## 4 Discussion
### Theoretical Consideration
\[\text{H}_{2}\text{CO}_{3}\rightleftharpoons H^{+}+\text{HCO}_{3}^{-}\] (r3)  
\[\text{HCO}_{3}^{-}\rightleftharpoons H^{+}+\text{CO}_{3}^{2-}\] (r4)  
\[K_{a1}=\frac{\left[\text{H}^{+}\right]\left[\text{HCO}_{3}^{-}\right]}{\left[ \text{H}_{2}\text{CO}_{3}\right]}(\text{degree of dissociation},\text{ }\alpha) \tag{4}\]  
\[K_{a2}=\frac{\left[\text{H}^{+}\right]\left[\text{CO}_{3}^{2-}\right]}{\left[ \text{HCO}_{3}^{-}\right]}\text{ (degree of dissociation},\text{ }\beta) \tag{5}\]  
\[\text{p}K_{a1}=-(-356.3094-0.06091964T+\frac{21834.37}{T}+126.8339\text{Log}(T )-1684915/T^{2}) \tag{6}\]  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]0  
Total absorbed CO\({}_{2}\)  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]1  
Dissociation equilibrium of water  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]2Charge balance (electric neutrality) in the liquid phase in the slurry  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]3  
From Equations (4)-(10), the concentration of each species in the liquid phase in the slurry is derived as a function of the proton concentration (\(c_{\mathrm{H}}\)), as follows. \[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]4  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]5  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]6  
The amount of \(\mathrm{CO}_{2}\) absorbed can be obtained as the function of \(c_{\mathrm{H}}\) and \(c_{0}\). Donnan equilibrium is considered between the liquid and the gel phases in the slurry. The dissociation reaction of the amine groups and its equilibrium constant, \(K_{a\mathrm{g}}\), (degree of dissociation = \(\beta_{\mathrm{g}}\)) are  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]7 (r5)  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]8  
Mass balance of amine groups  
\[p_{\text{CO2}}=Hx_{\text{CO2}}\text{ where }x_{\text{CO2}}=n_{\text{CO2}}/(n_{ \text{CO2}}+n_{H\text{2O}}) \tag{1}\]9  
Donnan equilibrium at the liquid-gel interface  
\[\ln{(H)}=11.181-1949.6/T+0.0822\ln{(T)}\text{ (in pure water)} \tag{2}\]0  
where \(c_{i,\mathrm{L}}\) and \(c_{i,\mathrm{g}}\) are the concentrations of \(i\) ion in the liquid and gel phases in the slurry, respectively. Donnan ratio and the valency of the \(i\) ion are indicated by \(p\) and \(z_{i}\), respectively. The charge balance (electric neutrality) in the gel phase in the slurry is given by the following:  
\[\ln{(H)}=11.181-1949.6/T+0.0822\ln{(T)}\text{ (in pure water)} \tag{2}\]1  
By substituting Equations (11)-(16) into Equation (17) of the charge balance, following Equation (18) is obtained. \[ \begin{array}{l}p^{2}\left[\mathrm{H}^{+}\right]_{\mathrm{L}}^{2}+p^{3} \left[\mathrm{H}^{+}\right]_{\mathrm{L}}\left(\left[\mathrm{X}_{0}\right]+K_{a,g}\right)-p^{2}\left(\left[\mathrm{H}^{+}\right]_{\mathrm{L}}\left[\mathrm{ HCO}_{3}^{-}\right]_{\mathrm{L}}+K_{w}\right)\\ -p(2\left[\mathrm{H}^{+}\right]_{\mathrm{L}}\left[\mathrm{CO}_{3}^{2-}\right]_ {\mathrm{L}}+K_{a,g}\left[\mathrm{HCO}_{3}^{-}\right]_{\mathrm{L}}+\frac{K_{a,g}K_{w}}{\left[\mathrm{H}^{+}\right]_{\mathrm{L}}})-2K_{a,g}\left[\mathrm{ CO}_{3}^{2-}\right]_{\mathrm{L}}=0\end{array}  \tag{18}\]  
From the experimental result shown in Figure 4, the influence of the cationic charge of amine groups on the pH of the slurry was suggested. [PERSON] et al. reported the pH change in the solution with the thermo-responsive ionic linear polymer [9].