## 3 Results
### Adsorption Isotherms  
The equilibrium adsorption data were analysed in linear form using Equations (3)-(6), which correspond to three different isotherm models of Langmuir, Freundlich, and Temkin. The mass balance relationship was used to calculate the amount of adsorbed Fe(III) (\(q_{e}\)) [10; 18; 19; 35]:  
\[q_{e}=\frac{(C_{0}-C_{e})V}{w} \tag{3}\]  
where \(C_{0}\) represents the initial Fe(III) concentration (mg/L), \(C_{e}\) represents the equilibrium Fe(III) concentration, \(V\) represents the volume of the Fe(III) solution (L), and \(w\) represents the quantity of adsorbent (g).  
As seen in Figure 12 depicting the findings, the isotherm is connected to the quantity of adsorption per weight at equilibrium (\(q_{e}\)) and the equilibrium concentration in the bulk fluid phase (\(C_{e}\)). As the concentration of Fe(III) grew, so did its adsorption.  
\[\frac{C_{e}}{q_{e}}=\frac{1}{q_{m}b}+\frac{C_{e}}{q_{m}} \tag{4}\]  
Figure 10: The comparison of FT–IR spectra between (a) C–PPF before adsorption and (b) C–PPF after adsorption.  
Figure 11: The SEM, elemental mapping analysis and EDX images of N–PPF (**a–d**) and C–PPF after adsorption (**e–h**).  
Figure 12: Adsorption isotherms of Fe(III) for adsorbents (**a**) N–PPF adsorbent; (**b**) C–PPF adsorbent.  
The slope and intercept for the plot of \(C_{e}\) versus \(\frac{C_{e}}{q_{e}}\) are parameters \(q_{m}\) and \(b\), respectively, where \(q_{m}\) (mg/g) is the maximum adsorption capacity of the adsorbent, and \(b\) (L/mg) is the Langmuir constant:  
\[logq_{e}=lgK_{f}+\frac{1}{n}logC_{e} \tag{5}\]  
The \(K_{f}\) and \(\frac{1}{n}\) values can be calculated from the intercept and slope for the plot of \(logq_{e}\) versus \(logC_{e}\), where \(q_{e}\) (mg/g) and \(K_{f}\) (mg/L) are the equilibrium adsorbed amount and the Freundlich constant, respectively:  
\[q_{e}=BlnA_{T}+BlnC_{e} \tag{6}\]  
where:  
\[B=\frac{RT}{b_{T}} \tag{7}\]  
The slope and intercept were used to calculate the \(B\) and \(A\) constants and display \(q_{e}\) against \(lnC_{e}\). The Temkin isotherm equilibrium binding constant (L/g) is denoted by \(A_{T}\). The Temkin isotherm constant is denoted by \(b_{T}\). \(R\) (8.314 J/(mol/K)) is the universal gas constant. \(B\) is a constant that is proportional to the heat of sorption (0/mol).  
The results of the adsorption data are given in Table 3 and Figure 13. The best-fit adsorption linear square model was determined according to the coefficient (\(R^{2}\)). The Fe(III) adsorption data are consistent with the Freundlich model. In addition, the Freundlich models of N-PPF and C-PPF have higher \(R^{2}\) values (0.9369 and 0.9768, respectively) than the Langmuir model, and may be used for heterogeneous systems where there is an interaction between the molecules adsorbed and the occurrence of multilayer coverage of Fe(III) on the adsorbent surface [23; 24]. As a result, on the biosorption surface, the adsorbed Fe(III) ions formed a monolayer. The N-PPF and C-PPF had \(q_{max}\) values of 0.0833 and 0.4147 mg/g, respectively [36]. The Temkin model has an \(R^{2}\) of 0.9325 and assumes that the sorption heat of all molecules reduces as the layer is covered, taking into account the influence of indirect contact between adsorbate molecules for C-PPF. Furthermore, the adsorption has the greatest energy distribution of a uniform bond due to the interactions between the adsorbent and the adsorbate [20; 27]. The \(B\) is +0.0163 and +0.0495 J/mol for N-PPF and C-PPF, respectively, implying that the heat of sorption, which is an indication of the heat of sorption, specifies a physisorption process of N-PPF and C-PPF [20; 36; 37; 38]. The values of n are measures of either adsorption intensity or surface heterogeneity, and they reflect the favourability of sorption, where \(n<1\), \(1<n<2\), and \(2<n<10\) represent poor, moderate, and favourable adsorption, respectively, and grow more heterogeneous as they approach 10 [39]. When the adsorbent surface is fairly heterogeneous, the value of \(n\) is 1.19 for C-PPF.