## 3 Results
### Adsorption Isotherms--Modeling Examination  
The data of adsorption of DB15 dye by HNT were analyzed by applying the adsorption isotherm models proposed by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON], isotherm models. To refine the outcomes and to make a differentiation between practically comparable information acquired by different isotherm models, SSE and \(\chi^{2}\) as two extra blunder capacities were consolidated in our examination.  
[PERSON] [27] proposed a model in which the adsorbent was accepted to have dynamic locales having practically uniform energies. This was additionally settled on the possibility that no horizontal collaboration happens between adsorbed atoms. A level in a two-dimensional diagram with harmony focus (\(C_{e}\)) as a free factor and \(q_{e}\) (Figure 7a) as a reliant variable describes the immersion of the adsorbent's active destinations on a superficial level [28]. This suggests that further adsorption cannot happen, and the possibility of multi-facet adsorption of the color is precluded. The experimental data, \(R^{2}\) = 0.94, \(q_{e}\) = 69.00 mg g\({}^{-1}\), and \(Q_{m}\) = 97.96 mg g\({}^{-1}\) for the DB15-HNT system, indicates a good fit for the Langmuir adsorption isotherm model. The separation factor (\(R_{L}\)) values of 0.035 and 0.419 indicate favorable adsorption of the DB15 dye onto HNT.  
In addition, the adsorption is more likely if the rise in the starting concentrations reduces the RL. This, however, goes against what is seen in Figure 7a. Furthermore, HNT has offered motivation to study different models by variations of 98.96 and 69.00 mg g\({}^{-1}\), \(Q_{m}\)/\(Q_{e}\) ratios.  
The Jovanovic model [29] tries to reduce the deviancies from the Langmuir isotherm Comparing the variance in the values of \(q_{e}\) = 69.00 mg g\({}^{-1}\) and \(Q_{m}\) = 85.67 mg g\({}^{-1}\) with respect to Langmuir isotherm model.  
The adsorption isotherm empirical model proposed by [PERSON] [30] describes adsorption by filling the pore of the adsorbent A value of 110.59 mg g\({}^{-1}\) for \(q_{e}\) con  
Figure 6: Structures of Direct Blue 15 dye in an aqueous medium.  
trasts with the [PERSON] and Jovanovic models (Table 2). Additionally, a value of 0.97 for the correlation coefficient (\(R^{2}\)) confirms that the process of adsorption is linear (Figure 7b).  
Figure 7: Cont.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & \\ \hline \hline
**[PERSON]** & **[PERSON]** & **[PERSON]** & **Dubinin–Radushkevich** \\ \hline \(Q_{m}\) & **97.96** & \(K_{F}\) & 24.17 & \(Q_{m}\) & 85.67 & \(Q_{s}\) & 110.59 \\ \hline \(K_{S}\) & 0.055 & \(n_{F}\) & 3.977 & \(K_{J}\) & 0.046 & \(K_{ad}\) & 4.46 \(\times\) 10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Two-parameter isotherms.  
Figure 7: (**a–d**) The curve fitting was conducted on adsorption data isotherm for DB15–HNT.  
isotherm model [34], respectively. The results are presented in Figure 7c. Radke-Prausnitz isotherm model [35]. The Redlich-Peterson isotherm model [36], the a \({}^{\prime}g^{\prime\prime}\) value of 1.063 as a correction exponent showing similarity to the Langmuir isotherm (see Figure 7d).  
Table 2 represents the results obtained with two-parameter isotherms, while, in Table 3, the parameters were calculated using three-parameter models. Finally, in Table 4, the curve-fitting results calculated for nine isotherm models are represented. Briefly speaking, the graphs plotted based on the nine models showed similar trends, including nonlinear and flat parts.