*3.7. Adsorption Kinetics*

Four kinetic equations namely pseudo 1st order, pseudo 2nd order, Elovich model and intra-particle diffusion models were used to analyze the adsorption data. The relationship between the amount of dye adsorbed on adsorbents and adsorption time was determined. Pseudo-first-order and pseudo-second-order equations are expressed in Equations (6) and (7) respectively as below;

$$\ln(\mathbf{q}\_{\text{g}} - \mathbf{q}\_{\text{t}}) = \ln \mathbf{q}\_{\text{g}} - \mathbf{k}\_{\text{l}} \mathbf{t} \tag{6}$$

$$\frac{\mathbf{t}}{\mathbf{q}\_{\rm t}} = \frac{1}{\mathbf{k}\_2 \mathbf{q}\_{\rm e}^2} + \frac{\mathbf{t}}{\mathbf{q}\_{\rm e}} \tag{7}$$

where qe (mg g<sup>−</sup>1) is the equilibrium adsorption and qt (mg g<sup>−</sup>1) is the amount of dye adsorbed after time t (min). K1 (min−1) and K2 (g mg<sup>−</sup><sup>1</sup> min−1) are the rate constants of pseudo-first-order and pseudo-second-order equations respectively.

The Elovich kinetic model can be expressed as shown below in Equation (8);

$$\mathbf{q}\_{\rm t} = \frac{1}{\beta} \ln(\alpha \beta) + \frac{1}{\beta} \text{Int} \tag{8}$$

where α (mg g<sup>−</sup><sup>1</sup> min−1) shows an initial rate of adsorption and β (mg g<sup>−</sup>1) is the desorption constant relating to the activation energy and the extent of surface coverage.

Intra-particle diffusion model is expressed in Equation (9);

$$\mathbf{q}\_{\rm t} = \,\,\mathbf{k}\_{\rm d}\mathbf{t}^{1/2} + \mathbf{c} \tag{9}$$

where Kd (g mg<sup>−</sup><sup>1</sup> min−1/2) represent the rate of diffusion constant and C (mg g<sup>−</sup>1) is the constant of boundary layer thickness.

The fitted curves of adsorption of AB40 onto Fe3O4, PANI and PANI/Fe3O4 composites are shown in Figure 11. The parameters of kinetics are summarized in Tables 3 and 4. The correlation factor of pseudo-first-order (*R*2), are 0.812, 0.885 and 0.881 for adsorption of AB40 onto Fe3O4, PANI and PANI/Fe3O4 composites. These values indicate that adsorption of AB40 does not follow pseudo-first-order kinetics [83]. Similarly *R*<sup>2</sup> of Elovich model for PANI/Fe3O4 composites is 0.707 and intra-particle diffusion model for Fe3O4 is 0.864 indicating that these models also do not fit well for the adsorption data of AB40 on all of the three adsorbents [84]. *R*<sup>2</sup> values of pseudo-second-order equation show that the adsorption kinetics are more accurately described by this model (Table 3). Moreover, the qe values calculated by the pseudo-second-order equation agree more closely with the adsorption isotherm values [85].

**Figure 11.** The Kinetics model (**a**) pseudo-first-order, (**b**) pseudo-second-order, (**c**) Elovich model and (**d**) intra-particle diffusion model for adsorption of AB40 on Fe3O4, PANI and PANI/Fe3O4 composite.


**Table 4.** Kinetics parameters for adsorption of AB40 on Fe3O4, PANI and PANI/Fe3O4 composite with Elovich model and intra-particle di ffusion model.
