# Mitigating Global Methane Emissions Using Metal-Organic Framework Adsorbents
## 2 Materials and Methods  
MOFs with experimentally determined crystal structures, densities, pore volumes, methane isosteric heats of adsorption (Q\({}_{\text{st}}\)), and methane uptakes at 298 K were evaluated for methane capture at low pressures (around 1 bar). When uptakes were not available, they were determined from adsorption isotherms obtained from experiments. A universal descriptor was utilized to calculate the entropy of adsorption [37]. The entropy change of adsorption of methane on a slab was approximated using [PERSON] and [PERSON] estimate [38]:  
\[-\Delta\text{S}_{\text{ads}}(\text{T})=0.3\;\text{S}_{\text{gas}}(\text{T})+3.3\;\text{R} \tag{1}\]  
For methane adsorption to mitigate global methane emission levels, the molecule gains energy that is measured by a change in enthalpy that is associated with interactions with the surface (\(\Delta\text{H}^{\circ}\text{ads}\)). However, the methane molecule will have restricted motion on the surface as compared to the gas phase which results in a loss of entropy:  
\[\Delta\text{G}^{\circ}\text{ads}=\Delta\text{H}^{\circ}\text{ads}-\text{T} \Delta\text{S}^{\circ}\text{ads} \tag{2}\]  
For associative adsorption of methane, the process is not activated and reversible. Adsorption of methane from the gas phase leads to a transition state (TS) of methane at a particular distance such that one degree of translational freedom is lost. For MOFs whose surfaces are curved, adsorbed methane was expected to be further restricted. This further restriction was expected to result in a further loss of entropy for the adsorbed state which makes the process less favorable by resulting in a larger Gibbs free energy of adsorption. Ultimately, thermodynamics suggests that this effect can be mitigated by decreasing the enthalpy of adsorption by designing the MOF to have more favorable interactions with methane. This can be achieved by reticular synthesis to fine tune the pore architecture [39], pore spacing [40], linker [41; 42], functional groups [43; 44], and hydrophobicity [45; 46], and the nature of ligands [47; 48] of the MOF. These features would result in more favorable enthalpies of adsorption which result from van der Waals forces were considered for MOFs. The Sackur-Tetrode equation was used to determine the translation entropy of methane molecules when they behaved as immobile adsorbents to calculate the entropy lost upon adsorption [49; 50; 51]:  
\[\text{S}^{0}_{\text{trans}}(\text{T})=\text{S}_{\text{Ar},298\;\text{K}}{}^{ \circ}+\text{R}\;\ln\left[\left(\frac{\text{m}}{\text{m}_{\text{Ar}}}\right) ^{\frac{3}{2}}\left(\frac{\text{T}}{299}\right)^{\frac{5}{2}}\right] \tag{3}\]  
\(\text{S}_{\text{Ar},298\;\text{K}}{}^{\circ}\) is the standard entropy of argon at standard pressure of 1 bar and temperature of 298 K in the gas phase. At these conditions, its value is 154.8 J mol\({}^{-1}\) K\({}^{-1}\). R is the universal gas constant and has a value of 8.314 J mol\({}^{-1}\) K\({}^{-1}\). m is the molecular weight of methane which is 16.04 g/mol and m\({}_{\text{Ar}}\) is the molecular weight of argon 39.948 g/mol. Finally, T is the temperature in units of Kelvin at which the translational entropy was calculated. When adsorbed methane is not immobile and has one degree of translational motion, the entropy was calculated by S\({}_{\text{trans}}\)/3. To increase the accuracy of the approximation, rotational entropy upon adsorption of methane was also considered on MOFs. This accounted for rotation of methane parallel to the pore surface,but does not include methane rotation perpendicular to the surface.