## 2 Materials and Methods
### Determination of Adsorption Equilibrium and Modeling of Adsorption Isotherms  
Prior to the adsorption test, used oil at concentrations (C\({}_{\text{o}}\)) of 100%, 50%, 25%, 12.5%, and 6.25% \(\
u/v\) was titrated with a KOH solution to determine the concentration of polar molecules (oleic acid) in the used oil. The concentration of C\({}_{\text{o}}\) was subsequently converted to mg/L, and the data are presented in Table 1.  
At different time intervals before reaching equilibrium, the purified used oil samples were titrated with KOH to measure the concentration of polar molecules in the used oil, denoted as C\({}_{\text{t}}\) (mol/L), which was then converted into mg/L. The adsorption process proceeded until the equilibrium adsorption capacity (C\({}_{\text{e}}\)) was achieved, which occurred at an equilibrium time of 30 min. The equilibrium concentration of used oil adsorbed onto the solid adsorbent (Q\({}_{\text{e}}\), mg/g) was determined by calculating the difference between the initial concentration of used oil (C\({}_{\text{o}}\)) and its concentration at equilibrium time (C\({}_{\text{e}}\)) using Equation (2). \[Q_{e}=\frac{(C_{0}-C_{e})V}{W} \tag{2}\]  
In this case, V represents the volume of the titration solution (0.05 L of adsorbate solution), while W denotes the mass of the adsorbent, fixed at 1 g.  
The interaction between the adsorbate and the adsorbent was evaluated through equilibrium analysis of the adsorption process, employing the C\({}_{\text{e}}\) and Q\({}_{\text{e}}\) parameters. The isothermal study of the adsorption process was conducted using the classical Langmuir and Freundlich models [28; 29; 30; 31], which describe monolayer and multilayer adsorption, respectively. The linearized expressions of these models are provided in Equations (3) and (4). \[\frac{C_{e}}{Q_{e}}=\frac{C_{e}}{Q_{m}}+\frac{1}{K_{L}Q_{m}} \tag{3}\]  
\[\log Q_{e}=\frac{\log C_{e}}{n}+\log K_{F} \tag{4}\]  
In the [PERSON] isotherm model, Q\({}_{\text{m}}\) (mg/g) denotes the maximum adsorption capacity, which is derived from the slope of the linearized plot. This parameter represents \begin{table}
\begin{tabular}{c c} \hline \hline
**C\({}_{\text{o}}\) (\% \(\
u/v\)) * & C\({}_{\text{o}}\) (mg/L) ** \\ \hline
100 (before dilution) & 3631 \\ \hline
50 & 3113 \\ \hline
25 & 1556 \\ \hline
12.5 & 1038 \\ \hline
6.25 & 519 \\ \hline \hline \end{tabular}  
*The used oil was diluted with hexane before adsorption. ** The initial oleic acid concentration in the used oil was determined by titration.  
\end{table} 
Table 1: Initial concentrations of used oil (C\({}_{\text{o}}\)) and the number of polar molecules of oleic acid in 5 g of used oil. the theoretical maximum amount of adsorbate that can be adsorbed onto the surface of the adsorbent. The parameters K\({}_{\rm{L}}\) (L/mg) and K\({}_{\rm{F}}\) (mg/g) are the Langmuir and Freundlich constants, respectively, related to the affinity of the adsorption sites. In the Freundlich equation, the parameter \(n\) represents the adsorption intensity, which serves as an indicator of the favorability of the adsorption process. A slope value of \(1/n<0.5\) suggests that the process is unfavorable for multilayer adsorption. To provide a more comprehensive analysis, additional adsorption isotherm models, including the Dubinin-Radushkevich (D-R) isotherm, have been incorporated into this study, alongside the Langmuir and Freundlich models. The Dubinin-Radushkevich (D-R) isotherm was employed to elucidate the adsorption mechanism on heterogeneous surfaces. This model facilitates the determination of the adsorption nature, distinguishing between physisorption and chemisorption, by evaluating the adsorption energy [32]. The mathematical expression of this isotherm is represented in Equation (5):  
\[\ln Q_{e}=-\beta\epsilon^{2}+\ln Q_{m} \tag{5}\]  
In this study, Q\({}_{\rm{m}}\) denotes the maximum sorption capacity, while \(\beta\) represents the Dubinin-Radushkevich (D-R) constant.