## 3 Results and Discussion
### Adsorption Kinetics and Isotherm
#### 3.3.2 Adsorption Isotherm  
The model of [PERSON] suggests that adsorbate molecules are retained on the solid surface of the biomaterial in a homogeneous state, resulting in the formation of a single layer (monolayer). The model formula is as follows [42]:  
\[\mathrm{q_{e}}=\frac{\mathrm{q_{max}K_{L}C_{e}}}{1+\mathrm{K_{L}C_{e}}} \tag{9}\]  
The linear form of Equation (9) gives the following equation [47]:  
\[\frac{\mathrm{C_{e}}}{\mathrm{q_{e}}}=\frac{1}{\mathrm{q_{max}}}\mathrm{C_{e} }+\frac{1}{\mathrm{K_{L}q_{max}}} \tag{10}\]  
where \(\mathrm{q_{e}}\) is the adsorption capacity of DR-81 at equilibrium (mg/g), \(\mathrm{q_{max}}\) is the theoretical maximum adsorption capacity of DR-81 (mg/g), and \(\mathrm{C_{e}}\) is the concentration of DR-81 at equilibrium (mg/L). The values of \(\mathrm{q_{max}}\), \(\mathrm{K_{L}}\), \(\mathrm{R_{L}}\), and R\({}^{2}\) are presented in Table 3.  
According to the results in Table 3, there is a convergence between the experimental adsorption capacity and the theoretical adsorption rate calculated by the Langmuir linear equation, with the correlation coefficient R\({}^{2}\) = 0.982, in contrast to the [PERSON] (R\({}^{2}\) = 0.941) and [PERSON] (R\({}^{2}\) = 0.97) models. The Langmuir model is the best model to represent the process of bios \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**PFO**} & \multicolumn{2}{c}{**PSO**} & \multicolumn{2}{c}{**IPD**} \\ \hline
**[DR-81]\({}_{0}\)** & **q\({}_{e}\)exp** & **q\({}_{e}\)cal** & **K\({}_{1}\)** & **R\({}^{2}\)** & **q\({}_{e}\)cal** & **K\({}_{2}\)** & **K\({}_{1}\)** & **C** & **C** \\
**(mg/L)** & **(mg/g)** & **(mg/g)** & **(min\({}^{-1}\))** & **R\({}^{2}\)** & **(mg/g)** & **(g/mg\({}^{-}\)min)** & **R\({}^{2}\)** & **K\({}_{1d}\)** & **(mg/g)** & **R\({}^{2}\)** \\ \hline
10 & 0.91 & 0.174 & 0.067 & 0.875 & 0.92 & 1.18 & 0.9994 & 0.024 & 0.739 & 0.750 \\
20 & 1.86 & 0.471 & 0.057 & 0.917 & 1.89 & 0.28 & 0.9988 & 0.063 & 1.391 & 0.853 \\
40 & 2.58 & 0.128 & 0.034 & 0.168 & 2.57 & 0.15 & 0.9993 & 0.053 & 2.214 & 0.587 \\
60 & 3.49 & 1.303 & 0.065 & 0.936 & 3.62 & 0.08 & 0.9988 & 0.167 & 2.279 & 0.942 \\
80 & 3.8 & 0.429 & 0.045 & 0.548 & 3.90 & 0.07 & 0.9989 & 0.134 & 2.927 & 0.620 \\
100 & 4.38 & 0.799 & 0.040 & 0.909 & 4.36 & 0.05 & 0.9978 & 0.125 & 3.37 & 0.953 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of kinetic models: PFO, PSO, and IPD. \begin{table}
\begin{tabular}{c c c} \hline \hline
**adsorption Model** & **Isotherm Parameters** & **R\({}^{2}\)** \\ \hline [PERSON] & \(\mathrm{d_{max}\,\mathrm{cal}}=4.65\) mg/g & \\  & \(\mathrm{K_{L}(L/mg)}=0.146\) & 0.982 \\  & \(\mathrm{R_{L}}=[0.406{-}0.064]\) & \\ \hline [PERSON] & \(\mathrm{K_{f}}=1.027\) mg/g & \\  & \(1/\mathrm{n}=0.364\) & 0.941 \\ \hline  & \(\mathrm{B_{T}}=0.364\) j/mol & \\ [PERSON] & \(\mathrm{A_{T}}=9.358\) L/g & 0.97 \\  & \(\mathrm{b_{T}}=6734.752\) mg/L & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the isotherm parameters for the biosorption of DR-81 onto cockleshells. In addition to the fact that the Langmuir model is the representative model of adsorption, another analysis to validate the model can be conducted using the separation factor represented by the following relation [48]:  
\[R_{L}=\frac{1}{1+K_{L}C_{0}} \tag{11}\]  
The value of \(R_{L}\) is between 0 and 1. \(R_{L}\) decreased gradually with the increase in concentration, showing that the latter favors the adsorption of DR-81 [49]. The results show that CS is effective in treating industrial textile wastewater and removing cationic and anionic dyes, as well as DR-81. Other criteria for the selection of this bio-adsorbent over other bio-adsorbents are durability, recyclability, and low cost. CS can be considered as a potential candidate for use in the removal of dyes present in liquid industrial textile effluents. Table 4 compares the adsorption capacities for a few biomaterials (modified and unmodified) with the CS used in this study.