture all the variables and interactions of the process. The overestimation of R\({}^{\circ}\) and the underestimation of Q\({}_{e}\) indicate that there are experimental factors or interactions between the variables that are not properly represented by the model. The significant difference between the experimental and predicted values for R\({}^{\circ}\) may be attributed to a limitation of the model in relation to the actual experimental conditions, such as nonlinear interactions or effects of unconsidered variables. For Q\({}_{e}\), the underestimation may reflect the complexity of the adsorption process, which is not fully addressed by the second-order polynomial model. Therefore, a revision of the model, including the consideration of additional variables or more complex interactions, is necessary to improve its accuracy. Furthermore, adjustments to the experimental design may be important to explore a wider range of factors that influence removal efficiency and adsorption capacity. However, in addition to the statistical aspects, it is also essential to consider all the experimental factors that may influence the results. The experimental data obtained were as expected, showing the desired behavior and aligning with the theoretical foundations of adsorption. The removal efficiency and adsorption capacity observed confirm that, despite the model's limitations, the underlying processes of adsorption occur as predicted, validating the experimental approach adopted. The integration of experimental and theoretical aspects is crucial for the development of more accurate models and for the optimization of adsorption processes. Based on the results presented, it was possible to conclude that the adsorption of paracetamol (PAR) on the GAC column is predominantly governed by specific interaction mechanisms, including electrostatic forces, hydrophobic interactions, hydrogen bonding (dipole-dipole), n-\(\pi\) interactions, van der Waals forces, \(\pi\)-\(\pi\) interactions, and pore-filling processes [51; 53; 54; 55; 56]. These mechanisms align with the Langmuir isotherm and pseudo-second-order kinetics, indicating strong and specific interactions between PAR molecules and the adsorbent surface. In contrast, the adsorption of MTZ is primarily driven by physical processes, exhibiting a lower adsorption affinity on activated carbon [57; 58]. The behavior aligns with the Freundlich isotherm and is characterized by pseudo-stationary kinetics, where the rate-limiting step is attributed to mass transfer resistance in the external film. These findings highlight the distinct adsorption behaviors of PAR and MTZ, demonstrating the varying roles of specific interactions and mass transfer dynamics in the adsorption process. The study underscores the effectiveness of GAC for removing pharmaceutical and organic compounds, providing valuable insights into the adsorption mechanisms that govern fixed-bed column systems.