## 3 Results  
### Precipitation  
Figure 1 shows the thermodynamic calculations of 30 mM Ca(OH)\({}_{2}\) in aqueous solution at 25 \({}^{\circ}\)C with different pH. The theoretical values were obtained by solving the matrix consisting of both equilibrium constants (Equations (2) and (3) from Table 1) and mass balances (Equations (4) and (5)). With Matlab, this set of equations were simultaneously solved with 100 steps in the whole pH range (0-14), thus giving the distribution of each species at different pH. In the case of Ca(OH)\({}_{2}\) solution, four species, Ca2+, CaOH+, OH-, and Ca(OH)\({}_{2}\)(s), were taken into consideration. Figure 1a shows the log concentrations of different species in 30 mM Ca(OH)\({}_{2}\) solution with different pH. Log concentrations were adopted due to the low concentrations of each species. As can be seen, with the increase of pH the concentration of Ca2+ first stays unchanged when pH \(<\) 12.6, while with the further increase of pH, the concentration of Ca2+ decreased slowly, and at the same time Ca(OH)\({}_{2}\)(s) emerged in the solution as a precipitate. CaOH+ ions started to appear at pH 6, the log concentration of which increased linearly until pH 12.6, after which the log concentration of CaOH+ ions started to decrease linearly. In Figure 1b, the fraction of calcium was calculated vs. pH, and a similar change of concentration was also observed for Ca2+, CaOH+, and Ca(OH)\({}_{2}\). The theoretical parameters of saturated Ca(OH)\({}_{2}\) are shown in Table 1; as can be seen, the theoretical pH of saturated Ca(OH)\({}_{2}\) is 12.45 with C\({}_{\text{Ca2}^{+}}\) = 19.48 mM. Similarly, the equilibrium distributions of saturated CaCO\({}_{3}\) solution can be calculated by solving the matrix composed of Equations (4)-(13) from Table 1 (thermodynamic parameters) and (6)-(8) (mass balances). The distributions of different species in CaCO\({}_{3}\) solution was also shown in Figure 2 and Table 1. As can be seen, for the saturated solution of CaCO\({}_{3}\), the equilibrium pH is 9.94 and C\({}_{\text{Ca2}^{+}}\) = 0.1415 mM. For the saturated CaCO\({}_{3}\) solution, if the pH is high enough (pH \(>\) 13), Ca2+ from CaCO\({}_{3}\)(s) will dissolve into solution and form Ca(OH)\({}_{2}\)(s), thus causing the increase of concentration of free Ca2+ ions. As for saturated CaSO\({}_{4}\) solution (Figure 3), the equilibrium pH is 7.08 and the equilibrium C\({}_{\text{Ca2}^{+}}\) = 12.03 mM. In the saturated solution of CaSO\({}_{4}\), the main precipitate would be CaSO\({}_{4}\):2H\({}_{2}\)O; as can be seen from Figure 3b, with the increase of pH CaSO\({}_{4}\):2H\({}_{2}\)O starts to decrease at pH 11 and the dominant precipitate becomes Ca(OH)\({}_{2}\)(s) at approximately pH 13. Therefore, for recovering CaSO\({}_{4}\):2H\({}_{2}\)O, the pH should be controlled properly. By comparing the data in Figures 1 and 2, it is clear that the equilibrium concentration of Ca2+ in CaCO\({}_{3}\) solution was significantly smaller than that in Ca(OH)\({}_{2}\) solution, which was caused by the much smaller solubility products (K\({}_{\text{sp}}\)) of CaCO\({}_{3}\), and should carbonation  
happen in the precipitation process the concentration of Ca\({}^{2+}\) should also decrease, which would, in turn, cause the increase of SO\({}_{4}\)\({}^{2-}\). Based on the results obtained from Figures 1-3, we conclude that the saturated CaCO\({}_{3}\) solution has the lowest equilibrium C\({}_{\text{Ca}^{2+}}\), and contacting with CO\({}_{2}\) might decrease the concentration of Ca\({}^{2+}\) ions, which will help us better understand the carbonation process discussed in Section 3.2.