## 3 Test Data Analysis
### Adsorption Isotherms-Modeling Analysis  
The study of the isotherm models wereintended to provide a view of the efficiency of HNT for the remediation of the dye for commercial applications. With a focus on the degree of economic advantages. The adsorption of BG dye onto HNT was evaluated employing the isotherm models of adsorption proposed by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON]. The main criteria of the adsorption isotherm study specified the selection of a model, \(q_{e}\) (test equilibrium), closely equivalent to \(Q_{m}\) (single layer adsorption  
Figure 4: Effect of (**a**) the pH, (**b**) initial dye concentration, (C\({}_{o}\)) with percent removal efficiency, (c) adsorbent dosage, and (**d**) temperature of BG-HNT. capacity), coefficient of determination (\(R^{2}\)) \(\geq\) 0.9. Data accuracy improvement was imbibed by including error correction factors, \(SSE\) and \(\chi^{2}\). [PERSON] [33] assumed uniform energy sites available on the adsorbent, with no transverse interactions among molecules. 2D graph (Figure 5a), of \(C_{e}\) (independent variable) and \(q_{e}\) (dependent variable), flattening specifies energy locality of the adsorbent surface saturation. This condition resists process continuation and rules out the occurrence of sorption in multilayers. The experimental data, \(R^{2}\) = 0.82, \(q_{e}\) = 87.00 mg g\({}^{-1}\), and \(Q_{m}\) = 124.15 mg g\({}^{-1}\) for the BG-HNT system, indicate a close fit to the [PERSON] adsorption isotherm model. The separation factor (\(R_{L}\)) values of 0.039 and 0.247 indicate the favorable adsorption of BG dye onto HNT. Figure 5: _Cont._  
Moreover, the adsorption was considered more favorable if the increase in the initial concentrations decreases the \(R_{L}\) value. This is in contradiction to our results displayed in Figure 5a. Additionally, the variances between the \(Q_{m}\) and \(q_{e}\) values of 124.15 and 87.00 mg g\({}^{-1}\), respectively, for HNT has given the impetus to explore other models. Contrary to the above discussion, [PERSON] professed diversity of surface sites with different adsorption energy and demonstrated relevance to multilayer adsorption [34]. \(n_{F}\) = 4.060 and \(1/n_{F}\) = 0.246 (Table 2) of HNT indicate that the course of adsorption is physical and suits the behavior of the Langmuir isotherm. Figure 5: BG-HNT adsorption fit with (**a**) [PERSON] and [PERSON]; (**b**) [PERSON] and Dubinin-Radushkevich; (**c**) [PERSON] and Brouers-Sotolongo; and (**d**) the [PERSON], Radke-Prausnitz, and Toth adsorption isotherm. The Jovanovic model [35] attempts to minimize the deviancies of the test results from the Langmuir isotherm model by introducing the exponential term \(K_{J}\). On comparing the differences in the values of \(q_{e}=87.00\) mg/g and \(Q_{m}=109.71\) mg/g with the values obtained by the Langmuir isotherm model, one may surmise that the values obtained in the Jovanovic isotherm were better compared to the Langmuir isotherm model. The adsorption isotherm empirical model proposed by [PERSON] [36] describes adsorption process through the filling of the pore of the adsorbent. \(q_{e}=110.59\) mg/g, comparatively with the [PERSON] and Jovanovic models, is more (Table 2). Furthermore, a value of 0.97 coefficient of determination (\(R^{2}\)) confirms that the adsorption process is linear (Figure 5b). To identify specific model(s) with a smaller gap between experimental \(q_{e}\) and Qm values, five isotherm models are Radke-Prausnitz, Vieth-Sladek, Redlich-Peterson, Brouers-Sotolongo, and Toth--were also studied. The importance of these models is described by us elsewhere [37]. To describe the heterogeneous adsorption system, the empirical mathematical equation was developed by [PERSON] [38]. Brouers-Sotolongo isotherm model [39] and the Vieth-Sladek isotherm model [40]. The results are presented in Figure 5c. The isotherm model of Radke-Prausnitz [41]. The Redlich-Peterson isotherm model [42].