## 2 Oxygen Phosphoric Acid Fractionation Factor for Low-Mg Calcite: Data and Discussion
### Selection of Data at Different Temperatures
Then, we used the following regression to estimate \(\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{298.15}}\):  
\[\alpha_{\mathrm{ACID(\Theta)}}^{\mathrm{T}}=\frac{\delta^{18}\mathrm{O}_{ \mathrm{CO}_{2}(\Theta)/\mathrm{FDB}}^{\mathrm{T}}+1}{\delta^{18}\mathrm{O}_{ \mathrm{0}/\mathrm{FDB}}+1}=bt\;(\frac{1}{\mathrm{T}^{2}}-\frac{1}{298.15^{2} })+at \tag{8}\]  
where the intercept \(a^{\prime}\) represents a weighted value of the fractionation factor at T = 298.15 K.  
In the literature, there is disagreement about the values of \(\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}\) at different temperatures (e.g., [4], natural calcite; [9; 20; 21] Islandic spar Mg-low calcite PSU-4, Pleistocene Key Largo formation Mg-low calcite LMC: 11 data items; [8] CaCO\({}_{3}\) Merck, synthetic carbonate: 3 data items; [10; 15] Calcite H, natural calcite with MgO = 0.77: 20 data items). In particular, the values at 25 \({}^{\circ}\)C have a large span, from about 1.01008 ([4], corrected to 1.01025 by [PERSON] and [PERSON] [20]) to 1.01050 [10]. Only [PERSON] et al. [15] report several data at different temperatures for the same sample; using their data (averages at temperatures 25, 50, and 75 \({}^{\circ}\)C, 3 data items), the regression type (8) gives intercept \(a^{\prime}=1.01040\pm 0.00001\) (standard error on the intercept), which represents a weighted estimate of \(\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{298.15}}\). Some authors (e.g., [22] and references therein, [17]) suggest the value 1.01025. In particular, [PERSON] et al. ([17] p. 276, abstract) state that they use 1.01025 because it is \"the most commonly accepted value for this quantity\" (see, for instance, [20; 23; 24]). Although this value is conventional, it is very important because it is used for defining the value of CO\({}_{2}\) gas produced through digestion of the standard NBS19 in 100% H\({}_{3}\)PO\({}_{4}\) at 25 \({}^{\circ}\)C.  
The Ratio \(\alpha_{\mathrm{ACID(CAL)}}^{T}/\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{298.15}}\) vs. Temperature  
Since generally the values \(\delta^{18}\mathrm{O}_{\mathrm{CAL}/\mathrm{FDB}}\) are not measured directly, the \(\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}\) values cannot be evaluated. Thus, comparison between the different data sets was generally carried out using the following regression, that, _theoretically_, represents a straight line passing for the origin:  
\[\frac{\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}}{\alpha_{\mathrm{ACID(CAL)}}^{ \mathrm{298.15}}}-1=\frac{\delta^{(18}\mathrm{O}_{\mathrm{CO}_{2}(\mathrm{CAL )}/\mathrm{FDB}}^{\mathrm{T}}+1)/(\delta^{18}\mathrm{O}_{\mathrm{CAL}}+1)}{( \delta^{18}\mathrm{O}_{\mathrm{CO}_{2}(\mathrm{CAL)}/\mathrm{FDB}}^{\mathrm{ 298.15}}+1)/\delta^{(18}\mathrm{O}_{\mathrm{CAL}}+1)}-1=\frac{(\delta^{18} \mathrm{O}_{\mathrm{CO}_{2}(\mathrm{CAL)}/\mathrm{FDB}}^{\mathrm{T}}+1)}{( \delta^{18}\mathrm{O}_{\mathrm{CO}_{2}(\mathrm{CAL)}/\mathrm{FDB}}^{\mathrm{ 298.15}}+1))}-1=b\;(\frac{1}{\mathrm{T}^{2}}-\frac{1}{298.15^{2}}) \tag{9}\]  
where is \(\frac{\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}}{\alpha_{\mathrm{ACID(CAL)}}^{ \mathrm{298.15}}}=1\) for T = 298.15 K. Regression (9) could exhibit an intercept \(a\) numerically different from zero, but, in any case, with very high probability for \(a=0\). We calculated a total of the \(15\) regressions  
\[\frac{\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}}{\alpha_{\mathrm{ACID(CAL)}}^{ \mathrm{298.15}}}-1=b_{\mathrm{CAL}}(\frac{1}{\mathrm{T}^{2}}-\frac{1}{298.15^ {2}})+a_{\mathrm{CAL}} \tag{10}\]  
for low-Mg calcites (Table 1). As expected (see Equation (9)), the probability for \(a_{\mathrm{CAL}}=0\) is very high.