*3.4. Adsorption Kinetics*

The adsorption surface, mass transfer, or intraparticle diffusion are different mechanisms involved in the adsorption process and, to study them, three kinetic models were employed to test the experimental data obtained in the adsorption of DB78 on eggshell. The determination coefficient values (R2) are essential to decide the best adjustment to the experimental data to the different models tested. The results for the adjustment to the pseudo-first-order, pseudo-second-order, and the intraparticle diffusion kinetic models are shown in Figure 4, and the main parameters for each model are presented in Table 1.

**Figure 4.** *Cont.*

**Figure 4.** (**a**) Pseudo-first-order model plots, (**b**) pseudo-second-order model plots, and (**c**) intraparticle diffusion model plots for the Direct Blue 78 adsorption onto eggshell at different concentrations of dye of 25 mg/L (•), 50 mg/L (-), 100 mg/L (-), 150 mg/L (), 200 mg/L (), 250 mg/L (Δ), and 300 mg/L ().



**<sup>1</sup>** PFOM: pseudo-first-order model; **<sup>2</sup>** PSOM: pseudo-second-order model; **<sup>3</sup>** IDM: intraparticle diffusion model.

The linearity of the Lagergren model (log(*qe* − *qt*) versus *t*) was graphed for 140 min of contact with the eggshell (Figure 4a). The *R*<sup>2</sup> values ranged from 0.800 to 0.995 (Table 1). Calculated values of *qe* were compared with the experimental data, and although some *R*<sup>2</sup> values were relatively high, the *qe* values calculated were not suitable. The obtained *R*<sup>2</sup> and *qecal* values indicated that the adsorption of dye onto eggshell did not follow the pseudo-first-order kinetics, even though some values were relatively high; consequently, this equation cannot be used to analyze the experimental results. Because of the obtained results, it was appropriate to fit the experimental data to the pseudo-second-order model. Figure 4b shows the results obtained after applying the Ho and McKay model. The plot of *t*/*qt* versus *t* produced straight lines for the entire measurement range.

The theorical *qe* values were identical to the experimental *qe* values obtained using the pseudo-second-order, as compared with those of the pseudo-first-order kinetic, indicating that DB78 adsorption onto eggshell followed the pseudo-second-order kinetic model. As can be observed from Table 1, the *R*<sup>2</sup> values of the pseudo-second-order kinetic model are higher than those of pseudo-first-order, the value was 1 in all cases analyzed. These results suggested that chemical adsorption was the rate-limiting step that controls this adsorption process. Chemisorption occurs when strong interactions, including hydrogen bonding and covalent and ionic bond formation, occur between the adsorbate and the solid surface. The endpoint for chemisorption is when all the active sites on the solid surface are occupied by chemisorbed molecules. Ehrampoush et al. observed similar kinetics in the adsorption of Reactive Red 123 dye onto eggshell [49], or in the adsorption of Acid Orange 51 onto the ground eggshell powder [16].

The adsorption is a process that follows many steps; firstly, it implicates a transport of dye molecules from the solution to the adsorbent surface, and then a diffusion to the interior of the eggshell could take place [32]. In order to understand the adsorption of DB78 dye onto eggshell, the kinetic of the adsorption process was analyzed using the intraparticle diffusion model, in order to determine if the intraparticle diffusion is the rate-limiting step in the adsorption. This effect was studied by plotting the amount of DB78 dye adsorbed versus the square root of time (Figure 4c).

Figure 4c shows the plot of *qt* versus *t* <sup>1</sup>/<sup>2</sup> for the intraparticle diffusion of DB78 for the eggshell and different concentrations of dye. Two different straight lines can be distinguished in Figure 4c for the range of concentrations analyzed, indicating that two or more forces are influencing the adsorption process; in this case, chemisorption and intraparticle diffusion played essential roles in the adsorption of DB78 onto eggshell (Figure 4c).

The intraparticle diffusion constant (*ki*) values are presented in Table 1. The values of *ki* and *C* were calculated from the slope and intercept of plots of *qt* versus *t* <sup>1</sup>/2. These *ki* values increased with increasing dye concentrations. The *R*<sup>2</sup> values were very different depending on the concentration, ranging from 0.704 to 0.995. The intraparticle diffusion model was not the rate-controlling step because the results did not pass through the origin. When this plot gives rise to a straight line, the adsorption process is controlled by intra-particle diffusion only. However, if the data present multi-linear plots, then two or more steps influence the adsorption process, as shown in Figure 4. Similar results were also reported for the adsorption of Acid Red 14 and Acid Blue 92 onto the microporous and mesoporous eggshell membrane [37].

Conversely, the intercept of each curve is proportional to the boundary layer thickness; a higher intercept indicates a higher effect. This value decreased with increasing dye concentrations for the eggshell; therefore, the intraparticle diffusion model was not the sole rate-controlling step for eggshell, confirming our previous suggestions [32].
