## 4 Discussion
### Theoretical Consideration  
Herein, the adsorption and recovery characteristics of the gels for CO\({}_{2}\) were examined based on an equilibrium model of the adsorption behaviour in the gel-slurry system. Various models have been shown for the equilibrium CO\({}_{2}\) sorption to amines in liquid systems [26; 27; 28; 29]. Equilibrium CO\({}_{2}\) sorption models have also been reported by [PERSON] et al. [30] for tertiary amines in liquid systems. Figure 6: Summary of the CO\({}_{2}\) adsorption/desorption test. The characteristics of the gel-slurry system in this study are the system in which the ionic gel is dispersed and the ionic group changes its equilibrium constant (p\(K_{\text{a}}\)), depending on the temperature and pH of the system. This system is expressed by the following model equations. In the model, gas-liquid equilibrium of CO\({}_{2}\) gas, mass balance in the liquid, and the gel phases in the slurry, dissociation equilibrium of carbonic acid and bicarbonate, electrical neutrality condition, ionic product of water, dissociation equilibrium of amine, and Donnan's equilibrium at the liquid-gel interface are taken into consideration. [PERSON]'s law was applied to the dissolution equilibrium (gas-liquid equilibrium) of CO\({}_{2}\). The influence of the vapour pressure of water, which reduces the partial pressure of CO\({}_{2}\), was considered in the gas-liquid equilibrium. The dissolution of CO\({}_{2}\) is an exothermic reaction; thus, the solubility of CO\({}_{2}\) decreases with increasing temperature. The amount of CO\({}_{2}\) that is in equilibrium with the partial pressure of CO\({}_{2}\) (\(p_{\text{CO2}}\)) in the gas phase was calculated by Equation (1). [PERSON]'s constant at each temperature was determined by Equation (2) in this calculation [27]. The saturated water vapour pressure was considered using the Tetens equation [31] in this calculation. \[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]  
\[\ln{(H)}=11.181-1949.6/T+0.0822\ln{(T)}\text{ (in pure water)} \tag{2}\]  
where \(H\) and \(T\) are the Henry constant and absolute temperature, respectively; \(n_{\text{H2O}}\) and \(n_{\text{CO2}}\) represent moles of water and dissolved CO\({}_{2}\) in the liquid phase, respectively. Most of the dissolved CO\({}_{2}\) exist in the form of CO\({}_{2}\) in the water system; however, the total amount of dissolved CO\({}_{2}\) and H\({}_{2}\)CO\({}_{3}\)(\(aq\)) defined by H\({}_{2}\)CO\({}_{3}\)* (Equation (3)) was used in the following equilibrium analysis based on the [PERSON]'s report [27]. The concentration of H\({}_{2}\)CO\({}_{3}\)* is equivalent to \(x_{\text{CO2}}\) from the mass balance. \[\text{CO}_{2}(\text{aq})+\text{H}_{2}\text{O}(\text{l})\rightleftharpoons H_{2}\text{CO}_{3}(aq),\text{ }K_{2}\text{ }\sim 650\text{ \@@cite[cite]{[\@@bibref{}{H2}{CO3}{}{}]}}\] (r2)  
\[[\text{H}_{2}\text{CO}_{3}\text{*}]=[\text{CO}_{2}(aq)]+[\text{H}_{2}\text{CO} _{3}(aq)] \tag{3}\]  
The dissolved CO\({}_{2}\) produces carbonic acid in water. The carbonic acid dissociates by Equation (r3) and (r4). The equilibrium constants for each reaction are shown by Equations (4) and (5). The temperature dependent dissociation constants of each dissociation reaction were calculated using Equations (6) and (7), reported by [PERSON] [28].