*2.6. Adsorption Kinetics*

In order to analyze the mechanism of dye adsorption onto eggshell, and to predict the rate at which a solute (dye) was removed from aqueous solution, three kinetic models could be employed: pseudo-first-order kinetic model [30], pseudo-second-order kinetic model [31], and intraparticle diffusion model [32].

The Lagergren's equation for pseudo-first-order kinetics is given by the following Equation (2) [30]:

$$
\log(q\_\varepsilon - q\_t) = \log q\_\varepsilon - \frac{k\_1}{2.303}t \tag{2}
$$

where *qe* and *qt* are the adsorption capacity at equilibrium and at time (*t*) (mg/g), respectively, and *k*<sup>1</sup> (min<sup>−</sup>1) is the rate constant of this model. From the linear plot of log (*qe* <sup>−</sup> *qt*) versus *<sup>t</sup>*, the rate constant (*k*1) can be obtained by the slope.

The linear form of the Ho and McKay rate equation for pseudo-second-order kinetics is expressed as Equation (3), as follows:

$$\frac{t}{q\_t} = \frac{1}{k\_2 q\_c^2} + \frac{1}{q\_c} t \tag{3}$$

where *qe* and *qt* are the adsorption capacity (mg/g) at equilibrium and at time (*t*), respectively, and *k*<sup>2</sup> (g/mg min) is the rate constant of this model and can be obtained from the intercept and slope of plot *t*/*qt* versus *t* [31].

In the adsorption experiments, it is mandatory to fit the experimental data to the intraparticle diffusion model in order to analyze in depth the adsorption behavior of DB78 on eggshell, and to know the rate-determining step in the liquid adsorption systems. In the adsorption of pollutants onto adsorbents, different stages are differentiated; in the first one, there is a transport of the dye from the solution to the adsorbent surface, followed by the diffusion into the adsorbent, which is usually a slow process [32]. In the diffusion model proposed by Weber and Morris [33], the rate can be expressed in terms of the square root time and can be determined as Equation (4), as follows:

$$q\_t = k\_i \sqrt{t} + \mathbb{C} \tag{4}$$

where *qt* is the adsorption capacity at any time *t* (mg/g); and *ki* is the rate constant of this model (mg/g min1/2) and its values can be calculated from the slopes of plots *qt* versus *t* <sup>1</sup>/2, where *t* is the time (min) and *C* is the intercept (mg/g).
