## 3 Irreversible Thermodynamics Approach
### The Kedem-Katchalsky Equations  
The system of Kedem-Katchalsky equations [68] is of great interest for the practical description of ion and water transfer through the membranes. These equations in differential form are written as follows [67]:  
\[J_{v}=-L_{p}\left(\frac{dp}{dx}-\sigma RTc_{s}v\frac{d\ln a_{\pm}}{dx}\right) +\beta i=-L_{p}\left(\frac{dp}{dx}-\sigma\frac{d\pi}{dx}\right)+\beta\;i \tag{4}\]  
\[J_{i}=-P\frac{dc_{i}}{dx}+\frac{it_{i}}{z_{i}F}-c_{i}L_{p}(1-\sigma)\frac{dp }{dx} \tag{5}\]  
\[\frac{d\varphi}{dx}=-\frac{i}{\kappa}-\frac{RT}{F}\left(\frac{t_{+}}{z_{+}} \cdot\frac{d\ln a_{+}}{dx}+\frac{t_{-}}{z_{-}}\cdot\frac{d\ln a_{-}}{dx}- \beta c_{s}vF\frac{d\ln a_{\pm}}{dx}\right)-\beta\frac{dp}{dx} \tag{6}\]  
where three thermodynamic fluxes are presented by \(J_{v}\), \(J_{i}\) and \(i\) (which are the volume and solute flux densities and the current density, respectively); \(c_{s}\) and \(c_{i}\) are the molar concentration of salt and ion \(i\), respectively, in the virtual solution of the membrane; \(p\) and \(\pi\) are the hydrostatic and osmotic pressures and \(\phi\) is the electric potential in the virtual solution, respectively; \(\
u=\
u_{+}+\
u_{-}\) is the stoichiometric number; \(d\) is the membrane thickness; subscripts \"+\" and \"\(-\)\" relate to cation and anion, respectively. Equations (4)-(6) can be applied to a CEM and an AEM, in which cations and anions act as counterions, respectively. Three thermodynamic forces in Equations (4)-(6) are as follows: mechanical, \(dp/dx\), electric, \(d\phi/dx\) and chemical. The latter is expressed through the gradient of osmotic pressure or that of concentration, which are linked between them [68]:  
\[\
abla\pi=RTc_{s}\
u\
abla\ln a_{\pm} \tag{7}\]  
where \(a_{\pm}\) is the mean ionic activity of the electrolyte. The major interest of the Kedem-Katchalsky equations is in the fact that their transport coefficients are those usually used in the characterization of membrane transport properties. That is why these six transport coefficients are called practical: hydraulic permeability (\(L_{p}\)); electro-osmotic permeability (\(\beta\)); diffusion permeability (\(P\)); transport number of the counter-ion (\(t_{+}\)); electrical conductivity (\(\kappa\)); and Staverman reflection coefficient (\(\sigma\)). The meaning of the latter can be understood from considering two limiting cases: \(\sigma=1\), when the membrane completely reflects the solute transported with convective flow across the membrane and \(\sigma=0\), when the retention is zero. The physical meaning of coefficients \(P\), \(\kappa\) and \(L_{p}\) is the same as that in the linear equations of Fick, Ohm and Darcy, respectively. These linear equations are easily derived from Equations (4)-(6), if only one of three driving force is kept non-zero. \(t_{+}\) is the fraction of electric charge carried by cation under the condition that the gradients of concentration and pressure are zero; \(\beta\), the electroosmotic permeability, should be determined as the proportionality coefficient between the volume flux and the current, when the gradients of concentration and pressure are zero. A form of Kedem-Katchalsky equations different from Equations (4)-(6) can be also used [68]: the pressure gradient in Equations (5) and (6) is replaced by the volume flux. This slightly changes the meaning of coefficients \(P\), \(\kappa\) and \(t_{+}\)[68]. Equations (4)-(6) form the basis of membrane characterization [37; 65; 71; 72; 73; 74; 75; 76], since each of the coefficients refers to one of the membrane transport properties. A transport coefficient should be measured when only one driving force is applied. For example, diffusion permeability \(P\) is to be obtained when \(\
abla p=i=0\). More details about measuring protocols can be found in Refs. [37; 67; 76]. As it was mentioned previously, one can pass from the Onsager to the Kedem-Katchalsky equations by a mathematical transformation, since the coefficients in these two systems of equations are linked [67; 77].