## 4 Results and Discussion
Figure 11a,b exposes the adsorption/desorption isotherms of CO\({}_{2}\) and CH\({}_{4}\) gases onto the sample S50 at 20, 40, and 60 \({}^{\circ}\)C. The adsorption capability of S50 towards each of CO\({}_{2}\) and CH\({}_{4}\) reduces noticeably when increasing temperature. The symbols and lines represent experimental points and DSL correlations, respectively. It was observed from Figure 11 that the experimental points are well fitted by the DSL model for both CO\({}_{2}\) and CH\({}_{4}\) gases; covering the whole range of pressure and temperature scales as well. Therefore, the DSL model is deemed to be an excellent presentation of experimental data over the whole ranges of pressure and temperature. Trials showed that two homogenous sites (i.e., \(J=2\)) are adequate to define the adsorption of CO\({}_{2}\) or CH\({}_{4}\) gases via the DSL model. The optimum fitting parameters, along with the corresponding \(LSSE\) and \(ARE(\%)\) values deduced from cumulative adsorption data of each component over the entire range temperature, are listed in Table S3 (Supplementary Materials). The unary constituent adsorption of CO\({}_{2}\) and CH\({}_{4}\) gases onto the sample S50 were fitted well by the DSL isotherm model, as described in the previous section. The adsorption equilibria of dual mixtures of CO\({}_{2}\) and CH\({}_{4}\) gases were forecasted utilizing both the PP (Equations (S2) and (S3)) and PN (Equations (S4) and (S5)) behaviors of the DSL model with merely the unary gas adsorption regression parameters listed in Table S3 (Supplementary Materials). No binary data were needed to forecast the dual adsorption equilibrium data. The predicted effects of the mole fraction of CO\({}_{2}\) in the adsorbed phase (\(x_{\text{CO}_{2}}\)) on the  
Figure 11: Experimental (symbols) and DSL fitting (lines) adsorption equilibrium points of (**a**) CO\({}_{2}\) and (**b**) CH\({}_{4}\) gases onto S50 at various temperatures. Solid and empty symbols refer, respectively, to adsorption and desorption points. gas-phase mole fraction of CO\({}_{2}\) (\(y_{\text{CO}_{2}}\)) and selectivity of adsorbing CO\({}_{2}\) relative to CH\({}_{4}\) (\(S_{\text{CO}_{2}\text{CH}_{4}}\)) are illustrated in Figure 12.  
Figure 12a,b show the predicted relationships between \(x_{\text{CO}_{2}}\) and \(y_{\text{CO}_{2}}\) for the PP and PN behaviors, respectively, at various temperatures, and at both minimal and high pressures (0.1 MPa and 0.5 MPa, respectively). It is noticed that \(y_{\text{CO}_{2}}\) increases with increasing \(x_{\text{CO}_{2}}\). However, Figure 12a exposes that the PP behavior always predicts higher \(x_{\text{CO}_{2}}\) than \(y_{\text{CO}_{2}}\) at all temperatures. Therefore, the PP predicts no azeotropic behavior between CO\({}_{2}\) and CH\({}_{4}\) gases at the full scale of temperature. At a high pressure (0.5 MPa), \(y_{\text{CO}_{2}}\) increases very slightly with increasing temperature. At any \(x_{\text{CO}_{2}}\), the value of \(y_{\text{CO}_{2}}\) at 0.5 MPa is always higher than that at 0.1 MPa. It is observed from Figure 12b that at a low temperature (293 K), the PN behavior predicts no azeotropic conditions in the full range of \(x_{\text{CO}_{2}}\) with coinciding curves at both low and high pressures (0.1 and 0.5 MPa, respectively). Furthermore, at medium temperature (313 K), an azeotropic behavior is predicted at \(x_{\text{CO}_{2}}\) of \(\sim\) 0.85 only at high pressure. Furthermore, at an elevated temperature (333 K), azeotropic behaviors are predicted at \(x_{\text{CO}_{2}}\) of \(\sim\)0.55 in states of both minimal and elevated pressures. At any \(x_{\text{CO}_{2}}\), the value of \(y_{\text{CO}_{2}}\) corresponding to the PN behavior (Figure 12b) is higher than that corresponding to the PP behavior (Figure 12a). Figure 12c,d illustrate the predicted relationships between \(x_{\text{CO}_{2}}\) and \(S_{\text{CO}_{2}\text{CH}_{4}}\) for the PP and PN behaviors, respectively, at different temperatures and at both low and high pressures (0.1 MPa and 0.5 MPa, respectively).