## 3 Results and Discussion
### Fixed-Bed Column Adsorption Evaluation
The R\({}^{2}\) value indicates that the model does not account for a sufficient amount of variability in the data, meaning that the relationship between the factors and the response (q\({}_{\text{max}}\) of the compounds) may not be accurately represented by the current model. Additionally, the experimental design, while based on established methodologies, may not fully capture the complexity or the interactions of the variables, leading to less reliable conclusions. Therefore, a reassessment of the model or potential refinement in the experimental design is necessary to improve the statistical robustness and ensure that the findings are both valid and reliable. However, since only significant factors were considered in the model, the difference in the values of q\({}_{\text{max}}\) also may be attributed to the conditions that were not statistically significant, which are inherently included in the experimental setup. Even though the interaction between W and Q is statistically significant only for the paracetamol (q\({}_{\text{maxPAR}}\)), as presented, a similar phenomenon is observed when considering metronidazole (Figure 5d-f). This insight underscores the complexity of multicomponent adsorption phenomena and emphasizes the need for tailored experimental and modeling approaches to effectively characterize and predict these interactions. Table 6 shows the comparison of the maximum adsorption capacity of organic molecules at different adsorbents and GAC present in the literature. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Compound** & **Q\({}_{\text{max}}\) (mg/g)** & **Reference** \\ \hline Paracetamol & 49.5 & Present study \\ \hline Metronidazole & 43.6 & Present study \\ \hline Atenol & 79.9 & [PERSON] et al. [70] \\ \hline Benzalkonium chloride & 26.3 & [PERSON] et al. [52] \\ \hline Caffeine & 7.3 & [PERSON] et al. [63] \\ \hline Tetracycline & 2.08 & [PERSON] et al. [71] \\ \hline Carbamazepine & 56.1 & [PERSON] et al. [72] \\ \hline Metronidazole & 32.2 & [PERSON] et al. [72] \\ \hline Salicylic acid & 205.1 & [PERSON] et al. [73] \\ \hline Isoproturon & 33.2 & [PERSON] et al. [70] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the maximum adsorption capacity (q\({}_{\text{max}}\)) of organic molecules at 25 \({}^{\circ}\)C in GAC present in the literature. The results in Table 6 highlight significant variations in the maximum adsorption capacity (qmax) of organic compounds on GAC, depending on the compound's physicochemical properties and specific interactions with the adsorbent surface. In this study, paracetamol (49.5 mg/g) and metronidazoleze (43.6 mg/g) showed moderate adsorption capacities compared to other compounds. For instance, salicylic acid (205.1 mg/g) [70] exhibited much higher qmax, likely due to stronger hydrophobic or electronic interactions with GAC, while tetracycline (2.08 mg/g) [74] had the lowest capacity, indicating less favorable interactions. Compounds such as atenol (79.9 mg/g) [75] and carbamazepine (56.1 mg/g) [69] displayed adsorption capacities closer to the values observed in this study. These findings emphasize that GAC interactions with organic compounds depend on factors like polarity, functional groups, molecular size, and hydrophobicity. Mathematical Modeling of Breakthrough Curves  
Table 7 presents the parameter values and statistical metrics derived using the MCMC method for the adsorption of paracetamol and metronidazoleze, respectively, under the optimal operational conditions (run order 1). The values presented in Table 7 are the posteriori means of the parameters, containing the lower and upper limits, according to the confidence interval applied (99%) in the present study. Figure 6 illustrates the breakthrough curve modeling for the simultaneous adsorption of the two pharmaceuticals in a fixed bed column during Experiment 1, with the Logistic and Gompertz models applied to the multicomponent adsorption system.