## 3 Results and Discussion
### Cu\({}^{2+}\) Removal Performance of the Biochar Sample
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Linear Parameters** & **BC300** & **BC450** & **BC600** \\ \hline \(q_{e},exp\) a (mg g\({}^{-1}\)) & 71.91 & 119.9 & 101.9 \\  & Pseudo-first-order & & \\ \(q_{e}\) (mg g\({}^{-1}\)) & 61.24 & 118.7 & 106.67 \\ \(k_{1}\) (min\({}^{-1}\)) & 0.00260 & 0.00269 & 0.00447 \\ \(R^{2}\) & 0.917 & 0.933 & 0.960 \\  & Pseudo-second-order & & \\ \(q_{e}\) (mg g\({}^{-1}\)) & 70.37 & 128.2 & 125.9 \\ \(k_{2}\) (g mg\({}^{-1}\) min\({}^{-1}\)) & 0.00019 & 0.000030 & 0.000037 \\ \(R^{2}\) & 0.995 & 0.844 & 0.958 \\ Liquid-film diffusion & & & \\ \(k_{fd}\) (min\({}^{-1}\)) & 0.00169 & 0.00233 & 0.00461 \\ \(R^{2}\) & 0.979 & 0.854 & 0.979 \\ Intra-particle diffusion & & & \\ \(k_{1}\) (mg g\({}^{-1}\) min\({}^{1/2}\)) & 2.081 & 1.906 & 5.646 \\ \(C_{1}\) (mg g\({}^{-1}\)) & 20.560 & 16.783 & 26.285 \\ \(R^{2}\) & 0.992 & 0.990 & 0.988 \\ \(k_{2}\) (mg g\({}^{-1}\) min\({}^{1/2}\)) & 0.844 & 4.7234 & 0.400 \\ \(C_{2}\) (mg g\({}^{-1}\)) & 39.2315 & 26.639 & 86.671 \\ \(R^{2}\) & 0.997 & 0.995 & 0.987 \\ \(k_{3}\) (mg g\({}^{-1}\) min\({}^{1/2}\)) & - & 1.342 & - \\ \(C_{3}\) & & 69.843 & - \\ \(R^{2}\) & - & 0.880 & - \\ \hline \hline \end{tabular}  
* Obtained values of \(q_{e}\) from the sampling and measurement of Cu\({}^{2+}\) in the suspension at 1440 min in the kinetics experiments.  
\end{table} 
Table 5: Kinetic parameters calculated from the pseudo-first- and pseudo-second-order models, in addition to the intra-particle diffusion and liquid film diffusion linear models. The multi-linearity of the intra-particle diffusion model indicates that Cu\({}^{2+}\) adsorption on biochar samples occurs by two or three phases, depending on the biochar sample. For BC300 and BC600, the first phase is generally associated with film diffusion that occurred mostly in the outer layer of the biochar samples, and the second portion of the line represents intra-particle diffusion into the pore structure of the biochar samples. The value of the constants \(k_{\rm l1}\) for BC 300 and BC600 samples are greater than \(k_{\rm l2}\), suggesting that the film diffusion of Cu\({}^{2+}\) occurred quickly until reaching saturation on the biochar samples, followed by an adsorption process inside the pores at a slower rate. In addition, the intercept value \(C\) is a constant related to the thickness of the boundary layer; the \(C\) values of BC300 and BC600 were higher than that of BC450 (Table 5), suggesting that surface diffusion played an important role as the rate-limiting step for the BCs produced at 300 and 600 \({}^{\circ}\)C [59]. These results agree with their similar pore diameter (18.1 and 19.9 nm for BC300 and BC600, respectively) and with the high correlation obtained for both BCs in the film diffusion model. Conversely, the intra-particle diffusion model of BC450 shows three straight lines associated with surface diffusion, followed by intra-particle diffusion, and finally reaching equilibrium [59]. The lower \(C\) value of BC450 compared to the other BCs suggests that the film diffusion was less important as a rate-limiting step, in agreement with a higher resistance to intra-particle diffusion associated with a smaller pored diameter [6].