# Highly Porous Organic Polymers for Hydrogen Fuel Storage
## 2 Theoretical Considerations  
Theoretical background is introduced from two perspectives: first, we present the fundamental thermodynamic requirements for adsorption enthalpy needed to store the maximum amount of hydrogen, as well as releasing/adsorbing H\({}_{2}\) molecules readily at ambient conditions as required by the DOE criteria; and second, we outline the enhancement of interaction between H\({}_{2}\) molecules and the host cavity, using molecular energy level considerations, necessary to achieve the overall adsorption enthalpy required. Probably the most promising theoretical work on the physisorption of H\({}_{2}\) from the thermodynamic perspective was carried out by [PERSON] et al., who were motivated by the conflicting reports on the capacity of H\({}_{2}\) storage in highly porous materials [20]. For instance, an isoreticular metal organic framework (IRMOF) material, IRMOF-8, was first reported to have an H\({}_{2}\) absorption capacity of 2.0 wt % H\({}_{2}\) at 298 K/10 bar [21]. However, subsequent computer simulations from [PERSON] et al. predicted a H\({}_{2}\) uptake of only 0.75 wt % at 77 K/1 bar for IRMOF-8, suggesting a significantly lower capacity [22]. In order to explore the theoretical limit for the maximum delivery of H\({}_{2}\), which is the difference between H\({}_{2}\) in a charged and discharged material, [PERSON] et al. used the Langmuir model to model H\({}_{2}\) adsorption inside macropores/mesopores (pores with diameters of 2-50 nm and of \(>\)50 nm, respectively) of materials to find the amount of H\({}_{2}\) adsorbed under equilibrium conditions, with an equilibrium constant \(K\) as shown in Equation (1). \[n=\frac{KPn_{m}}{1+KP}, \tag{1}\]  
where \(K\) is equilibrium constant of the adsorption, \(P\) is the pressure of H\({}_{2}\) gases, \(n\) is the amount of H\({}_{2}\) adsorbed and \(n_{m}\) is the quantity of active adsorbing sites in a material. So the delivery (\(D\)) of H\({}_{2}\) under two different pressures, \(P_{1}\) and \(P_{2}\), is:  
\[D(K,P_{1},P_{2})=n_{1}-n_{2}=\frac{KP_{1}n_{m}}{1+KP_{1}}-\frac{KP_{2}n_{m}}{1 +KP_{2}}. \tag{2}\]  
When \(P_{1}\) and \(P_{2}\) are fixed, the equilibrium constant \(K\) for the maximum delivery (\(D\)) can be found by setting \(dD/dK=0\) in Equation (2) to give Equation (3):  
\[K=\frac{1}{\sqrt{P_{1}P_{2}}}. \tag{3}\]According to the Gibbs equation:  
\[lnK=-\Delta G^{\circ}/RT=-\left(\Delta H^{\circ}-T\Delta S^{\circ}\right)/RT. \tag{4}\]  
By combining Equations (3) and (4), the optimal adsorption \(\Delta H_{ad}^{\circ}\) for the maximum delivery (\(D\)) is obtained as Equation (5):  
\[\Delta H_{opt}^{\circ}=T\Delta S^{\circ}=\frac{RT}{2}\ln\left(\frac{P_{1}P_{2 }}{P_{0}^{2}}\right). \tag{5}\]  
Equation (5) can be used to search for the optimal \(\Delta H_{ad}^{\circ}\) for different adsorption materials. On the other hand, for the same material, the optimal adsorption temperature \(\Delta T_{opt}\) can be determined by rearranging Equation (5) to give Equation (6):  
\[T_{opt}=\frac{\Delta H^{\circ}}{\left[\Delta S^{\circ}+\left(R/2\right)\ln(P_ {1}P_{2}/P_{o}^{2})\right]}, \tag{6}\]  
where \(\Delta S^{\circ}\approx-8R\) for H\({}_{2}\) adsorption in Langmuir model. If H\({}_{2}\) is charged/discharged at the pressures of 30/1.5 bar and the temperature of 77 K, \(\Delta H_{opt}^{\circ}\) is calculated to be -6.3 kJ/mol, close to -5.0 kJ/mol reported recently for a variety of porous materials at cryogenic temperature [23]. The difference between these values might come from the recharge \(P_{1}\) and discharge pressure \(P_{2}\). However, when storing H\({}_{2}\) under the same pressures, but at 298 K as DOE expects, the calculated \(\Delta H_{opt}^{\circ}\) is -15.1 kJ/mol. This value for the adsorption is optimal with respect to the affinity of hydrogen--strong enough to store a large amount of hydrogen gas at the charging pressure (\(\sim\)30 bar) but weak enough to release most of that hydrogen at the discharge pressure (\(\sim\)1.5 bar).