# Kinetics of Dye Adsorption

The kinetics of dyes adsorption onto adsorbent materials is prerequisite for choosing the best operating conditions for the full-scale batch process [27]. The study of adsorption dynamics describes the solute uptake rate and evidently this rate controls the residence time of the adsorbate uptake at the solid-solution interface. To design the adsorption system, this solute uptake rate plays the most significant role by determining the residence time required for completing the adsorption reaction and can be enumerated from kinetic analysis. Therefore the adsorption rate is an important factor for a better choice of material to be used as an adsorbent; where the adsorbent should have a large adsorption capacity and a fast adsorption rate [112]. In order to study the mechanism of sorption and potential rate determining steps, most of the adsorption studies used pseudo-first-order and pseudo-second- order models which are presented below:

## Lagergren Pseudo-First-Order and Pseudo-Second Order Kinetic Model

The linearized integral form of the pseudo-first-order model is generally expressed as [124] :

where *q _{t}* and

*q*are the adsorption capacity at time t and at equilibrium, respectively (mg g

_{e}^{-1}),

*k*is the rate constant of pseudo-first-order adsorption (min ') and

_{i}*t*is the contact time (min).

the slope and intercept of a linear plot of log (q_{e} *- q _{t})* versus

*t*will give the value of

*k*and q

_{e }respectively.

Similarly linearized form of Lagergren pseudo-second-order adsorption kinetic is as follows [125]:

where *k _{2}* (g mg

^{-1}min

^{-1}) is pseudo-second-order rate constant of adsorption. A plot between

*t/q*vs.

_{t}*t*gives the value of rate constant

*k*(g/mg min), initial sorption rate

_{2}*h*(mg/g-min) and also

*q*(mg/g) can be calculated.

_{e}The constant *k _{2}* is used to calculate the initial sorption rate h, at tâ€”>0, as follows:

Thus, the rate constant k_{2}, initial adsorption rate *h* and predicted *q _{e}* can be calculated from the plot of

*t/q*versus time

_{t}*t*using Eq. (2).