## 2 Oxygen Phosphoric Acid Fractionation Factor for Low-Mg Calcite: Data and Discussion
### Approximate Standard Uncertainty on \(\alpha_{ACID(CAL)}^{T}\)  
Considering that is \(\frac{\alpha_{\mathrm{ACID(CAL)}}^{298.15}}{\alpha_{\mathrm{ACID(CAL)}}^{298.15 }}=1\pm\mathrm{u}(\frac{\alpha_{\mathrm{ACID(CAL)}}^{298.15}}{\alpha_{\mathrm{ACID (CAL)}}^{298.15}})\)\({}_{\mathrm{tot}}\), with  
\(\mathrm{u}(\frac{\alpha_{\mathrm{ACID(CAL)}}^{298.15}}{\alpha_{\mathrm{ACID (CAL)}}^{298.15}})\)\({}_{\mathrm{tot}}\)\(\approx\)\(\sqrt{2\mathrm{x}\,\mathrm{u}^{2}\Big{(}\alpha_{\mathrm{ACID(CAL)}}^{298.15}})\)\({}_{\mathrm{0.11\%}}\) as calculated using Equation (15), we obtain:  
\[\mathrm{u}^{2}(\alpha_{\mathrm{ACID(CAL)}}^{298.15})_{\mathrm{13}}\approx \frac{\mathrm{u}(\frac{\alpha_{\mathrm{ACID(CAL)}}^{298.15}}{\alpha_{\mathrm{ACID (CAL)}}^{298.15}})}{\sqrt{2}}=0.078\%_{\infty} \tag{16}\]  
Since the values of the fractionation factors are close to the unit and, thus, is \(\mathrm{u}^{2}(\frac{\alpha_{\mathrm{ACID(CAL)}}^{T}}{\alpha_{\mathrm{ACID( CAL)}}^{298.15}})\)\({}_{\mathrm{tot}}\)\(\approx\)\(\mathrm{u}^{2}(\alpha_{\mathrm{ACID(CAL)}}^{298.15})\)\({}_{\mathrm{0.11\%}}\), the total uncertainty \(\mathrm{u}(\alpha_{\mathrm{ACID(CAL)}}^{T})\) at different temperature is  
\[\mathrm{u}(\alpha_{\mathrm{ACID(CAL)}}^{T})\approx\sqrt{\mathrm{u}^{2}(\frac {\alpha_{\mathrm{ACID(CAL)}}^{298.15}}{\alpha_{\mathrm{ACID(CAL)}}^{298.15}}) \mathrm{u}-\mathrm{u}^{2}(\alpha_{\mathrm{ACID(CAL)}}^{298.15})_{\mathrm{13}}}\]  
\[\approx-6.333\times 10^{-10}\;\mathrm{T}^{3}+6.431\times 10^{-7}\;\mathrm{T}^{2}-2.158\times 10^{-4}\;\mathrm{T}+2.403\times 10^{-2} \tag{17}\]  
For example, at 25 \({}^{\circ}\)C, 50 \({}^{\circ}\)C, and 75 \({}^{\circ}\)C we obtain about 0.08%, 0.09%, and 0.14%, respectively.  
Calibration with the Low-Mg Calcite Standard NBS19 and Standard Uncertainty on New Analytical Data \(\delta(^{18}O)_{CAL/PDB}\)  
Now, we calculate the uncertainty on the \(\delta^{18}\mathrm{O}_{CAL/PDB}\) value. Going back now to relation (7), the calibration line that is used to calculate \(\delta^{18}\mathrm{O}_{CAL/PDB}\) for a new sample of low-Mg calcite. Using the standard NBS 19, we write:  
\[\delta^{18}\mathrm{O}_{CAL/PDB}+1=\frac{\alpha_{\mathrm{ACID(NBS19)}}^{\mathrm{ T}}}{\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}}\;(\delta^{18}\mathrm{O}_{ \mathrm{CAL/PDB}}^{\mathrm{T}}+1) \tag{18}\]  
Thus, considered that \(\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}\), \(\alpha_{\mathrm{ACID(NBS19)}}^{\mathrm{T}}\), and \(\delta(^{18}\mathrm{O})_{\mathrm{CAL/PDB}}^{\mathrm{\#T}}+1\) have values close to the unit, approximately we obtain  
\[\mathrm{u}(\delta^{18}\mathrm{O}_{\mathrm{CAL/PDB}})\approx\sqrt{\mathrm{u}^{2}( \frac{\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}}{\alpha_{\mathrm{ACID(CAL)}}^{298. 15}})\mathrm{u}+\mathrm{u}^{2}(\delta^{18}\mathrm{O}_{\mathrm{CAL/PDB}}^{\mathrm{ \#T}})}\geq\mathrm{u}(\frac{\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}}{\alpha_{ \mathrm{ACID(CAL)}}^{298.15}})\]  
At 25 \({}^{\circ}\)C, 50 \({}^{\circ}\)C, and 75 \({}^{\circ}\)C, the value \(\mathrm{u}(\delta^{18}\mathrm{O}_{\mathrm{CAL/PDB}})\) never may be lower than 0.11%, 0.12%, and 0.16%. Assuming, for instance, that the uncertainty on \(\delta^{18}\mathrm{O}_{\mathrm{CAL/PDB}}^{\mathrm{\#T}}\) is about 0.2% (as it is the case for our laboratory), at 25 \({}^{\circ}\)C, 50 \({}^{\circ}\)C, and 75 \({}^{\circ}\)C we obtain \(\mathrm{u}(\delta^{18}\mathrm{O}_{\mathrm{CAL/PDB}})\) of about 0.23%, 0.26%, and 0.26%, respectively.  
We conclude that, because of the variable behavior of the different samples of low-Mg calcite during the acid dissolution and the uncertainty on the \(\alpha_{\mathrm{ACID(CAL)}}^{\mathrm{T}}\) value, the calibration with the standard NBS19 does not allow us to obtain results with standard uncertainty better than about 0.2%. This result is very important in the case we want to compare data obtained for different low-Mg calcites in the same laboratory or different laboratories.