## 3 Theory
### Equilibrium Adsorption Isotherms  
The forecasting of unary and binary adsorption equilibria relies on the precision of single component adsorption data and their description with a reliable adsorption isotherm model. Adsorption equilibrium data of the binary gas system (CO\({}_{2}\) and CH\({}_{4}\)) is forecasted in this work from single component (unary) adsorption isotherm fitting parameters by applying the extended dual-site Langmuir (DSL) isotherm model.  
The unary (single- constituent) DSL model [30] describes the adsorption of constituent \(i\) onto a nonhomogeneous surface that consists of two homogeneous, but energetically distinct, sites. Supposing that the adsorbent-adsorbate free energy onto each site is fixed, the adsorbed quantity of constituent \(i\) is determined from Equation (1)  
\[n_{i}=\left(\frac{n_{1,i}^{s}b_{1,i}P}{1+b_{1,i}P}\right)_{\text{site 1}}+\left(\frac{n_{2,i}^{s}b_{2,i}P}{1+b_{2,i}P}\right)_{\text{site 2}} \tag{1}\]  
where \(n_{1,i}^{s}\) and \(b_{1,i}\) correspond, respectively, to the monolayer saturation capacity limit and adsorption affinity on site 1; \(n_{2,i}^{s}\) and \(b_{2,i}\) are, correspondingly, the monolayer saturation limit and adsorption affinity on site 2, and \(P\) is pressure. The hypotheses of the traditional Langmuir model are considered valid to each site, and the two sites are assumed not to interact with each other [33]. In this formulation, the monolayer saturation limit for every constituent on each site is let to be dissimilar. The affinity parameter for the adsorption of component \(i\) is obtained from Equation (2)  
\[b_{j,i}=b_{j,i_{0}}\exp\left(\frac{E_{j,i}}{RT}\right) \tag{2}\]  
where \(j\) refers to the free-energy level (site 1 or site 2), \(E_{j,i}\) is the adsorption energy of constituent \(i\) on site \(j\), \(b_{j,i_{0}}\) is the pre-exponential factor (or adsorption affinity of constituent \(i\) on site \(j\) at infinite temperature, \(T\)) and \(R\) is the universal gas constant. The corresponding [PERSON]'s law constant can be estimated from Equation (3) as  
\[H_{i}=\lim_{P\to 0}\frac{n_{i}}{P}=\left(n_{1,i}^{s}b_{1,i}P\right)_{\text{ site 1}}+\left(n_{2,i}^{s}b_{2,i}P\right)_{\text{site 2}} \tag{3}\]  
In this formulation, \(j=1\) always refers to the adsorption site with a higher free energy level, and \(j=2\) always refers to the adsorption site with a lower free energy level. For unary gas adsorption, this causes the free energy of site 1 always superior than that of site 2.  
On the other hand, for binary gas adsorption, the two free energies can be attributed to either site 1 or site 2.  
For the extended DSL model, thermodynamic consistency necessitates the overall saturation capacity (\(n_{1,i}^{s}+n_{2,i}^{s}\)) to be similar for all the constituents [34]. Equivalently, the amounts \(n_{1,i}^{s}\) and \(n_{2,i}^{s}\) in extended DSL would not depend on the constituent [35]. Unary constituent adsorption equilibrium data for each of CO\({}_{2}\) and CH\({}_{4}\) gases were fitted to the DSL isotherm model (Equations (1) and (2)) by the method of the least sum of squared errors (\(LSSE\)) as shown in Equation (4). Moreover, the percentage average relative error (\(ARE\), %) between the measured and correlated quantities adsorbed is assessed from Equation (5).  
\[LSSE=minimum\left\{\sum_{i}^{N_{P}}\bigl{(}n_{cal}-n_{exp}\bigr{)}_{i}^{2}\right\} \tag{4}\]  
\[ARE(\%)=\frac{100\%}{N_{P}}\sum_{i}^{N_{P}}\biggl{|}\frac{n_{cal}-n_{exp}}{n _{exp}}\biggr{|}_{i} \tag{5}\]  
where \(i\) refers to the data point number, \(N_{P}\) refers to the total number of data points for every constituent, and the subscripts \"_exp_\" and \"_cal_\" symbolize, correspondingly, the experimental and correlated amounts. It is important to mention that the theoretical background of binary equilibrium adsorption isotherm is illustrated in the Supplementary Materials.