The potential energy curve and Langmuir isotherm of hydrogen adsorption by a truncated carbon sphere

Abstract

The potential energy curve for the adsorption of a hydrogen molecule by a truncated hollow sphere consisting of carbon atoms was evaluated by using the Lennard–Jones function for the pair interaction between a carbon atom and a hydrogen molecule. The sphere surface was regarded as a continuum with a uniform density identical to that of a graphite layer. The lower limit of the potential was found to be −200 meV when the sphere had a radius of 3.4 Å and no opening. By increasing the radius of the opening to 2.9 Å, the energy barrier for an incoming molecule disappeared and the lower limit increased to −150 meV, which is three times as deep as that observed for a graphite surface. The Langmuir isotherm for the truncated sphere of this size was evaluated based on the eigenvalues of the potential curve. We found that the pressure yielding a half occupancy was <50 bar at a temperature below 250 K. This indicates that a carbon pore with this shape and size can store a hydrogen molecule under mild conditions.

Appendix: List of symbols

Symbols	Description
$\epsilon_{{{\text{C}}{\text{-}}{\text{H}}_{2} }}$	Lennard–Jones potential depth between C and H2
$\epsilon_{n}$	The nth eigenvalue of the potential energy curve for the H2 adsorption
θ	Azimuthal angle of the surface of the truncated carbon sphere
θ a	Azimuthal angle of the opening edge on the truncated carbon sphere
θ(p, T)	Occupancy of the H2 adsorption site given by the Langmuir isotherm
ρ s	Surface density of the carbon sheet
$\sigma_{{{\text{C}}{\text{-}}{\text{H}}_{2} }}$	Lennard–Jones distance parameter between C and H2
a	Opening radius of the truncated carbon sphere
d	Radius of the truncated carbon sphere or the distance between H2 and an infinite single carbon sheet
d e	Equilibrium radius of the closed carbon sphere for the H2 adsorption
d g	Gravimetric H2 density of the substrate
dσ	Surface element of the carbon surface
d sg	Equilibrium distance between H2 and an infinite single carbon sheet
d v	Volumetric H2 density of the substrate
D	Potential depth of the H2 adsorption by the closed carbon sphere
$\overline{{N_{0} }} (T)$	Lower limit of the number density of the H2 adsorption sites
$\overline{{N_{\text{site}} }}$	Number density of the H2 adsorption sites
$\overline{{N_{{{\text{H}}_{2} }}^{\text{ads}} }} (p, T)$	Number density of adsorbed H2 molecules in the substrate
$\overline{{N_{{{\text{H}}_{2} }}^{\text{gas}} }} \left( {p, T} \right)$	Number density of gaseous H2 molecules
p 0(T)	Half-occupancy pressure of the Langmuir isotherm for the H2 adsorption
$p_{ \hbox{max} } (T)$	Pressure at which the substrate has the same H2 density as the gas
r	Distance between C and H2
z	Position of the H2 molecule adsorbed by the truncated carbon sphere
z ads(T)	Molecular partition function of the adsorbed H2 molecule
z gas(T)	Molecular partition function of the gaseous H2 molecule per unit volume
$V_{\text{LJ}} \left( r \right)$	Lennard–Jones potential
W	Integral form of the H2 adsorption energy
$W_{\text{sg}} \left( d \right)$	H2 adsorption energy by an infinite single carbon sheet
W(z, d, a)	H2 adsorption energy by the truncated carbon sphere