*3.5. Adsorption Equilibrium*

The adsorption process of DB78 dye by eggshell adsorbent can be analyzed by fitting the experimental data of adsorption equilibrium to different isotherm models to find the most suitable isotherm to describe the adsorption process. Three well-known models were applied in the present study: Freundlich, Langmuir and Temkin isotherms. The constant isotherm parameters obtained from linear regression describe the equilibrium characteristics of adsorption, and are presented in Table 2. The plot of ln*qe* versus ln*Ce* gave a straight line over the entire concentration range studied in the representation of the Freundlich isotherm, as may be observed in Figure 5a.


**Table 2.** Adsorption isotherm constants obtained for eggshell adsorbent.

**Figure 5.** Adsorption isotherms for Direct Blue 78 by eggshell (**a**) Freundlich isotherm, (**b**) Langmuir isotherm, (**c**) separation factor, and (**d**) Temkin isotherm.

Thus, the straight line obtained was used to calculate the parameters *KF*, *nF*, and *R*2. In this case, the *KF* was 3.02 and *nF* was 3.4 (Table 2). The process is favourable when the *nF* value was found in the range between 1 and 10, which was confirmed for eggshell. A good linear determination coefficient (*R*<sup>2</sup> = 0.991) (Table 2) was obtained. The Freundlich model was the most suitable to describe the adsorption process owing to the high determination coefficient obtained. The linear form of the Langmuir isotherm was obtained by plotting *Ce*/*qe* versus *Ce*, giving a straight line (Figure 5b). The Langmuir isotherm constants *aL*/*KL* and 1/*KL* were determined from the slope and intercept of plot (*Ce*/*qe* vs. *Ce*), respectively. *KL*/*aL* is *qmax* parameter, which is the maximum adsorption capacity of the adsorbent (mg/g) (Figure 5b). The parameters and constants obtained for Langmuir can be observed in Table 2. The maximum adsorption capacity of eggshell was 13 mg/g. Moreover, the value of *R*<sup>2</sup> was 0.975, but this value was lower than the Freundlich isotherm value.

This *qmax* value was similar to that of other adsorbents, for example, using chitosan in the adsorption of DB78 [15]; in the removal of various dyes using cellulose by-products, banana, and orange peels [50]; or in the adsorption of methylene blue dye using raw olive stone [20].

According to the results obtained, the Freundlich model fitted better than the Langmuir model. This may demonstrate the presence of heterogeneous adsorption sites on the eggshell surfaces. Similar results were analyzed for the adsorption of Reactive Red 123 dye using eggshell, in which the Freundlich and Langmuir isotherm models provided excellent fit with the highest *R*<sup>2</sup> value [49]. In another study, Pramanpol and Nitayapat carried out the adsorption of Reactive Yellow 205 dye using various components of eggshells and found a good concordance with the Freundlich model [51].

In the Langmuir isotherm, the value of *RL* determines if the adsorption process is favourable or unfavourable. A value of *RL* in the range between 0 and 1 indicates that the process is favourable. The values obtained for eggshell ranged from 0.281 to 0.032, confirming the adsorption process was favourable (Figure 5c). At low concentrations, the highest *RL* values are also observed, indicating that the adsorption is more favourable at those concentrations.

The plot of *qe* versus ln *Ce* shows the representation of the Temkin isotherm (Figure 5d). The Temkin constants *bT* and *aT* were determined from the slope and intercept, respectively. The *bT* value obtained was 1.41 and the *aT* value was 6.73, as shown in Table 2. A positive value of *bT* indicates that physical and chemical forces were involved in the adsorption process. The value of *R*<sup>2</sup> was 0.951. The obtained result is lower than the Freundlich and Langmuir determination coefficients, with the Freundlich model being the best isotherm to explain the experimental results.
