## 1 Introduction  
Human activities, particularly the combustion of natural gas and fossil fuels, are regarded as the main causes of global warming and accordingly the increment of carbon dioxide (CO\({}_{2}\)) emissions in the air [1; 2; 3]. Reducing the concentration of this gas before it is transmitted into the air is an urgent issue and has become an important challenge for the scientific community to safeguard the planet's environment. Carbon capture and storage (CCS) technology is regarded as a promising mechanism for the beneficial CO\({}_{2}\) concentration reduction in the atmosphere [4]. Cryogenic separation mechanisms [5], membrane-based processes [6], solid adsorbents adsorption [7] and aqueous amine absorption [8] are also carbon capture technologies that require even more research to remedy the shortcomings of the present step. Especially the synthesis of adsorbents, which must be cheap and appropriate for industrial conditions, and flue gas are main research questions. A broad choice of porous adsorbents is available for usage in pressure/temperature swing adsorption to segregate carbon dioxide from industrial flue gas, such as hydrotalcite [9], zeolites [10; 11; 12], mesoporous silica [13] and activated carbon [14]. Zeolite is a pore-pleentiful solid adsorbent with an electric field region because of their intrinsically negatively charged cations and their frameworks presenting important physical characteristics [15]. As a zeolite adsorbent, greatly crystalline ZSM-5 presents a great chemical/thermal stability and an elevated surface area and thereby is frequently utilized in the domain of gas separation and catalysis [16; 17; 18; 19]. The use of extra-framework cations in the zeolite equilibrates the negative charge in the silica-alumina framework; the various kinds of cations will also modify the electrostatic field, the structure of the internal pore and the crystallinity [20; 21; 22]. Moreover, these cations can ameliorate the application performance and depict a great effect on the physicochemical properties; however, concerning the CO\({}_{2}\) adsorption performance, the impacts of different cations outside the zeolite framework have been insufficiently investigated.  
It is noteworthy that the theoretical and experimental studies of the adsorption process on ZSM-5 zeolite are pertinent research topics with the object of ameliorating the CO\({}_{2}\) adsorption phenomenon related to these compounds. Particularly, the modeling utilizing the statistical physics can describe well the adsorption of carbon dioxide on ZSM-5 zeolites. For this study, the analysis of the adsorption process using the analytical model depended on the explanation of the energetic and steric parameters of the CO\({}_{2}\) adsorption for many adsorbents [23; 24].  
In this work, the impact of extra-framework cation in ion-exchanged ZSM-5 is analyzed and interpreted utilizing statistical physics formalism. Experimental isotherms are employed to determine the statistical physics parameters to study the CO\({}_{2}\) adsorption on this zeolite. Statistical simulation also makes it possible to analyze the interactions between the CO\({}_{2}\) gas and the zeolite surface and to estimate the corresponding energies.