## 3 Results and Discussion
### Isotherm Characteristics
It was clear that the value of \(\mathrm{K_{f}}\) was 49.52 mg\(\cdot\)g\({}^{-1}\); it was lower than the experimental adsorption efficiency of the yttrium ions by AHIBC. The value of R\({}^{2}\) was 0.68, whereas the 1/n was less than 1, indicating that the removal of the yttrium ions by the AHIBC adsorbent was favorable. The result reveals that the Y ions did not cover the adsorbent material heterogeneously. Furthermore, the experimental data were not associated with the Freundlich model. The Dubinin-Radushkevich (D-R) isotherm model distinguishes between both physical and chemical adsorption processes and defines the heterogeneity of adsorption of ions on the surface [42]. The next equation can calculate it:  
\[\mathrm{lnq_{e}}=\mathrm{lnq_{D}}-\mathrm{B_{D}}\,(\epsilon^{2}), \tag{12}\]  
where \(\mathrm{q_{D}}\) (mg g\({}^{-1}\)) is the monolayer capacity of D-R model, \(\mathrm{B_{D}}\) (mol\({}^{2}\cdot\)kJ\({}^{-2}\)) is a constant associated with the energy of sorption process, and \(\epsilon\) is a Polanyi potential correlated to the equilibrium concentration, and it was calculated by the next equation:  
\[\epsilon\ =\ \mathrm{RT}\ \ln(1+\frac{1}{\mathrm{C_{e}}}), \tag{13}\]  
where the value of R is 8.314 J\(\cdot\)mol\({}^{-1}\cdot\)K\({}^{-1}\) (gas constant), and T (K) is the absolute temperature. Figure 3c shows the graph of \(\mathrm{lnq_{e}}\) versus \(\epsilon^{2}\); the values of \(\mathrm{q_{D}}\) and \(\mathrm{B_{D}}\) are listed in Table 3 and are calculated from intercept and slope, respectively. The \(\mathrm{B_{D}}\) constant is the free energy (E) of the adsorption process per mole of the yttrium ions migrated and adsorbed on the surface of AHIBC and could be calculated by the following relation:  
\[\mathrm{E}\ =\ \frac{1}{\sqrt{2\mathrm{B_{D}}}}. \tag{14}\]  
As presented in Table 3, the E value was 0.94 kJ\(\cdot\)mol\({}^{-1}\) assuming the process of adsorption of the yttrium ions on the studied adsorbent was a physical nature [43]. Nevertheless, the correlation coefficient (R\({}^{2}\)) value was 0.789. Thus, the adsorption process of the yttrium ions on the AHIBC adsorbent was not corresponding to the Dubinin-Radushkevich adsorption isotherm. The Temkin adsorption isotherm postulates that the heat of adsorption of all molecules declines linearly with the increment in the concealment of the adsorbent surface. In addition, the adsorption process is described by a consistent distribution of binding energy, up to topmost binding energy [44]. The Temkin isotherm is calculated based on the following equation:  
\[\mathrm{q_{e}}\ =\ \left(\frac{\mathrm{RT}}{\mathrm{b_{T}}}\right)\ln\mathrm{K_{ T}}\ +\ \left(\frac{\mathrm{RT}}{\mathrm{b_{T}}}\right)\ln\mathrm{C_{e}}, \tag{15}\]  
where \(\mathrm{b_{T}}\) (kJ\(\cdot\)mol\({}^{-1}\)) is the Temkin constant associated with the temperature of the adsorption, while K\({}_{T}\) (L-g\({}^{-1}\)) is the equilibrium constant correlated to the utmost binding energy. The constants were calculated from the diagram of \(\mathrm{q_{e}}\) against lnC\({}_{e}\) and were given in Figure 3d and Table 3. It was obvious that the value of R\({}^{2}\) was 0.78, proposing that the yttrium ions adsorption process did not appropriate with the Temkin isotherm model.