## 2 Materials and Methods
### Adsorption Kinetic Models
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multicolumn{3}{c}{**Langmuir Model**} & \multicolumn{3}{c}{**Freundlich Model**} & \multicolumn{3}{c}{**Dubinin–Radushkevich (D–R)**} \\ \hline \(\mathbf{Q_{m}}\) & \(\mathbf{K_{L}}\) & \(\mathbf{R^{2}}\) & \(\mathbf{K_{F}}\) & \(\mathbf{(L/mg)}\) & \(\mathbf{\frac{1}{n}}\) & \(n\) & R\({}^{2}\) & \(\mathbf{\beta}\) & \(\mathbf{\frac{E}{(J/mol)}}\) & \(\mathbf{Q_{m}}\) & \(\mathbf{R^{2}}\) \\ \(-\)200 & \(-\)3.6036 & 0.1921 & 0.0080 &
\begin{tabular}{c} 1.1418 \\ (\textgreater{}0.5) * \\ \end{tabular} & 0.8758 & 0.9115 & 0.0736 & 2.61 & 59.15 & 0.5926 \\ \hline \end{tabular}  
* A favorable interaction for multilayer adsorption between the adsorbent and adsorbate is represented by \(1/n>0.5\)[40].  
\end{table} 
Table 4: Langmuir, Freundlich, and D–R parameters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\mathbf{C_{o}}\) (\% \(v/v\)) & \(\mathbf{C_{o}}\) (mg/L) & \(\mathbf{C_{e}}\) (mg/L) & \%FFA & \(\mathbf{Q_{e}}\) (mg/g) & \(\mathbf{C_{e}}\)/\(\mathbf{Q_{e}}\) & \(\mathbf{log}\)\(\mathbf{C_{e}}\) & \(\mathbf{log}\)\(\mathbf{Q_{e}}\) \\ \hline
100 & 3631 & 1692 & 1.69 & 96.95 & 37.45 & 3.56 & 1.99 \\ \hline
50 & 3113 & 1121 & 1.12 & 99.25 & 31.37 & 3.49 & 2.00 \\ \hline
25 & 1a value of 1.1418 (greater than 0.5), supports the conclusion that the Freundlich model provided a better representation of the multilayer adsorption process [41]. These findings suggest that the adsorption behavior of polar molecules in used oil on the heterogeneous surface of A-BP involves multilayer adsorption. This implies that higher concentrations of used oil can adsorb layer by layer without limitation to a monolayer. As shown in Figure 3c, the equilibrium data for polar molecule adsorption onto A-BP exhibited a superior fit to the Dubinin-Radushkevich (D-R) isotherm model compared to the Langmuir model, with an R\({}^{2}\) value of 0.5926. The adsorption capacity determined from the D-R model was 59.15 mg/g. Notably, the Langmuir model was not utilized for capacity estimation due to the occurrence of a negative slope. To determine the predominant adsorption mechanism--either chemisorption or physisorption--the adsorption energy (E) was estimated using the Dubinin-Radushkevich (D-R) equation. Specifically, the adsorption energy was calculated based on the D-R constant (\(\beta\)) using the relationship \(E=\frac{1}{\sqrt{2\times\beta}}\). Adsorption energy values typically range between 8 and 16 kJ/mol for chemisorption processes, whereas values below 8 kJ/mol indicate that physisorption is the predominant mechanism [33; 41]. In the present study, the adsorption energy (E) was determined to be 2.61 kJ/mol, suggesting that the uptake of polar molecules from used oil by A-BP primarily occurred via a surface physisorption process. The adsorption process is predominantly influenced by weak intermolecular interactions involving functional groups such as hydroxyl (-OH) and carboxyl (-COOH) on A-BP. These functional groups facilitate hydrogen bonding with polar molecules present in the used oil. This observation is further supported by the functional group analysis of the A-BP sample, as illustrated in Figure 4.  
Figure 3: Comparison of linearized isotherm models: (**a**) Langmuir, (**b**) Freundlich, and (**c**) Dubinin–Radushkevich (D–R) isotherms. The FTIR spectrum exhibits a peak at 3782 cm\({}^{-1}\), corresponding to O-H stretching vibrations and indicative of hydroxyl (-OH) groups commonly associated with cellulose, hemicellulose, and lignin in banana peel. The absorption band observed at 1578 cm\({}^{-1}\) is likely attributed to C=O stretching vibrations from carboxyl (-COOH) groups or the asymmetric stretching of carboxylate (COO\({}^{-}\)), which may have formed due to the acetic acid treatment. Furthermore, the absorption bands around 1424 cm\({}^{-1}\) can be ascribed to O-H bending in carboxyl (-COOH) groups or symmetric COO\({}^{-}\) stretching, further confirming the presence of carboxyl functionalities.