## 3 Results
### Adsorption Isotherm and Kinetics  
Modeling the adsorption results is essential to reveal the adsorption mechanism and thermodynamics. The equilibrium analysis determines the maximum dissolved oil adsorbed per unit mass of adsorbent. Our results were examined against Langmuir, Freundich, Temkin, and D-R isotherms. The regression coefficient R\({}^{2}\) was found, and an error analysis was carried out. Here, we provide the analysis of the results using the Freundlich model, as it provides the best fit for the experimental results. The details of the analysis of the results using these isotherm models are provided in the Supplementary Information.  
The Freundlich isotherm (Figure 4a) provided an excellent fit for the experimental adsorption data, indicating that the adsorption mechanism is not limited to the monolayer structure. Therefore, the amount of oil adsorbed on rGO-Fe\({}_{2}\)O\({}_{3}\) adsorbents was the total of adsorption on all sites, with more substantial energy binding sites at the beginning and exponentially decaying as the adsorption process moved towards completion.  
The parameters of the Freundlich isotherm model (Table 3) validate the excellent agreement between the model and the experimental results, indicating that oil adsorption on heterogeneous sites is the adsorption mechanism [6]. The heterogeneous nature of the adsorption process can be attributed to the different sizes of hydrocarbon molecules comprising the diesel oil and the presence of rGO and Fe\({}_{2}\)O\({}_{3}\) sites. The Freundlich parameter, n, was greater than 1, suggesting that the adsorption of emulsified oil onto rGO-Fe\({}_{2}\)O\({}_{3}\)-NC is favorable, and the mechanism is physical adsorption. The Freundlich model is the optimal isotherm to describe the physical adsorption mechanisms [50]. Moreover, the measured adsorption capacity was consistent with the model prediction, confirming that Freundlich is the correct isotherm for this process, indicating the multi-layer adsorption characteristic.  
Figure 3: Effect of the emulsion salinity on (**a**) the adsorption capacity and (**b**) the removal efficiency. Contact time = 30 min, initial concentration = 100 ppm, T = 22 \(\pm\) 1 \({}^{\circ}\)C, trembling speed = 350 rpm.  
Practical applications require understanding the kinetics of the adsorption mechanism. The adsorption kinetic study is essential to evaluate the performance of a batch and continuous flow adsorber. Additionally, it can verify the solute uptake rate to assess the residence time needed to complete adsorption. Numerous mathematical models describe the adsorption kinetics, classified as adsorption reaction and diffusion models. This study employed the most widely used sorption reaction models, pseudo-first-order (Lagergren model) and pseudo-second-order kinetic models, using the accomplished experimental results.  
The primary assumption of the pseudo-second-order kinetic model is that the rate-limiting step is potentially chemical adsorption, involving valent forces between the adsorbate and adsorbent [51]. In Figure 4, the R\({}^{2}\) > 0.999 indicates that this kinetic model can explain nearly 100% of the difference in the response variable [51; 52]. Therefore, the adsorption process of emulsified oil on rGO-Fe\({}_{2}\)O\({}_{3}\) follows pseudo-second-order kinetics. Thus, the adsorption process is hindered by intraparticle diffusion, boundary-layer, or external diffusion. The calculated adsorption rate was \(1.5\times 10^{-3}\)\(\frac{S_{adsorbent}}{min\cdot m_{gal}}\), and the predicted adsorption capacity was 661.6 mg/g.