## 3 Results and Discussion
### Isotherm Characteristics  
The adsorption isotherm is an effective, powerful technique used to analyze the retention process by ion exchange resin. The adsorption isotherm considers the essential necessity for resolving any adsorption system. The characterizations of adsorption isotherms provide the most valuable evidence, which indicates the diffusion mechanism of the adsorbed metal ions from the aqueous to the sorbent material when the equilibrium is attained. The investigation of the isotherm data by matching them to various isotherm patterns are the significant steps in determining a suitable model which can be used for designing the best purpose. The Langmuir isotherm model is one of the most ordinary models that express the equilibrium case and is applied to evaluate the highest efficiency of the adsorbent to shackle the Y(III). It presumed that the adsorption of Y(III) takes place on the homogenous surface of the adsorbent as a saturated monolayer of adsorbed Y(III) at the constant of adsorption energy, and no movement of Y(III) on the surface of the adsorbent material [40]. The Langmuir isotherm model is stated using the next equation:  
\[\frac{\mathrm{C_{e}}}{\mathrm{q_{e}}}\ =\ \frac{1}{\mathrm{q_{max}b}}\ +\ \left( \frac{1}{\mathrm{q_{max}}}\right)\mathrm{C_{e}}, \tag{9}\]  
where \(\mathrm{C_{e}}\) (mg:\(\mathrm{L^{-1}}\)) is the Y(III) concentration in solution at equilibrium, \(\mathrm{q_{e}}\) (mg:\(\mathrm{g^{-1}}\)) is the removal capacity per unit mass of the adsorbent at equilibrium, and \(\mathrm{q_{max}}\) (mg:\(\mathrm{g^{-1}}\)) is the highest adsorbed amount per unit mass of sorbent material (maximum uptake capacity); b (\(\mathrm{L\cdot mg^{-1}}\)) is a Langmuir constant related to the adsorption energy. Figure 3a demonstrates a graph of \(\mathrm{C_{e}}/\mathrm{q_{e}}\) versus \(\mathrm{C_{e}}\). The calculated maximum capacity was 172.41 mg:\(\mathrm{g^{-1}}\) (Table 3); it was much closer to experimental capacity (171.32 mg:\(\mathrm{g^{-1}}\)). In addition, the correlation coefficient (\(\mathrm{R^{2}}\)) value was closer to unity (0.99). It pointed toward the adsorption process of the yttrium ions being consistent with the Langmuir isotherm model and the adsorption of Y(III) being the monolayer on the surface of AHIBC sorbent. The dimensional separation factor (\(\mathrm{R_{L}}\)) is generally expressed in the feasibility of the Langmuir model as the following equation:  
\[\mathrm{R_{L}}\ =\ \frac{1}{1+\mathrm{bC_{r}}}, \tag{10}\]  
where \(\mathrm{C_{r}}\) (mg:\(\mathrm{L^{-1}}\)) is the reference of the Y(III) concentration in solution, and \(\mathrm{C_{r}}\) is generally the uppermost liquid phase concentration encountered in a single adsorption system. As listed in Table 3, the calculated value of \(\mathrm{R_{L}}\) was less than 1 and higher than zero. Consequently, it was a favorable adsorption process. The obtained result illustrated that the Langmuir isotherm model complied with the adsorption process of Y(III). The Freundlich isotherm model designated the yttrium ions adsorbed on the surface of the sorbent material. It was generally applied to study the surface energy, as well as the heterogeneity. It was assumed variable distribution energy for the yttrium ions adsorbed on active sites [41]. The following equation represents Freundlich isotherm:  
\[\mathrm{log}\,\mathrm{q_{e}}\ =\ \mathrm{log}\,\mathrm{K_{f}}\ +\ \frac{1}{ \mathrm{n}}\,\mathrm{log}\,\mathrm{C_{e}}, \tag{11}\]  
where \(\mathrm{q_{e}}\) (mg:\(\mathrm{g^{-1}}\)) is the amount of Y(III) adsorbed at equilibrium, and \(\mathrm{K_{f}}\) (mg:\(\mathrm{g^{-1}}\)) is the Freundlich constant associated with Y(III) adsorption capacity, whereas \(1/\mathrm{n}\) is a constant affiliated with the heterogeneity of the surface. The plot of \(\mathrm{log}\mathrm{q_{e}}\) versus \(\mathrm{log}\mathrm{C_{e}}\) produced a regression line (Figure 3b). The adsorption parameters are presented in Table 2; \(\mathrm{K_{f}}\) value was calculated from the intercept, and the value of \(1/\mathrm{n}\) was calculated from the slope.