2.5 Adsorption Isotherms Models

In order to explore the surfactants adsorption rules, the adsorption models were used, and the four adsorption isotherms were fitted.

Langmuir model was used to describe surfactants adsorption behavior onto calcium carbonate surface, which was expressed by as Eq. 2 (Ahmadi and Shadizadeh 2012; Monfared et al., 2015), where q_e is the quantity of the surfactant adsorption at the equilibrium per unit mass of calcite (mg/g), Q_m is the maximum uptake capacity (mg/g), Ce is the surfactant equilibrium concentration in solution (mg/L), K_L is Langmuir constant related to the adsorption site (mg/L).

The Langmuir model assumptions were as follows (Barati et al., 2016): The calcite surface should be considered homogeneous. Besides, surfactants molecules should have equal molar surface areas. Bulk and surface phase show an ideal behavior.

Freundlich model was based on the assumption that the calcium carbonate surface had a heterogeneous surface and different classes of adsorption sites covered the surface (Arabloo et al., 2015; Barati et al., 2016).

Freundlich model is expressed by Eq. 3 (Bera et al., 2013), where q_e is the quantity of the surfactants adsorption at the equilibrium per unit mass of calcium carbonate surface (mg/g), Ce is the surfactant equilibrium concentration in solution (mg/L), K_F Freundlich constants (L/mg), which showed the surfactants adsorption capacity, n is related to the adsorption intensity.

The assumption of the Temkin model was that during the adsorption process the heat of adsorption decreases linearly and is not a function of logarithmic (Saxena et al., 2018). The Temkin model is expressed as the Eq. 4 (Ahmadi and Shadizadeh 2015; Saxena et al., 2018), where q_e is the quantity of the surfactant adsorption at the equilibrium per unit mass of calcium carbonate (mg/g), Ce is the surfactant equilibrium concentration in solution (mg/L), B is the Temkin constant, K_T is the equilibrium binding (L/mg), respectively.

The Linear model was the most simplified model, which was expressed as Eq. 5 (Barati et al., 2016), where q_e is the quantity of the surfactant adsorption at the equilibrium per unit mass of calcium carbonate (mg/g), K_H was the linear constant (L/mg), Ce is the surfactant equilibrium concentration in solution (mg/L),