## 3 Results and Discussion
### Theory and Model
Through the original model, however, after the introduction of a, the R-square increases to 0.80-0.90, and the Sum of the Square's Total Variance (SST) value decreases by about two to four times. However, a considerable deviation from the actual value remains at saturation concentration. From the perspective of solution chemistry, the non-ideality caused by ion hydration is particularly significant in high-concentration solutions, where the solvent content is reduced, leading to an effective concentration that is higher than the actual concentration [63; 64; 65; 66; 67]. This factor is not accounted for in the framework of [PERSON]'s theoretical model [28; 29]. Therefore, it will be crucial to consider corrections based on hydration numbers in subsequent research. \[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]7 Ion activity coefficients in ion-exchange membranes as a function of external NaBr concentration obtained experimentally (ACLP-2% (\(\bullet\)), ACLP-3% (\(\blacklozenge\)), and ACLP-4% (\(\blacksquare\))) and predicted by [PERSON]’s limiting laws (ACLP-2% (yellow dashed line), ACLP-3% (teal dashed line), and ACLP-4% (indigo dash-dotted line)). As shown in (**a**), (**b**), (**c**), respectively, represent the introduction of correction parameter \(\alpha\); And the prediction results after introducing correction parameters \(\alpha\) and \(\beta\). As previously discussed, the deviation in the distribution between 0.003 mol\(\cdot\)kg\({}^{-1}\) and 0.1 mol\(\cdot\)kg\({}^{-1}\) may be attributed to the stabilization of the fixed charge concentration alongside the continued reduction in the mobile salt electrolyte concentration within the membrane. The right limit of [PERSON]'s theory--corresponding to the minimum predicted range of the activity coefficient--is not applicable to ion-exchange membranes. This theory assumes that the concentration of the mobile electrolyte in the membrane is at least three orders of magnitude lower than the concentration of the fixed group when the mobile electrolyte concentration is very low and no longer changing. In the context of [PERSON]'s model, this situation may correspond to a hypothetical collapse, where the mobile electrolyte can no longer provide sufficient shielding for the single linear charge chain, leading to the influence of multiple polymer chains on the ions in the membrane phase [10; 28]. Consequently, the ion activity coefficient should reflect the contributions of multiple polymer chains. Based on this hypothesis, according to the additive rule of ion activity coefficients, the relationship between the activity coefficient influenced by multiple polymer chains and the original activity coefficient can be expressed as follows:  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]8  
where \(\left[\gamma_{+}^{\mathrm{m}}\gamma_{-}^{\mathrm{m}}\right]_{\mathrm{mod}}\) represents the ion activity coefficient influenced by the modified multi-segment polymer chain, while \(\gamma_{+}^{\mathrm{m}}\gamma_{-}^{\mathrm{m}}\) denote the activity coefficients resulting from the line-to-point interaction of the single-segment polymer chain in the null hypothesis, the coefficient \(\beta\) serves as the linear electric field superposition correction coefficient. Under this assumption, \(\beta\) should be a function of the fixed group concentration \(m_{\mathrm{A}}^{\mathrm{m}}\) and the mobile salt concentration \(m_{\mathrm{s}}^{\mathrm{m}}\). By fitting the data from the NaCl experimental group shown in Figure 10, the empirical model expression for \(\beta\) in relation to \(m_{\mathrm{A}}^{\mathrm{m}}/m_{\mathrm{s}}^{\mathrm{m}}\) can be derived as follows:  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]9