Abstract. The present paper deals with a numerical solution of the two-dimensional problem of helium and methane molecules motion through an ultra-thin layer of a porous material composed of spherical nanoparticles of the same size. The interaction potential "nanoparticlemolecule" is obtained by integrating paired molecular interactions over the nanoparticle volume. Using the method of classical molecular dynamics, permeability of a layer having the size of about 10 -8 m is studied.

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The problem of helium isotope separation via bi-layer membranes of graphitic carbon nitride g-C3N4 has been studied. The probability of passing isotopes through the membrane is derived from solving the Schrödinger integral equation using Hermite polynomials. The potential energy of the membrane is calculated based on modified Lennard-Johnes potential. The separation degree of the 3He/4He reaches the value of 1045 due to the resonant effect.

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Nanoporous membranes are widely used in various fields, such as gas separation, water purification, catalytic processes, and the use of batteries in electrodes. Nowadays, hollow carbon spheres or nanowires are attracting attention of researchers and experimenters due to high adjustability of their mechanical and chemical properties. This makes it possible, among other things, to more accurately adjust permeability of membranes created from this material for various atoms and molecules, which ensures a good degree of gas separation. The mathematical simulation of gas separation via nanocapsule and hollow nanowire porous membrane is performed. Research has shown that such membranes are able to separate He/$$\text{CH}_4$$ CH 4 /$$\text{H}_2$$ H 2 /$$\text{N}_2$$ N 2 gas mixtures.