## 4 Discussion
### Oxygen Isotope Fractionation between CO\({}_{2}\) and Carbonate Minerals
As the reaction temperature increased, the oxygen isotope fractionation (1000 ln\(\alpha\)) decreased from 11.30 at 1 \({}^{\circ}\)C to 7.47 at 150 \({}^{\circ}\)C. The temperature dependence of the fractionation factor was expressed over the temperature range of 1-150 \({}^{\circ}\)C using the following equation:  
\[1000\ \text{ln}\alpha=0.494\ (10^{6}/\text{T}^{2})+4.64, \tag{1}\]  
where \(\alpha\) is the AFF between CO\({}_{2}\) and calcite, and T is in kelvin. Figure 3: Temperature dependence of the theoretically determined oxygen isotope fractionation factors between CO\({}_{2}\) and carbonate minerals between 0 \({}^{\circ}\)C and 600 \({}^{\circ}\)C (drawn using data from [13; 14; 15]). [PERSON] (2008) [13] and [PERSON] (1991) [14] are for the fractionation factors between CO\({}_{2}\) and calcite. Calcite through to siderite in the legend are the fractionation factors between CO\({}_{2}\) and each carbonate [15]. Figure 4 shows the temperature dependence of the oxygen isotope AFF (\(\alpha\)) between CO\({}_{2}\) and calcite and a least-squares straight line through the experimental data, which fitted the data well (R\({}^{2}\) = 0.996). The 1000 ln\(\alpha\) values are plotted against T\({}^{-2}\) in Figure 4. The use of T\({}^{-2}\) rather than T\({}^{-1}\) has been justified by theoretical considerations (e.g., [16; 17]). The reproducibility and temperature dependence were rather good, suggesting that each reaction accurately occurred at the intended temperature. Figure 5 shows the comparison of this study and previous theoretical studies [13; 14; 15] at low-to-intermediate temperatures. A maximum value is found in the theoretically determined fractionation factors at low temperatures between 50 \({}^{\circ}\)C and 100 \({}^{\circ}\)C. Figure 5 shows the 1000 ln\(\alpha\) values from several experimental studies. The experimental AFFs are expressed in a linear function of reciprocal squared temperature, and no maximum value is found at low temperatures, dissimilar to the theoretical data. There is a big difference in the theoretically and experimentally determined fractionation factors, though the latter is  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Run No. **} & \multirow{2}{*}{**T (\({}^{\circ}\)C)**} & \multirow{2}{*}{**10\({}^{6}\)/T\({}^{2}\) (K\({}^{-2}\))**} & \multicolumn{2}{c}{**5\({}^{18}\)Osmow**} & \multirow{2}{*}{**AFF (\(\alpha\))**} & \multirow{2}{*}{**1000 ln\(\alpha\)**} \\  & & & &  
not the equilibrium fractionation. [PERSON] et al. [9] and the present study did not conduct total oxygen isotope determination using the BrF\({}_{5}\) technique. Instead, the oxygen isotope fractionation factor between CO\({}_{2}\) and calcite of 1.01025 at 25 \({}^{\circ}\)C was assumed. Although this value is internationally recognized as trustworthy [3; 4; 7], this might fix the fractionation factor at 1.01025 at 25 \({}^{\circ}\)C. [PERSON] et al. [10] conducted the total oxygen isotope determination via the thermal decomposition of calcite, followed by the fluorination of the residual CaO using BrF\({}_{5}\). Their results are similar to those of [PERSON] et al. [9] and the present study (Figure 5), confirming that all the experimental AFFs are reliable. Figure 4: The 1000 In\(\alpha\) values calculated from the oxygen isotope AFFs (\(\alpha\)) between CO\({}_{2}\) and calcite are plotted against 10\({}^{6}\)/T\({}^{2}\). The temperature (\({}^{\circ}\)C) is shown on the upper axis. The regression line and R\({}^{2}\) (coefficient of determination) are also shown. Figure 5: Comparison of the 1000 In\(\alpha\) values of the experimentally and theoretically determined fractionation factors. Note that experimentally determined AFFs are not exactly the same as EFFs. The theoretical studies are [PERSON] 2008 [13], [PERSON] 1991 [14], and [PERSON] 1999 [15] in the legend. [PERSON] [10], [PERSON] [9], and ’This study’ in the legend are experimental studies. [PERSON] et al.