## 3 Results
### Kinetic Models of the Fe(III) Adsorption Process  
The kinetics of the Fe(III) adsorption process on N-PPF and C-PPF were studied using three kinetic models. Time-dependent experimental data on Fe(III) included pseudo-first order, pseudo-second-order, and intra-particle diffusion. Equations (8)-(10) are used to present them and integrate them into a linear form. \[log(q_{e}-q_{t})=logq_{e}-\frac{k_{1}}{2.303}t \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Biosorbent** & \multicolumn{2}{c}{**Langmuir Isotherm**} & \multicolumn{3}{c}{**Freundlich Isotherm**} & \multicolumn{3}{c}{**Temkin Isotherm**} \\ \hline  & \(q_{m}\) & \(b\) & \(R^{2}\) & \(K_{f}\) & \(N\) & \(R^{2}\) & \(A_{T}\) & \(B\) & \(R^{2}\) \\  & (mg/g) & (L/mg) & & (mg/g)/(mg/L)\({}^{1/n}\) & \(N\) & \(R^{2}\) & \(A_{T}\) & \(B\) & \(R^{2}\) \\ \hline N–PPF & 0.0833 & 0.9103 & 0.6516 & 36.9 & 1.66 & 0.9369 & 12.0709 & 0.0163 & 0.7781 \\ C–PPF & 0.4147 & 0.2579 & 0.6570 & 82.0 & 1.19 & 0.9768 & 6.7197 & 0.0495 & 0.9325 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of isotherm parameters for Fe(III) adsorption on C–PPF and N–PPF sorbents. the intra-particle diffusion rate constant. The equilibrium adsorbed quantities and adsorbed amounts at any moment are represented by the \(q_{e}\) (mg/g) and \(q_{t}\) (mg/g) values, respectively. The thickness of the boundary layer is represented by parameter \(c\) (mg/g). The border is effective if the intercept value exceeds \(c\). The experimental data are shown in Table 4 and Figures 14 and 15. Both the pseudo-first-order and pseudo-second-order reaction models agree well with the kinetic data of Fe(III) sorption. The initial Fe(III) sorption rate from C-PPF was determined to be 0.0090 mg/(g-min). The Fe(III) removal from C-PPF was discovered to be sluggish. On the other hand, the adsorption of Fe(III) onto C-PPF is better suited to the pseudo-first-order model (\(R^{2}\) = 0.9870) and the pseudo-second-order model (\(R^{2}\) = 0.9890). Both models exhibit a good match, but N-PPF is more suited to the pseudo-first-order model (\(R^{2}\) = 0.9616). Based on the experimental data for C-PPF and the pseudo-first-order model for N-PPF, the adsorption capacities were computed using the pseudo-first-order and pseudo-second-order models. The physisorption process was demonstrated by fitting the adsorption process using a pseudo-first-order model. The adsorption process was ideally suited to the pseudo-second-order model, proving chemisorption [40]. Similar behaviour has been observed for Fe(III) adsorption on other natural adsorbent materials [26; 27; 28]. The initial sorption rates of N-PPF and C-PPF were 0.0025 and 0.0090 mg/(g-min), respectively, indicating that the initial sorption rate of C-PPF is faster than that of N-PPF. The results of the intra-particle diffusion parameters are shown in Table 5 and Figure 16. The internal diffusion model was utilised to investigate the adsorption rate determination technique. The internal diffusion model parameters may be utilised to visualise the data between \(q_{t}\) and \(t(^{1/2})\). The fact that the plot from this model is a straight line that passes through the origin indicates that the adsorption process is controlled by intra-particle diffusion. However, the presence of two or more linear zones in the plot shows that the adsorption process occurs in stages [26; 27]. The intra-particle adsorption plots were found to be nonlinear and may be divided into three linear stages where the plots do not pass through the origin (\(c\
eq 0\)). This illustrates that, in addition to intra-particle diffusion being a rate-controlling step, the rate of adsorption may be regulated by other kinetic models, such as boundary layer diffusion or other processes with some control over the border layer [25; 26; 41]. The initial stage is characterised by a high diffusion rate (\(k_{it1}\)) due to the bulk diffusion or external diffusion, where ions are carried to the exterior surface of the adsorbent.