## 3 Results and Discussion
### Adsorption Isotherms of Biochar Samples
\(k_{DR}\) (mol\({}^{2}\) J\({}^{-2}\)) is the D-R constant related to the sorption energy, which allows estimation of the sorption energy \(E\) (kJ mol\({}^{-1}\)) from Equation (6); \(\epsilon\) is the Polanyi potential, calculated from Equation (7), where \(R\) (8.314 J mol\({}^{-1}\) K\({}^{-1}\)) is the gas constant, \(T\) (\(K\)) is the absolute temperature, and \(Cs\) (mg L\({}^{-1}\)) is the solubility of Cu\({}^{2+}\) (from the salt Cu(HNO\({}_{3}\))\({}_{2}\)\(\times\) 3H\({}_{2}\)O) at 23 \({}^{\circ}\)C. Finally, for the case of the Temkin isotherm, \(a_{T}\) (L mg\({}^{-1}\)) is the equilibrium binding constant, and \(b_{T}\) (J g mol\({}^{-1}\) mg\({}^{-1}\)) is the Temkin constant related to the adsorption heat. The equations are as follows:  
\[R_{L} = \frac{1}{1+k_{L}C_{0}} \tag{5}\] \[E = \frac{1}{\sqrt{2k_{DR}}}\] (6) \[\epsilon = RT\ \ln\left(\frac{C_{s}}{C_{e}}\right) \tag{7}\]  
The comparison of the analyzed isotherm models for several BCs is shown in Figure 3, and the parameters obtained are presented in Table 4. The obtained correlation coefficients suggest that the Langmuir model fits the experimental data better than the other models (R\({}^{2}\) = 0.994-0.999) (Figure 3, Table 4), indicating a predominantly homogeneous monolayer surface coverage process of Cu\({}^{2+}\) on the BCs, characteristic of chemical adsorption, considering the copper concentrations used [44]. Although by FTIR a heterogeneous composition of the functional groups composing the surfaces of the BCs was observed, the adsorption process on each of these different groups can be comparable and therefore assume a homogeneous behavior. The monolayer adsorption capacity (\(q_{m}\)) was the highest for BC600 (93.55 mg g\({}^{-1}\)), followed by BC450 (58.00 mg g\({}^{-1}\)) and BC300 (49.24 mg g\({}^{-1}\)). The \(K_{L}\) parameter indicates the affinity to the adsorption sites, and, in this sense, BC300 presented the greatest affinity, followed by BC450 and BC600 (23.50, 10.91, and 0.373 L mg\({}^{-1}\), respectively). The equilibrium separation factor \(R_{L}\) obtained in this work suggests a favorable adsorption process (0 \(<\)\(R_{L}\) < 1) [43] for all BCs used for Cu\({}^{2+}\) adsorption (Table 4). The value of the mean free energy required for adsorptive removal (\(E\)) was obtained from the D-R isotherm model [43]. When the value of \(E\) ranges between 8 and 16 kJ mol\({}^{-1}\), the process is by ion exchange; \(E\) > 16 kJ mol\({}^{-1}\) is indicative of a chemical adsorption process [45]. In this context, the \(E\) value for BC600 indicated an ion exchange process, whereas for BC450 and BC300, the values were 17.83 and 19.99 kJ mol\({}^{-1}\), respectively, suggesting a chemical adsorption process. In this context, it could be expected that an increase in temperature promotes a higher amount of retained copper assuming a prominent chemical adsorption in the process [46]. Conversely, the Temkin isotherm presented a higher correlation coefficient for Cu\({}^{2+}\) adsorption on BC600 (R\({}^{2}\) = 0.991), which revealed that the surface of BC600 could be heterogeneous, where not all the available bonding sites would have the same adsorption capacity [47]. The best fit of both the Langmuir and Temkin isotherm models for the Cu\({}^{2+}\) adsorption data for BC600 could be explained by the finding that although the surface consisted of heterogeneous sites, these were similar in adsorption phenomena [46], which could also explain the increased adsorption capacity of Cu\({}^{2+}\)[48]. Due to the variation in the operational and experimental conditions used, direct comparison with other studies of biochar application for copper adsorption should be considered with caution. However, the results obtained in the present study may be comparable in terms of Cu\({}^{2+}\) removal capacity, not considering the mechanisms for this purpose due to their different compositions and operational conditions of production.