## 3 Results and Discussion
### Ion Concentration and Activity Coefficient in AEMs
Considering these factors, molality (mol-kg\({}^{-1}\) H\({}_{2}\)O) is a more reasonable choice for describing ion concentration in the membrane. Once the unit selection is established, the standard state for the corresponding unit must also be determined. The ion activity factor essentially describes the deviation of the solution from the ideal state; therefore, the obtained ion activity factor should correspond to the standard state. The fixed groups in the ion-exchange membrane significantly influence the ion activity coefficient, which distinctly differs from that in solution. The ionic activity coefficient of the electrolyte in the membrane decreases with increasing solution concentration and approaches zero, while in the solution, it tends to approach one. This phenomenon is primarily attributed to the very low activity coefficient of counter-ions at low concentrations. To elucidate these trends, Figure 5c,d present the respective ionic activity coefficients of the co-ions and counter-ions in the membrane, obtained using Equation (7). This phenomenon can be reasonably explained by examining the changes in these coefficients with varying external solution concentrations. Considering the additional effect of the electric field from the fixed groups in the membrane on the ion non-ideality in the system, it is observed that, especially at low concentrations, nearly all counter-ions bind to the fixed groups, forming relatively localized ion pairs [5; 50]. This behavior deviates significantly from ideality compared to the standard state of the solution. Supporting evidence for this explanation is found in the relative magnitudes of the three AEMs used in the experiment, where the concentrations of fixed charge groups were regulated during preparation, as shown in Table 2: ACLP-4% > ACLP-3% > ACLP-2%. In contrast, the activity coefficients of counter-ions follow the trend: ACLP-4% < ACLP-3% < ACLP-2%, as illustrated in Figure 5c,d. However, as the concentration of the external solution increases, more mobile electrolyte-contributed cations fill the polymer chains and shield the charges of the fixed groups [10], which could weaken the non-ideality of the counter-ions. Consequently, the counter-ion activity coefficient begins to increase. Conversely, the influence of fixed charged groups in the membrane on the same ions primarily manifests as electrostatic repulsion. This repulsion diminishes the electrostatic attraction energy between anions and cations in the original electrolyte solution, a key factor in solution non-ideality, as described by the classical Debye-Huckel effect [14]. Consequently, the original non-ideality is reduced, leading to an increase in the ion activity coefficient. In such cases, the value of the ion activity coefficient may even exceed 1, as illustrated in Figure 5c,d, where the line \(\gamma_{i}^{\text{m}}=1\) is annotated. In the concentration range presented in the figures, the values of the co-ion activity coefficients remain above this line and tend to approach 1 as the concentration of the external solution increases. Unlike the counter-ions, a similar trend is observed in the activity coefficients of the same ion as in the solution. The differences in crosslinking degrees appear to be insignificant, suggesting that the fixed groups on the polymer chain do not affect co-ions as significantly as they do counter-ions. This can be attributed to the large number of counter-ions binding to the fixed groups. Therefore, despite significant differences in crosslinking degree and fixed group density among ACLP-2%, ACLP-3%, and ACLP-4%, the effects on the same ion remain similar.