## 3 Results and Discussion
### CAF Adsorption onto GAC
The importance of these parameters is corroborated by the F-values, where values higher than F\({}_{\text{calc}}\) indicate significant effects. The F\({}_{\text{calc}}\) for the model is 2.77, lower than the values of 69.01, 11.78, 63.67, and 9.40 for the parameters of linear and quadratic adsorbent dosage and linear and quadratic contact time, respectively. Furthermore, the effects of pH and the interaction of independent variables are not statistically significant, as evidenced by F-values lower than F\({}_{\text{calc}}\). So, as pH did not influence the system, it is possible to define the most viable pH to work in caffeine removal--in this case, the natural pH. An analysis of residual values was conducted, demonstrating their independence and adherence to a normal distribution, thereby validating the predicted model. The variance percentage obtained for the adjustment was 96.85%, and therefore, it was possible to conclude that the predicted model satisfied the experimental data obtained and can be used to plot and analyze the behavior of the independent variables for caffeine removal through response surfaces, as shown in Figure 1.  
The response surface plots illustrate that an increase in adsorbent dosage results in a reduction in caffeine C\({}_{\text{F}}\), indicating better effectiveness in removing the contaminant. Adsorbent dosages approaching the upper limit of 10 g L\({}^{-1}\) indicate the system's saturation point, suggesting that beyond this adsorbent concentration, the caffeine removal efficiency could remain stable. A longer contact time also results in higher removal rates, up to reaching the maximum adsorption capacity of the system. There is a nonlinear correlation between removal and contact time, particularly within the initial 60 min, which could be attributed to the abundant availability of active sites on the adsorbent surface at the onset of the process. The pH exhibited negligible impact within the studied conditions, likely due to the high pKa of caffeine (10.4), favoring adsorption across the evaluated pH range (3 to 11). Moreover, the GAC pHPCZ of 7.36 suggests a positively charged solid surface at lower pH values, which facilitates the adsorption of molecules such as caffeine, where the protonated form predominates when pH is lower than pKa (10.4) [7; 49]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Factor** & **SS R\({}_{\text{CAF}}\)** & **SS C\({}_{\text{F}}\)** & **DF** & **F-Value** & \(p\)**-Value** \\ \hline (1) pH (L) & 0.0044 & 1.7257 & 1 & 0.5048 & 0.5092 \\ (1) pH (Q) & 0.0002 & 0.0914 & 1 & 0.0267 & 0.8765 \\ (2) m\({}_{\text{ads}}\) (L) & 0.6134 & 235.9451 & 1 & 69.0123 & 0.0004 \\ (2) m\({}_{\text{ads}}\) (Q) & 0.1047 & 40.2711 & 1 & 11.7790 & 0.0186 \\ (3) t (L) & 0.5660 & 217.6919 & 1 & 63.6733 & 0.0005 \\ (3) t (Q) & 0.0836 & 32.1467 & 1 & 9.4027 & 0.0279 \\ 1 by 2 & 0.0018 & 0.6732 & 1 & 0.1969 & 0.6758 \\ 1 by 3 & 0.0007 & 0.2742 & 1 & 0.0802 & 0.7884 \\ 2 by 3 & 0.0058 & 2.2427 & 1 & 0.6560 & 0.4548 \\ Error & 0.0444 & 17.0944 & 5 & & \\ Total SS & 1.4124 & 543.2402 & 14 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ANOVA for the R\({}_{\text{CAF}}\) response function. Critical values derived from the experimental planning (presented in Table 5) reveal that the optimal adsorbent concentration peaks at the upper limit of 10 g L\({}^{-1}\), the ideal contact time stands at 116 min, and the optimal solution pH is natural. These outcomes align with prior studies [20,40,45,50,51], revealing the independent variables for the Fenton reaction, validating the successful optimization of adsorption parameters across diverse experimental conditions using the Box-Behnken design.