## 3 Irreversible Thermodynamics Approach
### The Kedem-Katchalsky Equations
The expressions linking \(\kappa\), \(t_{i}\) and \(P\) with the Onsager coefficients are as follows [67]:  
\[\kappa=(z_{+}^{2}L_{+}+z_{-}^{2}L_{-})F^{2} \tag{8}\]  
\[t_{i}=\frac{z_{i}^{2}L_{i}}{z_{+}^{2}L_{+}+z_{-}^{2}L_{-}}=\frac{z_{i}^{2}L_{ i}F^{2}}{\kappa},\ I=+,-, \tag{9}\]  
\[P=\frac{2 RTg}{c}\bigg{[}\frac{\kappa t_{-}t_{+app}}{F^{2}}+(L_{+-}-m_{s}M_{w}L _{-w})\bigg{]}\approx\frac{(z_{+}L_{+}t_{-}+|z_{-}|L_{-}t_{+})RT}{c} \tag{10}\]  
where  
\[L_{+}=L_{++}+(z_{-}/z_{+})L_{+-},L_{-}=L_{--}+(z_{+}/z_{-})L_{+-} \tag{11}\]  
refer to the individual ions \"\(+\)\" and \"\(-\)\" respectively; \(g=1+d\ln y_{\pm}/d\ln c\) is the activity factor written for the virtual solution [67; 78], \(y_{\pm}\) is the mean (molar) activity coefficient of electrolyte in the virtual solution; \(t_{+app}\) is the apparent transport number of counter-ion (cation in a CEM). According to the Scatchard equation, \(t_{+app}\) = \(t_{+}\ -mM_{w}t_{w}\), where \(m\) is the virtual solution molality (in eq/kg H\({}_{2}\)O), \(M_{w}\) is the water molar mass and \(t_{w}\) is the water transport number in the membrane. The frequently applied approximation (presented in Equation (10) by its right-hand side after \"\(\approx\)\") is found when assuming \(L_{+--}\) = \(m_{s}M_{w}L_{-w}\) = 0, \(t_{+app}\) = \(t_{+}\) and \(g\) = 1. Note that the central part of this equation (after sign \"=\") is presented for 1:1 electrolyte, whereas the right-hand side is for any single electrolyte; here \(c\) = \(|z_{i}|c_{i}\) is the electrolyte concentration in the virtual solution expressed in eq L\({}^{-1}\). Similar expressions for \(L_{p}\), \(\beta\) and \(\sigma\) are presented in Ref. [67]. The laborious and delicate experimental determination of Kedem-Katchalsky and Onsager coefficients can be facilitated when using an incomplete set of phenomenological coefficients. For instance, the Staverman coefficient \(\sigma\), which is very close to 1 in IEMs, is rarely used in characterization. Besides, additional approximate relations between the coefficients can be applied [67; 71; 77]. Such a relationship, presented by Equation (12), can be obtained if we assume that the difference between the \(L_{-+}\) coefficient and the \(m_{s}M_{w}L_{-w}\) term in Equation (10) is small compared to the \(L_{--}\) diagonal coefficient [67; 77]:  
\[P=\frac{2 RTxt_{-}t_{+app}g}{F^{2}c} \tag{12}\]  
Equation (12) is a generalization of the Nernst-Einstein equation for the case where the osmotic and electro-osmotic water transport are implicitly taken into account. This and similar equations have been applied and compared with experiment in different conditions [67; 72]. A slight deviation from Equation (12) was found [67] in the case of perfluorinated Nafion(r) 120 and MF-4 SK membranes. It was interpreted as a consequence of a relative hydrophobicity of these membranes according to [PERSON] and [PERSON] [72]. Equation (12) and similar relations are successfully implemented in the characterization of IEMs [67; 71; 73; 75; 79]. In practice, Equation (12) is more convenient to be applied in the form:  
\[t_{-}=\frac{F^{2}Pc}{2 RT\kappa t_{+app}g} \tag{13}\]  
The values of \(P\) and \(\kappa\) are easier to measure than the transport numbers. The apparent transport number, \(t_{+app}\), is found from the membrane potential measurements [55; 67]:  
\[J_{i}=-P\frac{dc_{i}}{dx}+\frac{it_{i}}{z_{i}F}-c_{i}L_{p}(1-\sigma)\frac{dp }{dx} \tag{5}\]0  
where \(E\) is the potential difference (PD), measured between two reversible electrodes situated from both sides of the membrane; \(a_{\pm}^{I}\) and \(a_{\pm}^{II}\) are the electrolyte activities in the left- and right-hand side bathing solutions; \(E_{max}\) is the theoretical value of \(E\) for an ideal membrane not permeable for co-ions. The activity factor, \(g\), can be found from the literature [75; 76; 78].