## 3 Results and Discussion
### Theory and Model
charge is described using the Debye-Huckel approximation, which yields an expression for the system's free energy and the product of the activity coefficients of ions in the membrane, expressed as follows [10; 28]:  
\[\gamma_{+}^{\mathrm{m}}\gamma_{-}^{\mathrm{m}}=\left(\frac{\frac{m_{\mathrm{A}} ^{\mathrm{m}}}{m_{\mathrm{A}}^{\mathrm{m}}}+1}{\frac{m_{\mathrm{A}}^{\mathrm{m }}}{m_{\mathrm{A}}^{\mathrm{m}}}+1}\right)\exp\left[\frac{-\frac{m_{\mathrm{A}} ^{\mathrm{m}}}{m_{\mathrm{A}}^{\mathrm{m}}}}{\frac{m_{\mathrm{A}}^{\mathrm{m} }}{m_{\mathrm{A}}^{\mathrm{m}}}+2\zeta}\right] \tag{13}\]  
For the linear polymer, the calculation of value \(\zeta\) is clear and determined [47]: However, for ion-exchange membranes, the dense cross-link structure and the volume ratio of the polymer in the medium complicate the discussion of \(b\) and \(\varepsilon\). Here, we will temporarily refer to an unverified calculation method proposed by [PERSON] et al. [10; 60]:  
\[b=l(1+\frac{x_{\mathrm{n}}}{x_{\mathrm{e}}}) \tag{14}\]  
\[\varepsilon=\varepsilon_{\mathrm{p}}\mathcal{X}_{\mathrm{p}}+\varepsilon_{ \mathrm{w}}\mathcal{X}_{\mathrm{w}} \tag{15}\]  
where \(x_{\mathrm{n}}\) is the fraction of neutral monomers among the total number of monomers in ion-exchange polymers, and \(x_{\mathrm{e}}\) is the fraction of charged monomers. \(l\) is the projection length of the polystyrene monomer, typically 2.5 A; \(\varepsilon_{\mathrm{p}}\) is the relative dielectric constant of ion-exchange polymers, which is generally around 6 for the polystyrene type; \(\varepsilon_{\mathrm{w}}\) is the relative permittivity of water, measured at 78.38. The values of \(\mathcal{X}_{\mathrm{p}}\) and \(\mathcal{X}_{\mathrm{w}}\) represent the volume fraction of polymers and water, and can be estimated by Equation (16) [61], taken from the equilibrium state of dissolution and swelling in pure water, as it has been demonstrated that changes in the volume fraction do not lead to significant deviations in the results:  
\[\mathcal{X}_{\mathrm{w}}=\frac{W\,\%}{W\,\%+\frac{\rho_{\mathrm{w}}}{\rho_{ \mathrm{p}}}}\]  
\[\mathcal{X}_{\mathrm{p}}{=1\,\mathcal{X}_{\mathrm{w}}} \tag{16}\]  
where \(\rho_{\mathrm{w}}\) is the density of water (about 1 g\(\cdot\)cm\({}^{3}\)); \(\rho_{\mathrm{p}}\) is the density of the polymer (the density of the polymer in this experiment is about 1.06 g\(\cdot\)cm\({}^{-3}\))  
In this estimation scheme, the value of \(b\) can be determined based on the ion-exchange capacity (IEC) of the ion-exchange polymer. The calculated values of \(b\) for ACLP-2%, ACLP-3%, and ACLP-4% are 0.73 nm, 0.70 nm, and 1.00 nm, respectively, using Equation (14). The values of \(\varepsilon\) calculated are 31.84, 24.89, and 17.15, according to Equations (15) and (16). Finally, the calculated values of \(\zeta\) are 2.41, 3.00, and 3.27. Because of the dissolvable swelling effect, the \(m_{\mathrm{A}}^{\mathrm{m}}\) will increase with the increase in the external solution concentration because of the decrease in water uptake; a more accurate calculation of \(m_{\mathrm{A}}^{\mathrm{m}}\) would be by Equation (17). \[m_{\mathrm{A}}^{\mathrm{m}}\,=\,m_{\mathrm{X^{-}}}^{\mathrm{m}}\,-\,m_{ \mathrm{Na^{+}}}^{\mathrm{m}} \tag{17}\]  
Based on the above discussion, the predicted values of the ion activity coefficients within anion-exchange membranes (AEM) derived from [PERSON]'s counter-ion condensation theory are illustrated in Figure 8a and Figure 9a. Overall, there is a reasonable agreement between the predictions made by [PERSON]'s theory and the experimental values. As a straightforward model that does not employ any fitting parameters and considers only the long-range electrostatic forces of line-to-point interactions, it can qualitatively and reasonably describe the trends in ion non-ideality in ion-exchange membranes. This is undoubtedly a significant success, despite the observed deviations from the experimental values.