## 3 Results and Discussion
### Optimization of the Sorption Process of Zn(II), Cu(II) and Pb(II) by the Bean Peel  
The results of the application of the Uniform Design are shown in Table 2 and were used to optimize the sorption process of Zn(II) and Cu(II) by the bean peel and to find the best mathematical equation to describe the data. The optimization for Pb(II) was not possible to realize by a similar means due to the decrease of the control concentration for pH 5.6 and 7.0 and the lack of some data points, as discussed above and evidenced in Figure 1 and Table 2. Therefore, for Zn(II) and Cu(II), the best subsets regression was applied to the data of the average of the removal percentages of each element, since this method compares all possible models that can be created based on a set of predictors (independent variables). For such, the predictors used were three first-order effects, namely the initial concentration of the element (\(C_{i}\)), contact time (\(t\)), and pH; three interaction effects, between the initial concentration and pH (\(C_{i}\times\) pH), between the initial concentration and contact time (\(C_{i}\times t\)), and between the pH and contact time (pH \(\times\)\(t\)); and three second-order effects, specifically \(C_{i}\)\({}^{2}\), pH\({}^{2}\), and \(t^{2}\). The regression analyses determined that the predictor pH\({}^{2}\) could not be estimated, and, therefore, it was not used for the regression analysis. By using the remaining predictors, it was possible to obtain the best model with up to eight predictors, considering statistical parameters, such as the \(p\)-value, coefficient of determination (\(R^{2}\)), adjusted \(R^{2}\) (\(R^{2}\)_adj_), and _Cp Mallow_'s for each model, and the \(p\)-value for the predictors in the model ([40]; see Supplementary Materials Tables S1 and S2). Results showed that all the models are significant at a 95% confidence level, with the exception of model 8 for the Cu(II), since \(p\)-values are lower than 0.05. On the other hand, the \(p\)-values of the predictors indicate that only models 1, 2, and 4 for the Zn(II) and 1, 2, and 3 for the Cu(II) include predictors that are significant in the model. The results of the \(R^{2}\) parameter were higher than 0.8, except for model 1, which indicates a good fit of the models. However, \(R^{2}\) is not the most suitable parameter for determining the best model since it increases with the increase of the number of predictors in the model and has stabilized from models composed of four factors. Regarding the \(R^{2}\)_adj_, it only increases if the added predictor improves the model, and its value closest to 1 is the most appropriate [41]. The _Cp Mallow_'s parameter is also relevant in the decision of the best model, presenting the predicted error, and it is desirable to be as low as possible [42]. Considering the aforementioned parameters, for Zn(II), the model constituted by four predictors was chosen as the best (model 4; \(R^{2}\)_adj_ = 0.900 and _Cp Mallow_'s = 1.7), while for Cu(II), the model composed of three predictors was chosen as the best (model 3; \(R^{2}\)_adj_ = 0.923 and _Cp Mallow_'s = 1.2). Regarding the best model chosen for Zn(II) removal, only the predictors of initial concentration, contact time, interaction between them, and the quadratic form of the time are present. These findings highlight, as expected, that the initial pH of the solution is not significant for the removal of zinc by the bean peel. On the other hand, the best model found for Cu(II) removal includes the predictors of contact time, pH, and the quadratic form of initial concentration. Table 3 shows a summary of the models found for Zn(II) and Cu(II) removals, with information relating to the predictors, sum of squares (\(SS\)), mean square (\(MS\)), degrees of freedom (\(DF\)), \(F\) and \(p\)-values, \(R^{2}\), and \(R^{2}\)_adj_. As shown in Table 3 for Zn(II), \(R^{2}_{adj}\) indicates that 90.0% of the experimental results can be explained by the model.