## 2 The Theoretical Description of Mass Transfer Through NG Membranes  
Firstly, let us consider the transport of molecules of the gas mixture through the NG membrane. The flux density of the gas molecules of component i passing through the NG membrane \(j_{\rm ir}\) can be presented as [5]. \[j_{\rm ir}=j_{\rm is}+j_{\rm id}, \tag{1}\]  
where the value of \(j_{\rm is}\) is related to adsorption of the gas molecules of component i on the graphene sheet, their diffusion along the graphene sheet to reach the pores and the penetration through the two-dimensional pores to the other membrane side, the value of \(j_{\rm id}\) is related to the direct penetration of gas molecules of component i through the pores of the NG membrane. We assume, similarly to [3], that the rates of adsorption and desorption of gas molecules are much higher than the diffusion rate of adsorbed molecules along the graphene sheet and the penetration rate of the molecules through the pores to the other side of the NG membrane, as well as that the adsorbed phase on the graphene sheet is in equilibrium with the free gas near the considered side of NG membrane. We suppose also that the number density of adsorbed gas molecules on the graphene sheet is sufficiently small and we can use the Henry law for adsorption of gas molecules, as well as that the size of the pores is greater than the size of the gas molecules and these molecules can penetrate through the two-dimensional pores in the NG membrane. The value of \(j_{\rm is}\) is related to the frequency of encounters of adsorbed molecules of component i with the two-dimensional pores on the unit surface of the graphene sheet \(Q_{\rm lip}\) and the probability of transfer (translocation) of the molecules after encounters with the two-dimensional pores onto the other side of the NG membrane (for the sake of simplicity, we assume that the two-dimensional pores are round and have the same diameter). In the case of localized mobile adsorption, for the molecule movement from one site to an adjacent one, the molecule must possess sufficient energy to pass over the energy barrier between the two sites [4]. If this barrier is low compared to the average energy of the molecule, surface diffusion of the adsorbed gas molecules on the graphene sheet can be considered on the basis of the model of the two-dimensional gas [6]. The value of \(Q_{\rm ip}\) can be found similarly to the method used in the theory of heterogeneous nucleation [13; 14]. In the case of localized mobile adsorption (when surface diffusion is realized by jumping mechanism), the value of \(Q_{\rm ip}\) in view of [13] can be written as  
\[Q_{\rm ip}=\pi dn_{\rm p}\frac{1}{4}n_{\rm ia}a_{0}\
u^{\prime}\exp\left(- \frac{E_{\rm is}}{kT}\right)=\frac{\varepsilon_{\rm p}}{d}n_{\rm ia}a_{0}\
u^ {\prime}\exp\left(-\frac{E_{\rm is}}{kT}\right). \tag{2}\]  
where \(d\) is the diameter of the two-dimensional pore, \(n_{\rm p}\) is the number density of the two-dimensional pores on the graphene sheet, \(n_{\rm ia}\) is the number density of the adsorbed gas molecules of component i on the graphene sheet, \(a_{0}\) is the jump distance for the adsorbed molecule (lattice parameter), \(\
u^{\prime}\) is the vibration frequency of the adsorbed molecule (the values \(a_{0}\) and \(\
u^{\prime}\) are assumed usually to be constant), \(E_{\rm is}\) is the activation energy for surface diffusion of adsorbed molecules of component i, \(k\) is the Boltzmann constant, \(T\) is the temperature (we assume further the system to be isothermal), \(\varepsilon_{\rm p}=n_{\rm p}S_{\rm p}\) is the porosity of the NG membrane, where \(s_{\rm p}=\pi d^{2}/4\) is the area occupied by the two-dimensional pore. The flux density of component i passing from one side of the graphene sheet onto its another side \(j_{\rm ia}\) can be written as  
\[j_{\rm ia}=Q_{\rm ip}\exp\left(-\frac{E_{\rm it}}{kT}\right), \tag{3}\]  
where \(E_{\rm it}\) is the effective activation energy for the molecule translocation from one side of the graphene sheet onto its another side.