Oxygen Isotope Phosphoric Acid Fractionation Factor and Uncertainty on the δ 18 O Measurements of Calcite

: At present, the isotope ratio 18 O/ 16 O (in the text expressed as δ 18 O) of carbonate (CO 2 − 3 ) is usually determined by isotope ratio mass spectrometry measuring the CO 2 gas produced dissolving the CO 2 − 3 -bearing substance in concentrated H 3 PO 4 . As with any analytical data, the δ 18 O values of carbonate are also affected by uncertainty which must be considered mainly when data from different laboratories are compared. Usually, scientiﬁc papers report only repeatability and/or reproducibility of analytical results, which, have scarce signiﬁcance for data comparison. With the aim of evaluating the overall uncertainty for new analytical data for low-Mg calcites, in this paper we reconsidered the δ 18 O data, which are reported in the literature. Two kinds of uncertainty must be taken into account: (1) due to the calibration of the delta values using international standards (prediction uncertainty), (2) that due to small differences in the chemistry of low-Mg calcites. The two uncertainties must be added when comparing data produced in the same or different laboratories. We found that this overall uncertainty cannot be lower than about 0.2‰. Thus, uncertainty lower than 0.2‰, sometimes reported in the literature, is misleading.


Introduction
Oxygen stable isotopes of carbonate (CO 2− 3 ) of several minerals are largely used in geology, environmental sciences, and archaeology. For instance, changes in seawater 18 O/ 16 O ratio and temperature are recorded by variations in the 18 O/ 16 O ratio of Ca-carbonates of marine fossils and microfossils, thus providing the basis for global chronostratigraphy ( [1] and reference therein). In archaeology, together with the oxygen isotopes of the phosphate (PO 3− 4 , HPO 2− 4 ) of bioapatite of human and animal bone and tooth remains, oxygen isotopes of the carbonate of bioapatite may give information on the diagenetic conditions of the remains and, thus, on their ability to furnish palaeoenvironmental indications ( [2,3], as pioneering works). However, as with any analytical results, the 18 O/ 16 O ratio values are affected by uncertainties that must be taken into account when different data are compared. In most papers, however, only data for precision (repeatability and/or reproducibility) are reported, values that have scarce significance for comparison of data obtained in the same laboratory with different calibrations and in different laboratories even more. A correct evaluation of uncertainty is essential: it must include prediction uncertainty due to calibration as well as the possible effect of chemical composition/crystal lattice of the different CO 2− 3 -bearing phases in analysis. The evaluation of this overall uncertainty for low-Mg calcites is the most important topic of this report.
The total oxygen of CO 2− 3 may be determined by decarbonatization/fluorination methods, as described, for instance, by Sharma and Clayton [4]. At present, this method is rarely used. Actually, starting from the fifty years of the last century [5], usually, the substances containing carbonate (e.g., carbonate minerals, CO 2− 3 -bearing apatite, et cetera) are dissolved in concentrated H 3 PO 4 . with production of CO 2 , which is analysed by isotope Appl. Sci. 2022, 12, 10094 2 of 10 ratio mass spectrometry (IRMS). During dissolution, only two oxygens of CO 2− 3 form CO 2 . Thus, the formation of CO 2 generates oxygen isotope fractionation between the original carbonate CO 2− 3 and the new formed CO 2 gas . At a given temperature T of dissolution, this fractionation is expressed by the "oxygen isotope phosphoric acid fractionation factor", α T ACID(θ) , which is defined as where θ is the generic substance containing CO 2− 3 , δ 18 O θ/RM and δ 18 O T CO 2 (θ)/RM are the isotopic values (In the following text, the parameter delta (δ) is defined according to IUPAC (International Union of Pure and Applied Chemistry): It is noteworthy that the international standards used for calibration in the isotope analysis of the carbonate group in different substances are low-Mg calcites (such as NBS 19, NBS 18, IAEA-CO-1, IAEA-CO-8, with MgO < 0.8% weight, [6]). For a standard (st) consisting of low-Mg calcite (CAL), we may write the ratio where δ 18 O T CO 2 (stCAL)/w is the measured value for CO 2 gas obtained from stCAL at temperature T referred to the working standard w of the laboratory (CO 2 in tank) and δ 18 O stCAL/PDB is the "true" value of delta referred to the primary international standard PDB. In the case the spectrometric response is ideally linear (consider the abundance ratio 18 18 O θ,m are the "true" value and the measured value, respectively, for the carbonate group of the substance θ, and k is a constant.), Equation (2) may be re-written as a calibration line where δ 18 O CAL/PDB is the estimated value for a generic calcite that behaves as the standard stCAL during dissolution in H 3 PO 4 , δ 18 O T CO 2 (CAL)/w is the measured value referred to the working standard w, and B T stCAL is a constant at temperature T. To improve the calibration, usually more than one standard of low-Mg calcite is used.
If the material θ in analysis is different from stCAL (e.g.,: calcite different from stCAL, dolomite, ankerite, CO 2− 3 -bearing apatite, et cetera), an apparent value is obtained. The value δ 18 O #,T θ/VPDB is an estimate of δ 18 O T θ/VPDB for the phase θ in the case it behaves as stCAL during dissolution. Thus, we may write By definition, is Thus, dividing (5) by (6), Equation (7) is very important for several reasons. Essentially, it tells us that (a) in the case the ratio is independent from temperature, then also changes with temperature, also the apparent value δ 18 O #,T θ/PDB changes with temperature and, thus, both the values α T ACID(stCAL) and α T ACID(Θ) are necessary to obtain δ 18 O θ/PDB . As already stated above, hereafter we consider only low-Mg calcite (CAL) from the literature: (a) we investigate the variation of its oxygen isotope phosphoric acid fractionation factor at different temperature, (b) we discuss the significance of the δ 18 O #,T CAL/PDB value obtained using international standard(s), (c), for δ 18 O T CAL/PDB , we evaluate the uncertainty u(δ 18 O T CAL/PDB ) related to the variable behaviour of different samples of low-Mg calcite during dissolution in phosphoric acid, and (d) we estimate the overall uncertainty due to (c) and to the prediction uncertainty related to the calibration line obtained with international standards. Owing to the small amount of data at disposal, the approach is very simplified and only approximate. However, in spite of this, we think that the results obtained allow us to draw some interesting conclusions.
Unfortunately, the authors frequently do not report single data, but averages at different temperatures, which reduces the statistical information. For this reason, we were compelled to use averages instead of single data (with the exception of Calcite H (1) in Table 1).  Some of the cited works demonstrate the low analytical repeatability of the fractionation factor determination at 25 • C (e.g., [13,15]). Kim et al. [15] report a spread of data that is larger at 25 • C than at higher temperature. These authors use the δ 13 C CAL/PDB values for the selection of the oxygen data on low-Mg calcite. Given a defined sample, they consider the average δ 13 C CAL/PDB ± standard deviation = −4.62‰ ± 0.06‰ for the carbon isotopes and eliminate the corresponding δ 18 O T CO 2 (CAL)/PDB value in case the sample is 10 3 |δ 13 C CAL/PDB − δ 13 C CAL/PDB | > 0.06. This drastic method is very "efficient", although not typical for the identification of aberrant values. We prefer to consider all the data for the reasons listed below.
Firstly, regarding elimination of data we follow the Burke's golden rule [18]: "No value should be removed from a data set on statistical ground alone" (p. 20) and "Delete extreme values only when a technical reason for their aberrant behavior can be found" (p. 23). Additionally, in case this opinion is not accepted, only data with very low probability of occurrence (indicatively with probability less than 0.05 or 0.01) should be eliminated.
The carbon δ 13 C data reported by Kim et al. [15] have abnormal distribution (Shapiro-Wilk test: p normal = 0.014). In this case, the standard deviation of the statistical sample does not express the spread of the data around the central value in the right way. In the case that the distribution is not normal, the use of simple non-parametric methods is suggested. For instance, the box and whiskers plot [19] indicates that no one value from Kim et al.'s data [15] for calcite is an "outlier".

Estimation of Oxygen Isotope Phosphoric Acid Fractionation
Factor, α T ACID(CAL) , at 298.15 K Using the estimated values reported by the authors, we verified that the linear relation between the data α T ACID(θ) vs T −2 (in kelvin when not indicated otherwise) generally gave lower standard error of regression than vs T −1 . Then, we used the following regression to estimate α 298.15 ACID(CAL) : where the intercept a represents a weighted value of the fractionation factor at T = 298. 15 [20]) to 1.01050 [10]. Only Kim et al. [15] report several data at different temperatures for the same sample; using their data (averages at temperatures 25, 50, and 75 • C, 3 data items), the regression type (8) gives intercept a = 1.01040 ± 0.00001 (standard error on the intercept), which represents a weighted estimate of α 298.15 ACID(CAL) . Some authors (e.g., [22] and references therein, [17]) suggest the value 1.01025. In particular, Kim 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 • C. 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: where is for low-Mg calcites ( Table 1). As expected (see Equation (9)), the probability for a CAL = 0 is very high. Moreover, according to Zar [25],  − 1 values are obtained ( Figure 1, Table 2 Table 2. From Figure 1, it is evident that for a given ( 1 T 2 − 1 298.15 2 ), the variance of  Table 1). The figure evidences the increasing of the data as the ( 1 T 2 − 1 298.15 2 ) value increases. ) ≈ −1.362 × 10 −10 T 3 + 1.348 × 10 −7 T 2 − 4.250 × 10 −5 T + 4.332 × 10 −3 (12) For instance, at 25 °C, 50 °C, and 75 °C, uncertainty is 0.032‰, 0. 077‰, and 0.125‰, respectively. This uncertainty is only related to the different behavior of the analyzed calcites during the acid dissolution. ) , and uncertainty u( + 0.000036 (13) The prediction uncertainty for an estimate of ≈ −7.244 × 10 −11 T 3 + 8.246 × 10 −8 T 2 − 3.061 × 10 −5 T + 3.914 × 10 −3 (14) For example, at 25 °C, 50 °C, and 75 °C we obtain 0.11‰, 0.091‰, and 0.090‰, respectively.
where X = any value of ( 1 T 2 − 1 298.15 2 ), X NBS19 = ( 1 T 2 − 1 298.15 2 ) values used in regression of Table 1, X NBS19 = average of ( 1 T 2 − 1 298.15 2 ) values used in regression, n = 5, number of couples of data used in regression, s(yx) NBS19 = 0.000068 standard error of regression with degree of freedom υ = (5 − 2) = 3, t α(2),ν = 1.197 Student's t for α(2) = 0.32 (two-tale) and ν = 3. We obtained: For example, at 25 • C, 50 • C, and 75 • C we obtain 0.11‰, 0.091‰, and 0.090‰, respectively. The Equations (11) and (13)  ) for a generic low-Mg calcite. The total uncertainty u( may be approximatively calculated, assuming that the dispersion of the regression lines reported in Figure 1 is representative of all the low-Mg calcites in nature. This, of course, is a restrictive condition that could lead to underestimation of the total uncertainty. The evaluation of the total uncertainty is given combining (12) and (14) For instance, at 25 • C, 50 • C and 75 • C, u(Y T 298.15 (CAL) ) tot is 0.11‰, 0.12‰, and 0.16‰ respectively. This total uncertainty is due both to the different behavior of calcites during the acid dissolution and to the substitution of 10a with 11a.

Approximate Standard Uncertainty on α T ACID(CAL)
Considering that is ) tot ≈ u 2 (α 298.15 ACID(CAL) ) + u 2 (α T ACID(CAL) ), the total uncertainty u(α T ACID(CAL) ) at different temperature is u(α T ACID(CAL) ) ≈ u 2 ( For example, at 25 • C, 50 • C, and 75 • C we obtain about 0.08‰, 0.09‰, and 0.14‰, 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 α T ACID(CAL) 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.