## 3 Results and Discussion
### Theory and Model
Therefore, we can state that [39; 62]:  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]0  
The \(p(l_{x})\mathrm{d}l_{x}\) for the probability of projection falls between \(l_{x}\) and \(l_{x}+\mathrm{d}l_{x}\); The polar coordinate system is chosen and \(\psi\) is used to represent the angle between the monomer and the \(X\)-axis, then \(\bar{l}_{x}=l\mathrm{cos}\psi\), and \(p(l_{x})\mathrm{d}l_{x}\) is proportional to solid angles that value is \(2\pi\mathrm{sin}\psi\mathrm{d}\psi\)[39; 62]; The sum of the solid angles in all directions is \(4\pi\), Therefore, this equation \(p(l_{x})\mathrm{d}l_{x}=1/2\mathrm{sin}\psi\mathrm{d}\psi\) is feasible, and can be substituted into Equation (20):  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]1  
However, if the root mean square of the projection of a single vector onto the \(X\)-axis is not zero, we can compute it in the same manner [39; 62]:  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]2  
Therefore, the root mean square value of the projection is [39; 62]:  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]3  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]4 (a) In the assumptions of [PERSON]’s theory, the polymer in a dilute polyelectrolyte solution is modeled as a chain of linear charges, with the solvent and counter-ions shielded from the effects of other nearby charge chains [28]. (b) The swollen ion-exchange membrane cannot be regarded as a dilute solution, and obviously cannot be approximated as a linear chain charge. The Rayleigh–[PERSON] model of an irregular free link chain is closer to the real situation [47], The resulting line charge density deviation is corrected by \(\alpha\). (c) The distance between the chain charges of the swelling ion-exchange membrane is much smaller than that of the dilute polyelectrolyte solution, which makes it difficult for the solvent and counter-ions to shield the nearby chain. This shielding may gradually fail at low concentrations, which is reflected by the electric field superposition of multi-segment chains, and leads to the further decrease in the ion activity coefficient that the deviation is corrected by \(\beta\). The parameter is used to correct the linear charge density in [PERSON]'s original model. Since the effective length of the monomer is shortened to \(1/\sqrt{3}\) of the original, the linear charge density increases by \(\sqrt{3}\) times compared with the original, which is the value of \(\alpha\):  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]5  
The \(\xi\) for ACLP-2%, ACLP-3%, and ACLP-4% were fixed to be \(\xi_{\text{mod}}\); respectively, 4.17, 5.20, and 5.66, \(\xi_{\text{mod}}\) were substituted into Equation (13); the predicted values of the ion activity coefficients are within the AEM, as shown in Figures 8b and 9b. \[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]6 Ion activity coefficients in ion-exchange membranes as a function of external NaCl concentration obtained experimentally (ACLP-2% (\(\bullet\)), ACLP-3% (\(\blacklozenge\)), and ACLP-4% (\(\blacksquare\))) and predicted by [PERSON]’s limiting laws (ACLP-2% (red dashed line), ACLP-3% (green dashed line), and ACLP-4% (black 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 \(\beta\). Compared to the original model, the adaptation of line charges significantly improved predictive performance in the external salt solution concentration range of 0.3 to 3 mol\(\cdot\)kg\({}^{-1}\). As shown in Tables 3 and 4, the correlation coefficient between the theoretical value and the actual value of ACLP-2%/3%/4% membrane is only 0.65-0.70.