## 3 Results and Discussion
### Adsorption Performances of DE and DE-LDH
#### 3.2.6 Adsorption Kinetics  
In the adsorption process, the adsorption kinetics are the keys to understanding the mechanism of sorption. To study the kinetics, the equations such as pseudo-first order kinetics, second-order kinetics and intra-particle diffusion models were plotted (Equations (S5)-(S7)) (Supplementary Materials) and tabulated for dye concentrations of 50, 100 and 250 mg/L (Figure 8 and Table 3). From the results, the pseudo-second order is well fitted with the data compared to the pseudo-first order model. The correlation coefficient was higher (R\({}^{2}\) = 0.999) for the pseudo-second order compared to the first order model. The values of q\({}_{\text{e,cal}}\) obtained from the pseudo-second order model were well matched with the q\({}_{\text{e,exp}}\) values obtained from the experiment studies. Therefore, based on fine data fittings with the pseudo-second order kinetic model of the CR adsorption on the adsorbent, it is understood that the sorption is a chemisorption.  
The rate controlling steps of CR adsorption in the adsorption process can be analyzed through [PERSON]'s intra-particle diffusion model and this could be used to understand the rate of adsorption at various concentrations (50, 100 and 250 mg/L) of CR onto the adsorbents (DE and DE-LDH). From the contact time studies, the removal efficiency of DE decreases with an increase in the time due to the smaller surface area and weak adsorption which leads to lesser adsorption; furthermore, increasing the time of contact could lead to the leaching of the dye from the surface. Based on time studies, an intra-particle diffusion model could not be used to describe the diffusion of the dye molecules onto  
Figure 8: Adsorption kinetic plots for different concentration of CR onto DE and DE-LDH.  
DE surfaces. From Figure 8, the diffusion plot of DE decreases with the increase in time for all the concentrations. Therefore, intra-particle diffusion was applied only for DE-LDH. The linear plot does not pass through the origin, suggesting that the intra-particle diffusion model is not a rate-limiting step. Two stages of the process were observed; the first step is attributed to the boundary layer diffusion, i.e., the diffusion of CR molecules from the bulk solution to the external surface of DE-LDH. The second step was due to the fast diffusion of dye molecules into the pores of DE-LDH. From Table S1 (Supplementary Materials), the diffusion rate constant Kint1 values are higher than Kint2, which reveals that the dye diffusion in the first step was quicker than the diffusion in the second step, as the surface of DE-LDH was freely accessible at the initial stage. Moreover, the diffusion rate constant (Kint) increases with an increase in dye concentration due to the higher dye concentration, which could rush the dye molecules from the bulk solution to the adsorbent surface. The boundary layer thickness (C) values increased from C1 to C2 due to the diffusion of dye molecules which occupy the surface, leading to low values of C1 and, later dye, molecules quickly entered the pores, leading an increase in the thickness of the boundary layer to C2. The C values increase with increasing dye concentration, which indicates that, at higher concentrations of dye, the adsorption rate is also high and this phenomenon is attributed to the increase in the boundary layer of the pores on the adsorbent surfaces.