## 3 Results and Discussion
### Nonlinear Fitting
Figure 5: Experimental data and calculated error functions for Langmuir isotherm fitting at (**a**) 303, (**b**) 313, (**c**) 323, and (**d**) 333 K.  
Figure 6: Experimental data and calculated error functions for Freundlich isotherm fitting at (**a**) 303, (**b**) 313, (**c**) 323, and (**d**) 333 K.  
Figure 8: Experimental data and calculated error function for Sips isotherm fitting at (**a**) 303, (**b**) 313, (**c**) 323, and (**d**) 333 K.  
Figure 7: Experimental data and calculated error functions for R-P isotherm fitting at (**a**) 303, (**b**) 313, (**c**) 323, and (**d**) 333 K.  
In comparison, the data trends of the R-P and Toth isotherms (Figures 7 and 9) showed better fitting that complies with the experimental data. However, the nonlinear regression functions yielded different residuals, generated from diverse error mechanisms, making it challenging to determine the so-called optimal or best set of isotherm constants [33]. To overcome this complication and enable a meaningful comparison of the error values, a statistical normalizing process called the sum of the normalized errors (SNE) is used, being a well-established approach among academics [26, 27]. The detailed method for SNE calculation is presented in the Supplementary Materials. The results obtained from the error analysis and normalized error values are presented in Supplementary Tables S4-S8. The bold numbers represent the dataset's minimum error values and minimum SNE. From the error analysis, we can see that the Freundlich and R-P isotherms have the lowest SNE values for all the error functions among all the temperatures (Tables S5 and S6, respectively). The lowest SNE value for the nonlinear regression of the Langmuir (Table S4) and Freundlich (Table S5) isotherms is observed for SSE and HYBRID, respectively, while for the R-P, Toth, and Sips models, both SSE and HYBRID were identical in most of the corresponding datasets (Tables S6-S8). Hence, it can be concluded that the SSE and HYBRID regression functions better represent the curve fitting and residual analysis for MB adsorption on AG. However, the curve fitting accuracy can be explained by examining the model's principle. To speculate the goodness of fit that satisfies the theory of isotherm models based on optimized error functions (SSE and HYBRID), the curves fitted to the experimental observation were plotted for all the studied models, as shown in Figures 10 and 11. Figure 9: Experimental data and calculated error functions for Toth isotherm fitting at (**a**) 303, (**b**) 313, (**c**) 323, and (**d**) 333 K.  
In particular, the parameter sets of the Freundlich model for all the error functions (Table S5) are significantly high, and the fitted curves (Figures 10 and 11) do not converge  
Figure 11: Experimental and isotherm model data predicted by minimizing the HYBRID error function at **(a)** 303, **(b)** 313, **(c)** 323, and **(d)** 333 K.  
Figure 10: Experimental and isotherm model data predicted by minimizing the SSE error function at (a) 303, (b) 313, (c) 323, and **(d)** 333 K.  
with the experimental data for all the error functions. A much higher \(A\) value (\(A>>1\)) associated with a small \(B\) value (\(B<<1\)) of the R-P isotherm parameter (Table S6) suggests that the isotherms tend to follow the Langmuir model more than the Freundlich model [68, 69]. Meanwhile, the error analysis of the Toth isotherm showed a slightly high SNE value compared to Freundlich and R-P. The obtained parameters provide meaningful information about the model's effectiveness for the present study. A high heterogeneity factor \(n_{T}\) (\(n_{T}>1\)) for the optimal SNE functions signifies the wide distribution of binding energy, with a sufficient availability of adsorption sites on the adsorbent surface, so that chemisorption is the progressive action. Moreover, an \(n_{T}\) value greater than 1 also indicates that the curvature of the isotherm behaves more like the Freundlich type [69].