# Application of Three-Dimensional Porous Aerogel as Adsorbent for Removal of Textile Dyes from Water
## 3 Results
### Modeling Results  
#### 3.2.1 Adsorption Equilibrium  
The [PERSON] and Freundlich isotherms were fitted and the adsorption characteristics calculated for analyzing the aerogel. The model parameters (Table 3) were calculated by using nonlinear and linear forms of the [PERSON] and Freundlich isotherms. \begin{table}
\begin{tabular}{c c c c c} \hline
**Type of Dye** & **Concentration of Dye, mg/L** & **Type of Adsorbent** & **Adsorption Capacity, mg/g** & **References** \\ \hline Congo red, Naphthol green B, Rhodamine B and Methylene blue & 0.1; 0.5; 1.0; 2.0; 5.0; 10.0; 50.0; 100.0; 200.0 & Cellulose aerogel & 0.013–7.698; 0.020–8.768; 0.024–13.538 & [PERSON] et al., 2024 [20] \\ \hline Methyl orange & 200.0 & Cellulose aerogel & 1013.11 & [PERSON] et al., 2023 [24] \\ \hline Acid green 25 and Crystal violet & 70.0 & Graphene oxide-doped silica aerogel & 20.25; 41.46 & [PERSON] et al., 2023 [25] \\ \hline Methyl orange & 20.0 & Cellulose/polyethyleneimine composite aerogel & 980.39 & [PERSON] et al., 2024 [26] \\ \hline \end{tabular}
\end{table} 
Table 2: Values of adsorption capacity for specific textile dyes and other parameters. The constant values for the linearized models were obtained by using the slope and intercept values of the plot of \(\rm C_{e}/q_{e}\) versus \(\rm C_{e}\) of the Langmuir model for the dye Congo red (Figure 5) and by using the slope and intercept values of the plot of \(\rm log\ q_{e}\) versus \(\rm log\ C_{e}\) of the Freundlich isotherm for the dye Congo red (Figure 6). The maximum adsorption capacity for Congo red obtained by using the linearized Langmuir model was 14.684 mg/g, and that obtained with the nonlinear model was 14.589 mg/g. The obtained values of \(n\) suggest favorable adsorption. The constant values for the nonlinear models were obtained by using the nonlinear curve-fitting method in MS Excel solver 2023. The obtained values of the parameters were used in the models, and the adsorbed amount of adsorbate was calculated. Both (linearized and nonlinear) Langmuir and Freundlich isotherms for Congo red are presented in Figure 7 below. \begin{table}
\begin{tabular}{c c c} \hline
**Model** & **Parameters** & **Determination Coefficient** \\ \hline \multicolumn{3}{c}{Congo red} \\ \hline Linearized Langmuir model & \(q_{m}=14.684\) mg/g & \(R^{2}=0.95\) \\  & \(K_{L}=0.099\) L/mg & \\ Nonlinear Langmuir model & \(q_{m}=14.589\) mg/g & \(R^{2}=0.95\) \\  & \(K_{L}=0.083\) L/mg & \\ Linearized Freundlich model & \(n=1.447\) & \(R^{2}=0.70\) \\  & \(K_{f}=0.955\) mg/g & \(R^{2}=0.70\) \\ Nonlinear Freundlich model & \(n=2.213\) & \(R^{2}=0.95\) \\  & \(K_{f}=1.892\) mg/g & \\ \hline \multicolumn{3}{c}{Naphthol green B} \\ \hline Linearized Langmuir model & \(q_{m}=9.833\) mg/g & \(R^{2}=0.99\) \\  & \(K_{L}=0.041\) L/mg & \(R^{2}=0.99\) \\ Nonlinear Langmuir model & \(q_{m}=11.655\) mg/g & \(R^{2}=0.99\) \\  & \(K_{L}=0.026\) L/mg & \\ Linearized Freundlich model & \(n=1.245\) & \\  & \(K_{f}=0.326\) mg/g & \(R^{2}=0.91\) \\ Nonlinear Freundlich model & \(n=1.653\) & \(R^{2}=0.99\) \\  & \(K_{f}=0.597\) mg/g & \\ \hline \multicolumn{3}{c}{Rhodamine B} \\ \hline Linearized Langmuir model & \(q_{m}=10.320\) mg/g & \(R^{2}=0.98\) \\  & \(K_{L}=0.037\) L/mg & \\ Nonlinear Langmuir model & \(q_{m}=11.627\) mg/g & \(R^{2}=0.99\) \\  & \(K_{L}=0.023\) L/mg & \\ Linearized Freundlich model & \(n=1.326\) & \(R^{2}=0.84\) \\  & \(K_{f}=0.323\) mg/g & \\ Nonlinear Freundlich model & \(n=1.872\) & \\  & \(K_{f}=0.691\) mg/g & \(R^{2}=0.99\) \\ \hline \multicolumn{3}{c}{Methylene blue} \\ \hline Linearized Langmuir model & \(q_{m}=15.798\) mg/g & \(R^{2}=0.98\) \\  & \(K_{L}=0.052\) L/mg & \\ Nonlinear Langmuir model & \(q_{m}=16.739\) mg/g & \(R^{2}=0.99\) \\  & \(K_{L}=0.039\) L/mg & \\ Linearized Freundlich model & \(n=1.318\) & \\  & \(K_{f}=0.620\) mg/g & \(R^{2}=0.77\) \\ Nonlinear Freundlich model & \(n=1.984\) & \(R^{2}=0.98\) \\  & \(K_{f}=1.407\) mg/g & \\ \hline \end{tabular}
\end{table} 
Table 3: Determination coefficients and parameters calculated for models.