*2.7. Isotherm Analysis*

The interaction between dyes and the adsorbent materials is described using different theoretical models, known as adsorption isotherms. These isotherms are essential in the optimization of the adsorption process [34,35]. Equilibrium isotherm equations were used to describe experimental sorption data [36] and the parameters of equilibrium isotherms provide useful information on adsorption mechanisms, affinity of the adsorbent, and surface properties [20]. Different isotherms were used to analyze the adsorption equilibrium in this study: The Freundlich, Langmuir and Temkin models.

The Freundlich isotherm model suggests heterogeneity in the adsorption sites and takes into account that the adsorption occurs at sites with different energy of adsorption. This isotherm is obtained from the linear form of the Freundlich expression Equation (5) [36]:

$$
\ln q\_{\varepsilon} = \ln K\_{\text{F}} + \frac{1}{n\_{\text{F}}} \ln \mathbb{C}\_{\varepsilon} \tag{5}
$$

where *Ce* (mg/L) and *qe* (mg/g) are the liquid phase concentration and solid phase concentration of dye at equilibrium, respectively; 1/*nF* is the heterogeneity factor; and *KF* is the Freundlich constant (L/g) related to the bonding energy. 1/*nF* and *KF* values were calculated from the slope and intercept of plots ln*qe* versus ln*Ce*, respectively. The values of 1/*nF* indicate the type of adsorption process: favorable (0 < 1/*nF* < 1), unfavorable (1/*nF* >1), or irreversible (1/*nF* = 0) [37].

The Langmuir isotherm model assumes that the adsorption process happens at specific homogeneous sites on the adsorbent. This model is probably the most employed adsorption isotherm and is used successfully for the adsorption of contaminants from water solutions. The linearized form of Langmuir model can be given as follows Equation (6) [33,38]:

$$\frac{C\_{\varepsilon}}{q\_{\varepsilon}} = \frac{1}{K\_{L}} + \frac{a\_{L}}{K\_{L}} C\_{\varepsilon} \tag{6}$$

where *Ce* is the dye concentration at equilibrium in solution (mg/L), *qe* is the adsorption capacity (mg/g) at equilibrium time, and *KL* (L/g) and *aL* (L/mg) are the Langmuir isotherm constants. The constants *KL* and *aL* can be calculated from the intercept (1/*KL*) and the slope (*aL*/*KL*) of the linear plot between *Ce*/*qe*, and *Ce*. *qmax* is the maximum adsorption capacity of adsorbent (mg/g) and is determined by *KL*/*aL*.

The fundamental characteristic of Langmuir isotherm is the separation factor, which is a dimensionless constant (*RL*), and is given as follows Equation (7) [39]:

$$RL = \frac{1}{1 + a\iota \mathcal{C}\_{\bullet}} \tag{7}$$

where *Co* is the initial dye concentration (mg/L) and *aL* is the Langmuir constant related to the energy of adsorption (L/mg). The calculated *RL* values indicate the type of adsorption: unfavourable process (*RL* > 1), linear (*RL* = 1), favourable (*RL* between 0 and 1), or irreversible (*RL* = 0) [40].

The Temkin formula determines that the decrease of adsorption heat with coverage is linear because of some adsorbate/adsorbent interactions. The adsorption is characterized by uniform distribution of bond energies, up to a maximum bond energy [41]. The linear form of Temkin isotherm's equation can be expressed as Equation (8), as follows:

$$q\_{\varepsilon} = \frac{RT}{b\_T} \ln a\_T + \frac{RT}{b\_T} \ln \mathcal{C}\_{\varepsilon} \tag{8}$$

where *T* is the absolute temperature (K); *R* is the universal gas constant (8.314 J/mol K); *aT* is the constant of Temkin isotherm (L/g); *bT* is the Temkin constant related to the heat of adsorption (J/mol); and *qe* and *Ce* are the equilibrium concentration of DB78 on eggshell (mg/g) and in the solution (mg/L), respectively. The Temkin constants *aT* and *bT* values can be calculated from the slope and intercept of straight line plot of *qe* versus ln*Ce.*

## *2.8. Thermodynamic Study*

The thermodynamic analysis is needed to conclude whether the adsorption process of the dye onto eggshell is exothermic or endothermic. In order to gain further insight related to these experiments, it is essential to calculate the value of Δ*H*◦ (standard enthalpy change), Δ*S*◦ (standard entropy change), and Δ*G*◦ (Gibbs standard free energy change). The values for the different thermodynamic parameters can be calculated using the thermodynamic equilibrium coefficient obtained at different concentrations and temperatures. The Δ*G*◦ value is the fundamental parameter to elucidate the spontaneity of the adsorption process, and reactions occur spontaneously when the value of Δ*G*◦ is negative [34].

Considering the exchange adsorption, the equation employed to calculate the *K*◦ value at different temperatures was as follows:

$$K^\circ = K\_p \times M\_{\text{adsorbit}} \times 55.5 \tag{9}$$

where *Kp* is the equilibrium constant at time *t* (L/g), *Madsorbate* is the molecular weight of DB78 (g/mol), and 55.5 (mol/L) is the mole concentration of water [42].

Using the *K*◦ values obtained from the previous equation, the Gibbs free energy was determined using the following equation:

$$
\Delta G^{\circ} = -RT\ln K^{\circ} \tag{10}
$$

To confirm the results for the Gibbs free energy, the Van't Hoff [12] equation was graphed (ln*K*◦ vs. 1/*T*). Plotting ln*K*◦ versus 1/*T* gives a straight line with slope and intercept equal to −Δ*H*◦/*RT* and Δ*S*◦/*R*, respectively, where *R* is the universal gas constant (8.314 J/mol K) and *T* is the absolute temperature in Kelvin (K). Using this representation, the Gibbs free energy was calculated again using the following equation:

$$
\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ \tag{11}
$$
