## 3 Results and Discussion
### Optimization of the Sorption Process of Zn(II), Cu(II) and Pb(II) by the Bean Peel
In addition, the model has high significance since \(F_{crit}\) (3.36) \(<\)\(F_{(4,11)}\) (25.69) and \(p\)-value \(<\) 0.05. The results also show that the predictors considered in the model have high significance due to presenting \(p\)-values lower than 0.05, except \(C_{i}\times t\), which has slightly more (0.052). The coefficient values highlight that an increase in the initial concentration and contact time leads to a higher Zn(II) removal. Therefore, the equation that explains the relationship between the removal percentage of Zn(II) and the predictors is as follows:  
\[\text{Zn(II) removal (\%)}=-\,3.7+0.0718C_{i}+33.83t-2.890t^{2}-0.01331C_{i}\times t \tag{4}\]  
where \(C_{i}\) is the initial zinc concentration (\(\mu\)g/L) and \(t\) is the contact time (h). Figure 2 shows the response surface model obtained for Zn(II) removal. The relationship between the predicted removal percentages and the experimental data showed to be linear with a high correlation (\(R^{2}\) = 0.9362; see Supplementary Materials Figure S2), confirming that the chosen model is adequate to estimate the zinc removal from an aqueous solution by contact with the biomaterial. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(\mathcal{S}\)S & \(\boldsymbol{DF}\) & \(\boldsymbol{MS}\) & \(\boldsymbol{Coefficient}\) & \(\boldsymbol{F}\) & \(\boldsymbol{p}\)-\(\boldsymbol{Value}\) & \(\boldsymbol{R^{2}}\) & \(\boldsymbol{R^{2}}_{adj}\) \\ \hline
**Zn(II)** & & & & & & & & \\ Corrected & 2203.57 & 4 & 550.89 & – & 25.69 & \(<\)0.001 & 0.936 & 0.900 \\ Cond & - & - & - & \(-\)3.7 & - & - & - & - \\ \([C_{i}]\) & 188.50 & 1 & 188.50 & 0.0718 & 8.79 & 0.021 & - & - \\ \(t\) & 692.20 & 1 & 692.20 & 33.83 & 32.28 & \(<\)0.001 & - & - \\ \(t^{2}\) & 354.60 & 1 & 354.60 & \(-\)2.890 & 16.54 & 0.005 & - & - \\ \([C_{i}]\times t\) & 116.98 & 1 & 116.98 & \(-\)0.0133 & 5.46 & 0.052 & - & - \\ Error & 150.09 & 7 & 21.44 & - & - & - & - & - \\ Corrected total & 2353.66 & 11 & - & - & - & - & - & - \\ \hline
**Cu(II)** & & & & & & & & \\ Corrected & 4683.7 & 3 & 1561.23 & – & 44.71 & \(<\)0.001 & 0.944 & 0.923 \\ model & - & - & - & 87.80 & - & - & - & - \\ Constant & 1828.3 & 1 & 1828.31 & 7.56 & 52.36 & \(<\)0.001 & - & - \\ pH & 2434.7 & 1 & 2434.67 & \(-\)11.62 & 69.73 & \(<\)0.001 & - & - \\ \([C_{i}]^{2}\) & 432.8 & 1 & 432.81 & 0.000058 & 12.40 & 0.008 & - & - \\ Error & 279.3 & 8 & 34.92 & - & - & - & - & - \\ Corrected total & 4963.0 & 11 & - & - & - & - & - & - \\ \hline \end{tabular}
\end{table} 
Table 3: Summary of the models with 4 predictors for Zn(II) and 3 predictors for Cu(II), presenting the \(SS\), \(DF\), \(MS\), coefficients, \(F\) and \(p\)-values, \(R^{2}\), and \(R^{2}_{adj}\). Figure 2: Response surface model obtained for Zn(II) removal (%) as a function of time (h) and initial concentration (\(\mu\)g/L), accordingly with Equation (4). Table 3 also shows that the chosen model for copper is representative of the copper experimental data, since \(R^{2}\)_adif_ indicates that 92.3% of the results obtained can be explained by the model. In addition, \(F_{crit}\) (4.07) \(<\)\(F_{(3,8)}\) (44.71), and the \(p\)-value of the model is lower than 0.05, as well as the \(p\)-values of the predictors, indicating that the model has high significance. Through the coefficients in the equation, it is possible to observe that an increase in the contact time and concentration leads to a higher removal, while an increase in pH leads to a decrease in the copper biosorption capacity. Therefore, the relationship between the Cu(II) removal and the predictors is given by the following equation:  
\[\text{Cu(II) removal (\%)}=87.80+7.56t-11.62\text{pH}+0.000058C_{i}^{2} \tag{5}\]  
where \(t\) is the contact time (h), pH is the initial pH of the solution, and \(C_{i}\) is the initial concentration of Cu(II). Figure 3 shows the response surfaces obtained from Equation (5) for Cu(II) removal (%), showing the interaction between the removal percentage, time (h), pH, and initial concentration (\(\mu\)g/L).