Fugacity

For a classical non-condensing real gas in external potential, the number density of the gas is given by a virial expansion[35]:

$$\jot9pt \frac{n}{V}=\sum_{j \geq 1}jb_{j} z^{j} ,$$ (1.5)

where $n$ is the number of particles, $V$ is the volume of the system, and $z=\frac{f}{kT}$ is the activity of the gas, where $f$ is the gas fugacity. The first two cluster integrals $b_{j}$ in the expansion 1.5 are given by[35]:

$$\jot9pt b_{1}=\frac{Z_{1}}{V},$$ (1.6)

$$\jot9pt b_{2}=\frac{1}{2V} \left(Z_{2}-Z_{1}^{2}\right).$$ (1.7)

Assuming that the adsorption potential $v$ is additive with the $N$-particle contribution to the interaction energy $u$, the $N^{th}$ configuration integral Z$_N$ is given by[35]:

$$\jot9pt Z_{N}=\mathop{\int}_{\mathbf{V}} exp\left\{-\frac{1}{kT}\... ...r_{i}\right) +u_{N}\left(r_{1}...r_{N}\right)\right]\right\}\,dr_{1}...dr_{N} .$$ (1.8)

Substituting Eq. 1.8 into 1.7, results to:

$$\jot9pt b_{2}=\frac{1}{2V}\mathop{\int}_{\mathbf{V}}exp\left\{-\f... ...p\left[-\frac{1}{kT}u_{2}\left(r_{1}-r_{2}\right)\right]-1\right\}dr_{1}dr_{2}.$$

If the adsorbing potential $v$ is slowly varying compared to the two-particle interaction energy $u_2$ ($u_2$ decays to zero at large distances) and $u_2$ is isotropic, the integral 1.9 can be replaced by:

$$\jot9pt b_{2}\approx\frac{1}{2V}\mathop{\int}_{\mathbf{V}}exp\lef... ...{2}\left\{exp\left[-\frac{u_2 \left(r_{12}\right)}{kT}\right]-1\right\}dr_{12}.$$

Finally, if the adsorbing potential is slowly varying in the entire integration volume, 1.10 can be further approximated by:

$$\jot9pt b_{2}\approx\frac{1}{2V}{\left\{ \mathop{\int}_{\mathbf{V... ...{2}\left\{exp\left[-\frac{u_2 \left(r_{12}\right)}{kT}\right]-1\right\}dr_{12}.$$

Therefore, the number of particles in the simulation volume $V$ is approximately given by:

$$\jot9pt n \approx {\left\{\frac{f}{kT} \mathop{\int}_{\mathbf{V}}exp\left[-\frac{v\left(r_{1}\right)}{kT}\right]dr_{1}\right\}}$$

$$+ \int^{\infty}_{0}4{\pi}r_{12}^{2}\left\{exp\left[-\frac{u_2 \left... ...t}_{\mathbf{V}}exp\left[-\frac{v\left(r_{1}\right)}{kT}\right]dr_{1}\right\}}^2$$

$$+ ...$$ (1.9)

That is, for a slowly varying adsorbing potential the quantity:

$$\jot9pt f'=f\mathop{\int}_{\mathbf{V}}exp\left(-\frac{v\left(r_{1}\right)}{kT}\right)dr_{1}$$ (1.10)

plays the role of the fugacity. The contributions of the intermolecular interactions within the gas itself are, to the first approximation, the same as in the free gas at the same number density. In the case of the classical ideal gas ($u_2$=0; $f=p$), the integral in Eq.1.13 is the classical equilibrium constant for the ideal gas adsorption. In order to estimate the real-gas adsorption capacity of a nanostructure, there are two quantities required: On one hand this is the ideal gas adsorption constant K$_{eq}$. On the other hand the estimation of the accessible volume within the structure $V_{ads}$, with a slowly varying adsorption potential. Given the external pressure $p_{ext}$, the effective pressure of the absorbed gas inside the nanostructure $-$ the internal pressure $-$ is $p_{int}=K_{eq}\,p_{ext}$. The experimental real gas equation of state, with the internal pressure $p_{ext}$ gives all further information on the guest$-$host system, including the volumetric and the gravimetric storage densities. Thus, despite the ``frozen'' host structure used in the simulation, H2 gas properties are given by the experimental equation of state[36] which covers the conditions for a conventional H2-storage applications.