## 3 Results and Discussion
### Ion Concentration and Activity Coefficient in AEMs
\(\gamma_{i}^{\text{m}}\), \(\gamma_{i}^{s}\) are the ionic activity coefficient of ion \(i\) in the membrane and solution, respectively. \(m_{i}^{\text{m}}\), \(m_{i}^{s}\) are the molarity of ion \(i\) in the molality of membrane and solution, respectively. \(\varphi^{\text{m}}\), \(\varphi^{\text{m}}\) are the potential in the membrane and in solution; \(z_{i}\) is the charge of the ion and \(F\) is the Faraday constant (\(F=96\),\(485\) C-mol\({}^{-1}\)). The standard chemical potential inside and outside the membrane is chosen to be consistent, and the case of monovalent ions in the absence of an electric field is considered in this experiment. The relationship between the activity coefficients is as follows [51; 52]:  
\[\gamma_{i}^{\text{m}}m_{i}^{\text{m}}=\gamma_{i}^{\text{s}}m_{i}^{\text{s}} \tag{7}\]  
If the ionic activity coefficient of the electrolyte is discussed, then:  
\[\left(\gamma_{\text{Na}^{+}}^{\text{m}}m_{\text{Na}^{+}}^{\text{m}}\right) \left(\gamma_{\text{X}^{-}}^{\text{m}}m_{\text{X}^{-}}^{\text{m}}\right)= \left(\gamma_{\text{Na}^{+}}^{\text{s}}m_{\text{Na}^{+}}^{\text{s}}\right) \left(\gamma_{\text{X}^{-}}^{\text{s}}m_{\text{X}^{-}}^{\text{s}}\right) \tag{8}\]  
Figure 4: The molal concentrations of co-ions, \(m_{\text{Na}^{+}}^{\text{m}}\), and counter-ions, \(m_{\text{X}^{-}}^{\text{m}}\) (X = Cl, Br), in the anion-exchange polymer are presented as a function of external electrolyte concentration for NaCl **(a,b)** and NaBr (**c,d**). In solution, the average activity coefficient \(\gamma\,^{\rm s}_{\pm\,\,\rm NaX}\) has the following relationship with the ion activity coefficient [53]:  
\[\gamma\,^{\rm s}_{\pm\,\,\rm NaX}=\left(\gamma^{\rm s}_{\rm Na^{+}}\gamma^{\rm s }_{\rm X^{-}}\right)^{\frac{1}{2}} \tag{9}\]  
Therefore, Equation (8) can be written as follows [10]:  
\[\gamma^{\rm m}_{\rm Na^{+}}\gamma^{\rm m}_{\rm X^{-}}=\frac{\left(\gamma\,^{\rm s }_{\pm\,\rm NaX}\right)^{2}\left(m^{\rm s}_{\rm NaX}\right)^{2}}{m^{\rm m}_{\rm Na ^{+}}m^{\rm m}_{\rm X^{-}}}\;;\;(\rm X=Cl,\;Br) \tag{10}\]  
In the formula, the ion concentrations in the membrane, \(m^{\rm m}_{\rm Na^{+}}\) and \(m^{\rm m}_{\rm X^{-}}\), have been experimentally measured, while the concentration of the external solution, \(m^{\rm s}_{\rm NaX}\), has been predetermined. The average ion activity coefficient \(\gamma\,^{\rm s}_{\pm\,\rm NaX}\) in the external solution is calculated using [PERSON]'s model. Finally, the ion activity coefficient in the membrane is derived, and its dependence on the external solution concentration is illustrated in Figure 5a,b. To facilitate a proper comparison between the electrolyte activity factors in the solution and in the membrane phase, it is essential to emphasize that any unit used to describe ion concentration (e.g., molarity in mol\(\cdot\)L\({}^{-1}\) or amount fraction in \(x\%\)) is theoretically valid. However, in solution thermodynamics, molality (mol\(\cdot\)kg\({}^{-1}\) H\({}_{2}\)O) concentration  
Figure 5: The influence of external electrolyte concentration \(m^{\rm s}_{\rm NaX}\) on the activity coefficients of the membrane in NaCl (**a**) and NaBr (**b**) is presented and obtained experimentally via Equation (10). Additionally, the respective ionic activity coefficients of the co-ions \(\gamma^{\rm m}_{\rm Na^{+}}\). (**c**) and counter-ions \(\gamma^{\rm m}_{\rm X^{-}}\). (**d**) in the membrane were determined experimentally using Equation (7). is more commonly employed, particularly in the Pitzer theoretical model for predicting the activity coefficient. Furthermore, measuring the mass of the solvent absorbed by the membrane is more accurate than measuring the volume; the validity of the concept of \"volume of absorbed solution in IEM\" is still debated, as highlighted by researchers such as [PERSON] and [PERSON] [54; 55; 56]. Using Fourier-transform infrared spectroscopy (FTIR) and differential scanning calorimetry (DSC) [55; 56], they have confirmed that not all water molecules in the membrane phase form hydrogen bonds in the same manner as those in the solution.