## 3 Results and Discussion
### Theory and Model
\(\gamma_{+}^{\text{m}}\gamma_{-}^{\text{m}}\) was calculated by substituting \(\alpha\zeta\), \(\beta\) and the concentration data of \(m_{\text{A}}^{\text{m}}/m_{\text{s}}^{\text{m}}\) in the membrane phase into Equation (26). The agreement between the ion activity coefficients predicted by the modified model and the experimental values is illustrated in Figures 8c and 9c. The predicted model shows that the behavior of ion activity coefficients in ACLP-2%, ACLP-3%, and ACLP-4% AEMs closely aligns with the experimental values, marking a significant improvement  
Figure 10: The empirical expression of \(\beta\) as a function of \(\ln(m_{\text{A}}^{\text{m}}/m_{\text{s}}^{\text{m}})\) is obtained by fitting the experimental data of ACLP-2%, ACLP-3%, and ACLP-4% equilibrium with NaCl solution. According to the hypothesis of the modified model, the mobile salt concentration affects the shielding of the surrounding chain charge. When the value of \(m_{\text{s}}^{\text{m}}\) decreases, the value of \(m_{\text{A}}^{\text{m}}/m_{\text{s}}^{\text{m}}\) increases. In this case, the shielding gradually weakens and shows the chain electric field superposition scenario shown in Figure 7c, which further reduces the ion activity coefficient, the resulting reduction factor is captured by the correction factor \(\beta\); However, in high concentration solutions, the value of \(m_{\text{s}}^{\text{m}}\) will be very large, providing enough shielding for nearby chain charges and making the value of \(m_{\text{A}}^{\text{m}}/m_{\text{s}}^{\text{m}}\) very low, in which case \(\beta=1\) and returning to the situation of Figure 7b. over the original model; the introduction of the parameters \(\alpha\) and \(\beta\) makes the R-square increase from 0.65-0.70 to above 0.94, and the value of SST drops at least one order of magnitude. This result supports the assumption of multiple interactions involving line charges and electric fields to a certain extent. The empirical parameters \(\alpha\) and \(\beta\) succinctly capture the differences in ion non-ideality between ion-exchange membranes and polyelectrolyte solutions. Two commercially available ion-exchange membranes AR103 (\(\xi=2.21\)) and CR61 (\(\xi=1.83\)), which were known linear charge densities [10], were used to further explore the applicability of the modified model. Equation (26) and the above algorithm were used to estimate the ionic activity coefficient values of NaCl in the membrane, and these data were compared with the experimental values and the values predicted by [PERSON]'s original model. The results are shown in Figure 11.  
The experimental methods and data processing are the same as in the previous paper for AR103 as a typical AEM similar in structure to ACLP. However, for the cation-exchange membrane (CEM) CR61; Na\({}^{+}\) is the counter-ion and Cl\({}^{-}\) is the co-ion, but this difference does not affect the treatment of [PERSON] theory and the improved model because the model theory actually abstracts any situation as an interaction between line charges and point charges and does not need to know the specific chemical composition information about line charges and point charges; this means that the model has the theoretical potential to be generalized for ion-exchange membranes that are not specially designed and contain only a polymer chain structure with uniformly distributed charged fixed groups. For the moment, preliminary experimental results confirm this view, and a reasonable agreement was observed between the predicted values of the modified model shown in Figure 11 and the experimental values, where only the \(\xi\) was changed. In addition, the fitting correction parameter \(\beta\) has not yet been derived theoretically, in a simple, easy-to-manipulate way to induce the trend of the ionic activity coefficient of certain components of solutes, which differ significantly in crosslinking degree, moisture content, and fixed group density.