## Materials and Methods
### Methods  
The thermodynamic calculations were carried out with Matlab(r) R2010a software using thermodynamic parameters of all possible reactions. At the first stage, spent H\({}_{2}\)SO\({}_{4}\) from the electrolyte sample was precipitated with Ca(OH)\({}_{2}\) powder until the solution was almost saturated calcium hydroxide, then the carbonation processes were simulated, both with and without pre-filtration of the precipitates resulted from the precipitation process. All simulations were performed under the condition that the temperature was kept at 25 \({}^{\circ}\)C at all times. Equations (1)-(3), (4)-(13), and (14)-(21) from Table 1 were used in the calculation of the thermodynamic equilibrium distributions of the saturated solutions of Ca(OH)\({}_{2}\), CaCO\({}_{3}\), and CaSO\({}_{4}\).2H\({}_{2}\)O, respectively, while Equations (21)-(29) and (30)-(43) from Table 1 were utilized to calculate the thermodynamic distributions of different species during the precipitation process and carbonation process after filtration of the precipitates. Equations (44)-(47), together with Equations (23), (24), (26)-(29), (34), (35), (37), and (39) from Table 1, were utilized to calculate the carbonation process without the filtration of precipitates resulted from the precipitation process. Equations (48)-(55) and (56)-(61) from Table 1 were used to calculate the thermodynamic distributions of Pb\({}^{2+}\) in precipitation and carbonation processes, respectively. All species (aqueous species and precipitates) at different pH can be obtained by solving the matrix of equilibrium constants and mass balances.  
It is worth pointing out that in the modeling process of carbonation, the system was considered as a closed system with a certain amount of CO\({}_{2}\) injected into the system. This was adopted due to the fact that CO\({}_{2}\) was the gas and the amount of CO\({}_{2}\) consumed cannot be easily obtained.  
Due to the fact that calcium, sulfate, and carbonate are all divalent ions with quite low ionic activity coefficients (\(\gamma\)), which are \(\gamma_{\text{Ca2}^{+}}\) = 0.28 (\(I\) = 0.7, 25 \({}^{\circ}\)C and 1 atm), \(\gamma_{\text{SO4}^{-}}\) = 0.12, and \(\gamma_{\text{CO3}^{2-}}\) = 0.20 (\(I\) = 0.7, 25 \({}^{\circ}\)C and 1 atm), respectively, the activity corrections are needed in the thermodynamic modeling process. The definition of ionic strength (\(I\)) and activity coefficient are shown in Equations (1) and (2) [24]:  
\[I=1/2\sum m_{i}Z_{i}^{2} \tag{1}\]  
\[\gamma_{i}=\frac{a_{i}}{m_{i}} \tag{2}\]  
where \(m_{i}\) is the molality of an aqueous species, \(i\) is the corresponding aqueous species, \(Z_{i}\) is the charge of \(i\), \(a_{i}\) is activity of species \(i\), and \(\gamma_{i}\) is the activity coefficient of species \(i\). In this study, a simplified Helgeson-Kirkham-Flowers model was used to calculate the activity coefficient, as shown in Equation (3) [25; 26; 27]:  
\[Log\gamma_{i}=\ -AZ_{i}^{2}\bigg{(}\sqrt{\frac{I}{1+B\sqrt{I}}}\bigg{)}-Log \Big{(}1+0.018\sum m_{k}\Big{)}+bl \tag{3}\]  
where \(A\) is the Debye-Huckel slope, \(B\) is the distance of the closest approach, \(b\) is a solute specific parameter, and \(k\) is all solute species in the aqueous solution.