## 3 Results and Discussion
### Nonlinear Fitting  
The nonlinear method is a draft process in which a trial-and-error action is functional under the testing conditions. With contemporary technological progress, the processing of nonlinear regression has become effective using computer-based solving features. The present study was carried out to perform nonlinear data fitting through Microsoft Excel's solver add-in with a spreadsheet. The curve fitting results for the experimental and predicted equilibrium data of all five isotherm models using nonlinear methods at consecutive temperatures of 303-333 K are shown in Figure 4, and the parameters obtained from the nonlinear forms are included in Table 4.  
Figure 4: Nonlinear curve fitting of isotherm models at (**a**) 303, (**b**) 313, (**c**) 323, and (**d**) 333 K.  
Since the nonlinear forms use identical \(x\)- and \(y\)-axes for the experimental and calculated equilibria, the data fitting for the nonlinear method eliminates the nonconformity that arise from the iteration of linear transformation [23]. However, statistically nonlinear regression is not error-free either, as the variables of an isotherm equation do not possess the same error structures [30,67]. The constructed fitted curves have distinctive features due to their dissimilarity based on the model principles and empirical predictions (Figure 4a-d). The present study showed comparable data fitting for the Langmuir and Sips models, as they overlapped each other for almost every testing temperature, and the value of \(R^{2}\) remained in the range of 0.973-0.982. A lower coefficient value was observed for the Freundlich isotherm. In contrast, for the R-P and Toth models, with the highest coefficients among all the models, the curves over-imposed and traversed the experimental entity. These anomalies in isotherm model fitting violate an ideal fitting relationship. Moreover, nonlinear regression using \(R^{2}\) cannot represent nonlinear interactions, as the regression coefficient \(R\) describes the intensity and direction of linear relationships between two numerical variables [27]. Thus, the statistical error functions are more reliable for the development and comparison of suitable nonlinear regression methods. In a single-component isotherm system, statistical error functions evaluate the equilibrium data fitting of the isotherm by minimizing or maximizing the errors distributed based on the data convergences. In the present study, for all the isotherm models studied at the tested temperatures (303-333 K), we carried out nonlinear curve fitting along with five error functions to analyze the adsorption systems. The interpretation of the functions and the method of data regression are presented in Supplementary Table S3. Figure 5 compiles the trends of the experimental and derived isotherms for all four solution temperatures, minimizing the error distribution using the error functions depicted in Table S3. It can be seen from the figures (Figures 5-9) that the curve fitting for a particular isotherm varies depending on the chosen error functions. This is because the fitted curves have distinct characteristics based on the model principles and empirical predictions. It can be noted for the Langmuir isotherm (Figure 5) that the lower variance of this theoretical isotherm derived using the error function indicates a better fit, while the [PERSON] isotherm (Figure 6) shows poor fitting compared to the other models.