## 3 Developing of the Model for Mass Transfer Through GO Membranes  
Graphene oxide is used for the production of highly effective membranes for the purification of water and separation of gas mixtures [2]. The GO membrane can be considered as a composite, comprised of pristine and oxidized parts of the system consisting of graphene sheets. The GOmembranes have two types of pores: the slit-like pores between the GO sheets and the small holes in the sheets related to defects in the GO sheets [2]. The empty space between the non-oxidized regions of GO monolayered sheets can be assumed lying in the range 0.4-0.9 nm. Distance between the point-like defects in the graphene sheets can be estimated as 10.0-13.3 nm [11]. The proposed in the present paper model for the description of mass transfer in GO membranes is related to following main points (which can serve as a kind of the scheme to describe the theoretical model and its novelty): the transport of gas molecules in GO membranes is realized mainly through the slit-like pores formed by graphene sheets. Mass transfer in these pores can be affected by the processes such as adsorption and desorption of gas molecules, surface diffusion of adsorbed molecules on the pore walls, and the gas molecule scattering by the pore walls. The external fields (e.g., resonance laser radiation) can affect the parameters characterizing the aforementioned processes that allows to develop new methods controlling mass transfer in GO membranes without modification of the membranes. The proposed model gives also the ability to explain some experimental results given in [12]. The effective diffusion coefficient for the activated Knudsen diffusion (or in other words translational diffusion) of component i of a gas mixture in the nanoscale slit-like pore of the GO membrane with the flat parallel pore walls \(D_{\text{ie}}\) according to [15] can be written as  
\[D_{\text{ie}}=\frac{1}{3}H_{\text{ie}}v_{\text{i}}\exp\Bigl{(}-\frac{E_{\text {id}}}{kT}\Bigr{)}, \tag{10}\]  
where \(H_{\text{ie}}=H-d_{\text{im}}\) is the effective accessible distance between the pore walls in relation to molecules of component i (the accessible pore width [15]), \(d_{\text{im}}\) is the kinetic diameter of molecule of component i, H is the real width of the slit-like pore, \(E_{\text{id}}\) is the energy barrier for the activated Knudsen diffusion, which depends on the pore width. It is worth noting that the pre-exponential factor in Equation (10) in the case of the cylindrical pore is equal to \(d_{\text{ie}}v_{\text{i}}/3\), where \(d_{\text{ie}}\) is the effective diameter of the cylindrical pore in relation to molecules of the component i (\(d_{\text{ie}}=dc-dim\), where \(d_{\text{c}}\) is the real diameter of the cylindrical pore [16]). It follows from the aforementioned that the greater the molecule size (the kinetic diameter \(d_{\text{im}}\)) the smaller the effective pore diameter and respectively the diffusion coefficient. The value of \(E_{\text{id}}\) in Equation (10) decreases with a decrease in the value of \(d_{\text{im}}/H\). As indicated in [12], the activation energy for the diffusion of gas molecules in the cylindrical pore is small (close to zero) at \(d_{\text{im}}/d_{\text{c}}<0.6\) (in the case of the slit-like pore we should to use the value of \(d_{\text{im}}/H\) instead of \(d_{\text{im}}/d_{\text{c}}\)). It is worth to note that Equation (10) corresponds to the case of diffuse molecule scattering by the pore walls. For the sake of simplicity, further we assume that the value of \(E_{\text{id}}\) in Equation (10) is small (close to zero) and \(H_{\text{ie}}=H\). Under these assumptions and Equation (10), we can estimate the diffusion coefficient of the classic Knudsen diffusion for the slit-like pore as  
\[D_{\text{i,Kn}}=\frac{1}{3}Hv. \tag{11}\]  
The experimentally observed excess of the measured value of the flux of light gas molecules passing through the carbon nanotubes in relation to the value of the flux of gas molecules according to the classic Knudsen diffusion was discussed in [17].