## 4 Discussion
### Equilibrium vs. Kinetics  
\[\text{CaCO}_{3}+\text{H}_{3}\text{PO}_{4}\Rightarrow\text{CaHPO}_{4}+\text{CO} _{2}+\text{H}_{2}\text{O}. \tag{2}\]  
In the above reaction to obtain CO\({}_{2}\) from calcite, only two-thirds of the calcite oxygen converts into CO\({}_{2}\), and one-third of it remains in the reactant fluid. This may cause a kinetic isotope fractionation. [PERSON] and [PERSON] [5] conducted CO\({}_{2}\) generation experiments by changing the amount of calcite with respect to the amount of phosphoric acid and observed that a nonnegligible amount of CO\({}_{2}\) remained in the phosphoric acid after the reaction. This phenomenon can cause oxygen isotope exchange between CO\({}_{2}\) and phosphoric acid; however, variations in the oxygen isotope ratios of evolved CO\({}_{2}\) with CO\({}_{2}\) yields (amount recovered as CO\({}_{2}\) for reactant calcite) were within 0.1% in the range of 83-100% yield [5]. In addition to this, no reaction was observed between CO\({}_{2}\) and phosphoric acid in Equation (2) [11] because no measurable shift in the 8\({}^{18}\)O of the CO\({}_{2}\) was observed [18]. These findings may suggest the working hypothesis that the AFF is apparently close to EFF between CO\({}_{2}\) and calcite at the same temperature. Next, the following inferences are made.  
Oxygen isotope fractionation experiments at low temperatures are extremely difficult because it is difficult to attain isotopic equilibrium between minerals and water at low temperatures. The experimentally determined apparent oxygen isotope fractionation factors between amorphous silica and water in the temperature range of 34-93 \({}^{\circ}\)C were obtained by measuring the oxygen isotope ratios of amorphous silica precipitated from the geothermal waters of power plants [19]. The temperature dependence of the fractionation factor (\(\alpha\)) between amorphous silica and water at 34-93 \({}^{\circ}\)C is given [19] by  
\[10^{3}\text{ ln}\alpha=3.52\;(10^{6}\text{T}^{-2})-4.35. \tag{3}\]  
The fractionation factor in Equation (3) is on the low-temperature extrapolation line of the oxygen isotope equilibrium fractionation between quartz and water obtained experimentally [20] at high temperatures (>250 \({}^{\circ}\)C). When quartz is precipitated from water, the initial formation of \"absorbed silica,\" which is a reaction intermediate between dissolved and amorphous silica in the process of the particle growth of amorphous silica, is followed by rearrangement and interlinkage to form quartz [21]. Although there is a large structural difference between quartz and amorphous silica, there is no significant oxygen isotope fractionation between quartz and amorphous silica [19]. Therefore, the oxygen isotope fractionation between amorphous silica and water would be the rearrangement and interlinkage of \"absorbed silica,\" which is the same as the fractionation between quartz and water [21]. The mineral precipitation environments in many epithermal vein-type deposits, including silica precipitation in geothermal environments, would be close to the isotopic equilibrium with the coexistent hydrothermal fluid.  
The experimentally determined AFFs obtained in this study look like they were based on a certain principle because of the data consistency (R\({}^{2}\), the coefficient of determination, is near unity in Figure 4). To ensure that the fractionation factors in this study were reliable, several experimental fractionation factors at low-to-intermediate temperatures are shown in Figure 5. There appears to be high consistency among the experimental data. Although experiments under equilibrium conditions are not possible at low temperatures, the AFF at low temperatures would be similar to that under equilibrium conditions.