## 3 Test Data Analysis
### Adsorption Isotherms-Modeling Analysis
Table 2 has a \"\(g\)\" value of 1.638 as the correction exponent, which shows similarity to the Langmuir isotherm (Figure 5d). The data obtained using five three-factor models are tabulated in Table 2. The study of nine isotherm models and the evaluation of statistical parameters were obtained in Table 3. In brief, the graphs obtained by nine models have the similarity that it contains two parts, viz., a nonlinear part and a plateau. The former indicates that the dye molecule adheres to the active site of the porous H.N.T., and the latter characterizes the saturation of the adsorption process. Considering the values of \(R^{2}\), S.S.E., and \(\chi^{2}\), the Redlich-Peterson isotherm model fits best. Figure 6: BG adsorption on HNT fittings for (**a**) 50 \(\upmu\)g/mL, (**b**) 100 \(\upmu\)g/mL, and (**c**) 150 \(\upmu\)g/mL at thermal variations. Analysis of the adsorption kinetics data confirmed multiple levels of linearity, which, in turn, suggests multiple mechanisms. Higher concentrations and higher temperatures lead to higher adsorption rates, which lead to different linear routes. However, the process of adsorption gets stabilized over time. This was observed in the film diffusion model [45]. From Figure 7 and Table 5, the diffusion constant values of \(R^{\prime}\) of a liquid film agree with high \(R^{2}\) values. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Initial Concentration (\(\mu\)g mL\({}^{-1}\))**} & \multirow{2}{*}{**Temp (K)**} & \multicolumn{4}{c}{**Pseudo-First-Order**} & \multicolumn{4}{c}{**Pseudo-Second-Order**} \\ \cline{3-10}  & & \(\mathbf{q_{c}\_{c,\textit{opt}}}\) & \(\mathbf{q_{c,\textit{pred}}}\) & \(\mathbf{q_{c,\textit{pred}}}\) & \(\mathbf{K_{1}}\) & \(\mathbf{R^{2}}\) & \(\mathbf{\chi^{2}}\) & \(\mathbf{q_{c,\textit{pred}}}\) & \(\mathbf{K_{2}}\) & \(\mathbf{R^{2}}\) & \(\mathbf{\chi^{2}}\) \\ \hline \multirow{4}{*}{50} & 303 & 40 & 35.96 & 6.63 \(\times\) 10\({}^{-2}\) & 0.94 & 0.59 & 44.26 & 1.64 \(\times\) 10\({}^{-3}\) & 0.97 & 3.94 \\  & 313 & 44 & 117.53 & 2.28 \(\times\) 10\({}^{-2}\) & 1.00 & 149.57 & 48.08 & 1.79 \(\times\) 10\({}^{-3}\) & 0.96 & 0.32 \\  & 323 & 47 & 45.09 & 5.32 \(\times\) 10\({}^{-2}\) & 0.96 & 0.80 & 57.85 & 9.04 \(\times\) 10\({}^{-4}\) & 0.97 & 0.48 \\ \hline \multirow{4}{*}{100} & 303 & 86 & 38.86 & 1.02 \(\times\) 10\({}^{-1}\) & 0.91 & 8.51 & 44.68 & 3.12 \(\times\) 10\({}^{-3}\) & 0.91 & 0.48 \\  & 313 & 89 & 43.90 & 4.39 \(\times\) 10\({}^{-1}\) & 0.93 & 1.54 & 65.88 & 1.71 \(\times\) 10\({}^{-3}\) & 0.93 & 1.65 \\  & 323 & 93 & 43.32 & 1.07 \(\times\) 10\({}^{-1}\) & 0.94 & 1.58 & 49.70 & 2.94 \(\times\) 10\({}^{-3}\) & 0.94 & 0.76 \\ \hline \multirow{4}{*}{150} & 303 & 110 & 77.79 & 1.95 \(\times\) 10\({}^{-1}\) & 0.91 & 0.36 & 82.43 & 5.33 \(\times\) 10\({}^{-3}\) & 0.95 & 0.07 \\  & 313 & 117 & 85.27 & 4.76 \(\times\) 10\({}^{-2}\) & 0.94 & 1.65 & 111.58 & 4.01 \(\times\) 10\({}^{-4}\) & 0.97 & 0.92 \\ \cline{1-1}  & 323 & 122 & 96.90 & 4.04 \(\times\) 10\({}^{-2}\) & 0.96 & 4.99 & 127.72 & 2.95 \(\times\) 10\({}^{-4}\) & 0.90 & 3.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Unsettled and hypothetically anticipated boundaries for the adsorption energy formats.