## 4 Discussion
### Theoretical Consideration
When the p\(K_{a,g}\) becomes small due to the temperature, the pH difference between the interior and exterior of the gel, and the pH value itself in the system, decrease considerably with the increase in temperature, as was observed experimentally in the slurries under study. The concentrations of ions in the gel phase are higher than in the liquid phase in the slurry due to the Donnan equilibrium (=existence of the fixed charge groups), resulting in the larger change in the proton concentration depending on the temperature and the p\(K_{a,g}\). Figure 8a indicates the influence of \(\beta\) on the pH inside (pH\({}_{L}\)) and outside gel (pH\({}_{g}\)). The pH in the liquid and gel phases increases with an increase in \(\beta\). The charge density (concentration of amine) in the gel is independent, irrespective \(\beta\). Thus, the pH\({}_{g}\) is independent to \(\beta\) in this model. Furthermore, the amount of absorbed CO\({}_{2}\) depends on \(\beta\), as shown in Figure 8b. The influence of \(\beta\) on the absorption amount of CO\({}_{2}\) decreases as the temperature increases. In the model that considers the influence of the surface charge on the pH in the slurry, the pH\({}_{L}\) in the liquid phase in the slurry increases depending on the amount of charge in the slurry, which, in turn, depends on the surface area and the size of the gel in the slurry. A large \(\beta\) is equivalent to large amounts of protonated amine groups that affect the charge balance in the liquid phase in the slurry. As the pH\({}_{g}\) inside gel cannot be controlled and monitored directly, the presence of the liquid phase in the slurry is important for effectively controlling the pH\({}_{g}\) for the CO\({}_{2}\) separation and recovery in this process. As discussed above, pH\({}_{g}\) in the gel significantly affects the CO\({}_{2}\) adsorption performance. From the pH dependency of the dissociation curves, an effective operational condition can be proposed based on the dissociation curves shown in Figure 9. When pH\({}_{g}\) \(>\) 8, electrostatic adsorption sites (i.e. the amount of protonated amine in the gels) decrease with the increase in pH\({}_{\text{g}}\) in DMAPAA (pK\({}_{\text{a}}\) = 10.35) due to the deprotonation of the amine groups. Thus, pH\({}_{\text{g}}\) \(<\) 8 is preferable to keep the amount of electrostatic adsorption sites at maximum. At pH\({}_{\text{g}}\) \(>\) 8, the CO\({}_{3}\)\({}^{\text{-}}\) ions increase due to the pH dependency of the bicarbonate dissociation equilibrium (\(\beta\)). This also reduces the total adsorption amount of CO\({}_{2}\) in the gel due to the charge compensation of the CO\({}_{3}\)\({}^{\text{-}}\) ions that are twice larger than the HCO\({}_{3}\)\({}^{\text{-}}\) ions in charge. At pH\({}_{\text{g}}\) \(<\) 8, the amount of HCO\({}_{3}\)\({}^{\text{-}}\) decreases with the decrease in pH\({}_{\text{g}}\) in the gel. Therefore, for the gel with DMAPAA, the condition with pH\({}_{\text{g}}\) = 8 will achieve the largest adsorption amount from the viewpoint of pH. To desorb CO\({}_{2}\) in the gel effectively, pH\({}_{\text{g}}\) should be decreased below the pK\({}_{\text{a}}\) line to increase \(Q\). As shown in this model, the pH in the gel depends on the ion concentration in the solution, the fixed charge concentration and the Donnan ratio. The cyclic capacity of the gel is effectively usable by controlling the solution pH based on these conditions. In the case of the DAC process, it requires the small regeneration and small desorption energy, a large capacity of the adsorbent for the low concentration CO\({}_{2}\) in the atmosphere, and the long life of the adsorbent. In this study, low temperature desorption/regeneration (small desorption/regeneration energy) for dilute CO\({}_{2}\) were achieved. However, the slurry consumes a large amount of heat in the temperature swing of the slurry due to its water content.