## 3 Results and Discussion
### Sorption Study  
The sorption potential of chemically modified CH-ANP was tested against copper, lead, and cadmium in an aqueous environment. The cations sorption occurred on the surface of sorbent when coming in contact with cation solution and the cations partitioned between the sorbent/solution interface. The amount of cations on the sorbent increased with time until a dynamic equilibrium was established; this phenomenon is mostly determined from sorption isotherm. In this study, both linear and nonlinear models for Langmuir, Freundlich, and Temkin isotherms were deployed, and the results obtained are summarized in Table 2[35]. The results showed that the chemically modified chitosan (CH-ANP) has a higher sorption capacity for copper, lead, and cadmium as compared to the pristine chitosan. The isotherms revealed that the sorption of cations on the surface of sorbent increased with time until surface saturation. The CH-ANP demonstrated higher sorption capacity for copper (2.82 mmol g\({}^{-1}\)) than cadmium (1.96 mmol g\({}^{-1}\)) and lead (1.60 mmol g\({}^{-1}\)) based on Langmuir sorption isotherm type 1, which shows its higher selectivity for copper as compared to the other metals, as shown in Table 3[36]. The CH-ANP contains nitrogen, which acts as a borderline basic center with relatively high electronegativity, and polarizability favors linkage with the borderline copper atom in contrast to the soft metals lead and cadmium [37]. The raw chitosan exhibited lower sorption capacity for the copper (1.40 mmol g\({}^{-1}\)) than cadmium (1.35 mmol g\({}^{-1}\)) and lead (1.19 mmol g\({}^{-1}\)) in comparison with the chemically modified materials. The quantity of cations sorbed and the remaining supernatant at equilibrium were determined from an isotherm. It also gives information about the interaction between the sorbent and sorbate molecules. The four different Langmuir isotherms types in both linear and nonlinear regression analyses were performed along with experimental data with Freundlich and Temkin models to obtain the best results. It is seen that the anticipated values of constant and the values of correlation coefficients for every one of the four linear forms of Langmuir isotherms are very contrasting, as shown in Table 3. The correlation coefficient value acquired from Langmuir isotherm type 1 was more precise than those of the other linear equations. The obtained correlation coefficient values displayed best fits with the Langmuir type 1 equation to the exploratory information in correlation with direct types of the Langmuir isotherm, as shown in Table 3. Standard Error (SE), chi-square, and \(\chi^{2}\) values were extremely little for the Langmuir model type 1, as compared to other tested conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Isotherm** & **Nonlinear Form** & **Linear Form** & **Plot** \\ \hline Langmuir type 1 & \(Nf=\frac{NsbCs}{1+bCs}\) & \(\frac{Cs}{Nf}=\frac{Cs}{Ns}+\frac{1}{NsB}\) & \(\frac{Cs}{Nf}\) \\ \hline Langmuir type 2 & \(\frac{1}{Nf}=\left[\frac{1}{NsB}\right]\frac{Cs}{1+\frac{1}{Ns}}\) & \(\frac{1}{Nf}\) & \(\frac{1}{Nf}\) \\ \hline Langmuir type 3 & \(Nf=Ns-\left[\frac{1}{B}\right]+\frac{Nf}{Cs}\) & \(Nf\) & \(\text{vs}\) \\ \hline Langmuir type 4 & \(\frac{Nf}{Cs}=bNs-bNf\) & \(\frac{Nf}{Cs}\) & \(\text{vs}\) \\ \hline Freundlich & \(Nf=KfCs^{\frac{1}{2}}\) & \(\text{log}\) & \(Nf-\text{log}\) & \(kf+\frac{1}{1}logCs\) \\ \hline Temkin & \(Nf=\ln\left(K_{T}Cs\right)^{\frac{1}{n_{T}}}\) & \(Nf=n_{T}\ln K_{T}+n_{T}\ln Cs\) & \(Nf=n_{T}\ln K_{T}+n_{T}\ln C\) \\ \hline Standard error & \(SE=\sqrt{\frac{1}{n-P}\sum\limits_{i=1}^{m}\left(N_{f\text{exp}i}-N_{f\text{ crit}}\right)^{2}}\) & \\ \hline Chi-square & \(x^{2}=\sum\limits_{i=1}^{\infty}\frac{\left(N_{f\text{exp}i}-N_{f\text{ crit}}\right)^{2}}{N_{f\text{crit}}}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sorption isotherm model, the linear and nonlinear representations and respective plot graphical forms.