# Equilibrium Dye Adsorption Isotherms

Adsorption isotherms are crucial in understanding the mechanism of adsorption and the analysis of the isotherm data by fitting them to different isotherm models is an important step to understand the suitable model that can be used for design purposes. Among several isotherm models presented in the literature [139], Langmuir and Freundlich models are the earliest and simplest known relationships describing the adsorption equation [140] and therefore most widely used to describe the adsorption isotherm.

## Langmuir Adsorption Isotherm Model

The Langmuir model of adsorption is obtained on the basis of ideal assumption that the intermolecular forces decrease rapidly with distance, and consequently, predicts the existence of monolayer coverage of adsorbate at the outer surface of the adsorbent, which is assumed to be structurally homogeneous [141]. The activities of the surface sites are proportional to their concentration and the number of sorption sites is fixed; no further adsorption can take place after the saturation point. For solid/liquid systems, the linear form of the Langmuir equation is:

where *q _{e}* is the amount of dye adsorbed at equilibrium time (mg/g),

*C*is equilibrium concentration of dye in solution (mg L

_{e}^{-1}),

*q*is maximum adsorption capacity (mg/g) and

_{m}*K*is isotherm constants for Langmuir (L mg

_{a }^{-1}).

The linearized form of Langmuir isotherm can be written in two different forms:

**A) Langmuir-I Isotherm Model**

The slop and intercept of plot between *C _{e}/q_{e}* vs.

*C*will give

_{e}*q*and

_{m}*K*respectively.

_{a}**B) Langmuir-II Isotherm Model**

In this form *q _{m}* and

*K*are determined from plotted between

_{a}*1/q*vs. 1/C

_{e}_{e}. The separation factor RL can be found from Langmuir isotherm which is:

where *K _{a}* is the Langmuir constant and

*C*is the initial MB dye concentration (ppm).

_{0}R_{l} indicates the nature of adsorption [142] as indicated below:

- • unfavorable R
_{L}> 1; - • linear R
_{L}=1; - • favorable 0 < R
_{L}<1; - • irreversible R
_{L}= 0.