###### Abstract  
The theory of electrolyte solution provides a precise description of the thermodynamic state and non-ideality of electrolyte solutions, allowing for the accurate prediction of the crystallization separation process of Salt Lake brine. Analogously, we attempt to describe the non-ideality of ions in ion-exchange polymers based on [PERSON]'s Counterion Condensation Theory, which was originally used to describe the thermodynamics of polyelectrolyte solutions, has amply proven the potential to extend to ion-exchange polymers. In this article, equilibrium solvent and solute concentrations in aminated cross-linked polystyrene AEM were determined experimentally as a function of external NaCl concentration, and ion activity coefficients in the membranes were obtained via a thermodynamic treatment. With the recombination and empirical parameters added to [PERSON]'s model, the ion activity coefficient of NaCl and NaBr in the aminated cross-linked polystyrene AEM can be accurately described in concentration ranges of 0.01 mol\(\cdot\)kg\({}^{-1}\)-3 mol\(\cdot\)kg\({}^{-1}\). Compared with the original model, the Coefficient of Determination between the improved model and the experimental data was increased from 0.65 to 0.95. The Residual Sum of Squares is reduced by about one order of magnitude, significantly improving the Manning model's adaptability when applied to AEM. ion-exchange membrane; ion activity coefficient; ion separation 2024113601390/s/s/11203601390/1203601390  
to the phase equilibrium of salt-water systems. Since the introduction of the Debye-Huckel limiting law in 1923 and the subsequent development of widely used models, such as Pitzer theory and the peNRTL model [14; 15; 16], significant advancements have been made in the accuracy, predictability, and extensibility of thermodynamic theories applicable to salt-water systems [17; 18; 19; 20]. However, many researchers tend to focus on the properties and effects of materials while neglecting the thermodynamic information of ions in ion-exchange polymers, leading to a lack of comprehensive thermodynamic frameworks in the field of separation using ion-exchange materials. In the theory of electrolyte solutions, ion activity coefficients are commonly employed to describe the non-ideality of ions in solution [21]. This non-ideality typically encompasses multiple influences, including long-range electrostatic interactions, short-range repulsion between ions, ionic association effects, and ionic hydration effects [22; 23; 24]. These factors can be summarized as a single dimensionless parameter, \(\gamma\), which is generally a function of component concentration, temperature, and pressure. Similarly, other researchers have sought to characterize the non-ideality of ions in ion-exchange polymers using [PERSON]'s theory [10; 11; 12; 13], originally developed to describe the thermodynamics of polyelectrolyte solutions and its potential for extension to ion-exchange polymers has been well-established. These studies encompass commercial ion-exchange membranes such as CR61, CSE, AXM, and AR103 [10; 11], as well as ion-exchange polymer materials like XLPEGDA and AMPS/PEGDA hydrogels [13; 25]. In these investigations, the ion distribution within these ion-exchange polymers in electrolyte solutions was experimentally determined, and the experimental values of the ion activity coefficients in the swollen polymers were obtained through specific thermodynamic calculations, primarily utilizing the theory of the Donnan equilibrium [26; 27], which is based on the principle of chemical potential equilibrium between two phases. These experimental values were then compared with the theoretical predictions of the Manning model and its derivatives [28; 29]. Generally, these models are effective within a specific range, typically between 0.1 and 1 mol-L\({}^{-1}\)[10; 11; 12; 13].