The prediction method for standard enthalpies of apatites using the molar volume, lattice energy, and linear correlations from existing experimental data

Experimental data of thermodynamic state functions and molar volume for phosphate, arsenate, and vanadate apatites containing Ca, Sr, Ba, Pb, end Cd at the cationic positions Me2+ and F, OH, Cl, Br, and I at the halide position X were collected. The apatite supergroup splits into distinct subgroups (populations) constituted by Me10(AO4)6X2 with the same Me2+ cations and tetrahedral AO43− anions but with different anions at the X position. Linear relationships between various parameters within apatite subgroups are observed. The prediction method for standard enthalpies of apatites (ΔHºf,el) is based on regression analysis of the linear correlations within the subgroups between ΔH°f,el of apatites and their molar volume Vm, lattice energy UPOT, and ΔH°f,el of their anions AO43− or X−. This allowed to predict 22 new ΔH°f,el values for apatites and materials with an apatite structure. The prediction precision is comparable to the experimental uncertainty obtained when reproducing experimental data using calorimetric measurements or dissolution experiments and can be applied to a wider range of apatites than other methods.


Introduction
Quantitative geochemical calculations are not possible without thermodynamic databases. Considerable advances in the quantity and quality of these databases have been made since the early days of Lewis and Randall (1923), Latimer (1952), and Rossini et al. (1952). According to Oelkers and Shott (2018), the emergence of thermodynamic databases can be considered one of the greatest advances in geochemistry of the last century. Thermodynamic data have been used in basic research and for countless applications in computational modelling, computer simulations, waste management, and policy-making. The challenges today are to evaluate thermodynamic data for internal consistency and to reach a most reliable properties. The present work focuses on the enthalpy of formation from elements (ΔH°f ,el ) of minerals and synthetic compounds belonging to the apatite supergroup.
The natural apatites and apatite-based materials are a class of compounds with the stoichiometry Me 10 (AO 4 ) 6 X 2 , where the Me-site is occupied by larger monovalent (Na + , K + , etc.), divalent (Ca 2+ , Sr 2+ , Ba 2+ , Pb 2+ , Cd 2+ , etc.), or trivalent (La 3+ , Y 3+ , Ce 3+ , Sm 3+ , etc.) cations, the A-site is occupied by a smaller metal, metalloid or nonmetal (P 5+ , As 5+ , V 5+ , Si 4+ , etc., often accompanied by carbonate anion CO 3 2− ), and the X-site is filled by halides, hydroxides, or oxides (F − , Cl − , Br − , I − , OH − , O 2− , etc., also often accompanied by a carbonate anion CO 3 2− ) (e.g., Rakovan and Hughes 2000;Pan and Fleet 2002;Pasero et al. 2010;Tait et al. 2015;Ptáček 2016;Hughes and Rakovan 2018;Pieczka 2018;Rakovan and Scovil 2021). Due to the extremely rich array of possible substitutions in each of the highlighted positions, the possible end-members alone are over 200 types, indicating that this is currently the most numerous supergroup of minerals and compounds (Baker 1966;Oelkers and Valsami-Jones 2008; Communicated by Mark S Ghiorso. Table 1 provides a compilation of the thermodynamic data available in the literature (based on experiments and "ab initio" calculations) for stoichiometric Me 10 (AO 4 ) 6 X 2 apatites (phosphate, arsenate, and vanadate with different Me 2+ and X − ), such as the standard enthalpy of formation from elements ΔH°f ,el , the standard entropy S°2 98.15K , the specific heat capacity C°p ,m , the molar volume V m , and the solubility constant K sp,298.15K . The Gibbs free energy of formation (ΔG°f ,el ) is not included to maintain consistency in the thermodynamic data presented. ΔG°f ,el values available in the literature are mostly calculated from approximations or using different, often mixed thermodynamic databases, which contributes significant scatter. Therefore, the compilation and variability analysis of the ΔG°f ,el data for apatites should be discussed in a separate paper.
The observed discrepancies in the data are likely due to the varying crystallinity states, polymorphs (either hexagonal or monoclinic, mostly not identified in literature reports), nonstoichiometry, hydration state and/or the presence of undetected impurities. A lower degree of crystallinity, for example, may favor somewhat less negative values of ΔH°f ,el (Craig and Rootare 1974). The difference between the hexagonal (P6 3 /m) and monoclinic (P2 1 /b) symmetries results in different positioning of the X − anions along the apatitic channels (giving rise or not to a mirror plane) but does not correspond to a large ion rearrangement. Therefore, the energetics of formation are not expected to be very different (although not identical), allowing both polymorphs to be considered equal. Drouet (2015) and Puzio et al. (2022) previously reported that thermodynamic state functions for apatites vary in a regular, mostly linear manner, depending on various physicochemical parameters of their components, such as the ionic radius of X -, the electronegativity of X -, the ionization energy of X, and others. A current and complete review of the data presented in Table 1 allows such trends and relationships to be clearly observed. For example, for a given X − anion (from among OH − , F − , Cl − , or Br − ), the formation of apatite is less exothermic (the enthalpy of formation ΔH°f ,el is less negative) when apatite contains a heavier element, such as As or V instead of P and Cd or Pb instead of alkali metals (Fig. 1A). In contrast, this relationship is not observed when alkali metals (Ca 2+ , Ba 2+ or Sr 2+ ) are substituted in the Me 2+ position. It is clearly apparent from the graph that apatites form distinctly separate subgroups (Fig. 1A). Here, a subgroup is defined as a population of apatites with the same substitution at position Me and A but with different substitutions at position X (where X = F, Cl, Br, I, OH) e.g., subgroup of Table 1 Experimental-based and "ab initio" literature data available for apatite end-members with the general chemical formula Me 10 (AO 4 ) 6 X 2 (where Me = Ca, Ba, Sr, Pb, Cd; A = P, As, V, and X = F, OH, Cl, Br, I), at T     Ca 10 (PO 4 ) 6 X 2 . The correlation of ΔH°f ,el of apatites with molar volume of apatite (V m ) is also apparent (Fig. 1B). So far, such relationships can be found within P-apatites. Gaps in experimental data do not allow a complete picture of these relationships for As-or V-apatites.

Correlation of V m with ionic radius of halogen anion X
The molar volume V m is not yet known for all apatites e.g., Ca 10 (VO 4 ) 6 I 2 , Cd 10 (AsO 4 ) 6 I 2 or Ba 10 (AsO 4 ) 6 Br 2 (Table 1). Glasser and Jenkins have proposed a method to calculate missing V m values based on the sum of contributions of internally consistent single-ion volumes (Jenkins and Glasser 2003;Glasser and Jenkins 2008). The use of their method gives promising and accurate results with the uncertainty not exceeding ± 11% compare to experimental V m (Glasser and Jenkins 2008). Over the last 15 years, many of the experimental diffraction data have been published for not only phosphate but also arsenate and vanadate apatites. This allows the calculation of more experimental V m values and verification of this approach.
In this work, we propose a different procedure for predicting V m values for apatites whose structure has not yet been determined or for potential apatite-based structures predicted by Wang (2015) and Hartnett et al. (2019). The method is based on the linear correlation of the V m value with the ionic radius (R i ) of the halides present at the X position (Fig. 2). In this procedure, all available experimental data of apatites and their synthetic analogs (exptlV m ) are divided into apatite subgroups based on the same substitution at Me 2+ and AO 4 3− positions but different X. The subgroups should be considered separately within the X substitutions excluding OH (X = F, Cl, Br, I), e.g., Ca 10 (PO 4 ) 6 X 2 , Pb 10 (PO 4 ) 6 X 2 , Ca 10 (AsO 4 ) 6 X 2 , Pb 10 (AsO 4 ) 6 X 2 , Ca 10 (VO 4 ) 6 X 2 , Pb 10 (VO 4 ) 6 X 2 , etc. A complete dataset within apatite subgroups exists for the Pb 10 (AsO 4 ) 6 X 2 , Pb 10 (VO 4 ) 6 X 2 and Cd 10 (VO 4 ) 6 X 2 (Fig. 2). Both visual inspection and Pearson correlation coefficient along with R 2 values greater than 0.99 indicate positive linear correlations. This positive correlation of V m vs. R i allows for interpolation and extrapolation within other apatite subgroups. Linear correlation was assumed for all apatite subgroups based on linearity within subgroups with the most available experimental data. If there are at least two known values of exptlV m within a subgroup, the parameters a and b of the linear regression between exptlV m and R i of the halides can be calculated. The unknown values of V m are predicted from the relationship (determined separately for each subgroup of apatites): (1) predV m = a × R i + b where predV m is predicted molar volume of apatite and R i is the ionic radius of element X (X = F − , Cl − , Br − , and I − ; Table SI 1). Linear regression coefficients a and b are listed in Table SI 2. The results of calculations are presented in Fig. 2 as empty marks. Predicted molar volumes (predV m ) are summarized in Table 2. These volumes will be used in calculations below as data equal to the experimental ones.
A comparison of the values obtained using the approach presented here (predV m ) with those obtained using the Glasser-Jenkins (2008) method (calcV m ) and with the experimental values is presented in Table 2. Precision of prediction was estimated by the relative percentage difference. The difference between exptlV m and the same values calculated from the regression does not exceed 0.5% for any apatite considered. In contrast, the differences determined for the values calculated by the Glasser-Jenkins method are up to 10% for calcium phosphate apatites, 30% for lead phosphate apatites, or 20% for cadmium phosphate apatites. This large difference is partly because the volumes used by Glasser and Jenkins (2008) for Pb 2+ and Cd 2+ cations were not corrected (calibrated) but taken directly from Marcus (1987). This indicates that greater precision in predicting V m values was achieved using the approach presented in this work.

Correlation of lattice energy U POT with V m
U POT is the energy change upon the formation of one mole of an ionic compound from its constituent ions in the gaseous state. Experimental lattice energy (exptlU POT ) can be determined using Born-Haber thermochemical cycles described in detail by Flora et al. (2004b). For those apatites for which experimentally determined ΔH°f ,el is available, the exptlU POT values are summarized in Table 3. The thermochemical data necessary to determine exptlU POT are given in Table SI 3.
The lattice energies listed as exptlU POT in Table 3 were obtained from the lattice enthalpy ΔH latt by correcting for the difference between enthalpy and lattice energy U POT (Jenkins 2005). ΔH latt involves correction of the U POT term by an appropriate RT (where R is the gas constant and T is the temperature in K; Jenkins and Liebman 2005). For U POT extraction from the Born-Fajans-Haber cycle (which is essentially an enthalpy-based thermochemical cycle) the ΔH latt must be transformed using an extension discussed by Jenkins et al. 1999. Finally, for F-, Cl-, Br-and I-apatites, ΔH latt = U POT, so we do not present ΔH latt values separately (Jenkins et al. 1999).
Lattice energy can be calculated also as calcU POT using the improved Kapustinskii equation, a generalized version of which was given by Glasser and Jenkins (2000). This equation for an isostructural family of minerals requires no parameters other than the molar volume V m (in nm 3 ) and is reduced to the form: Flora et al. (2004b) used this equation to calculate calcU-POT values for phosphate apatites. We have extended these calculations to As-and V-apatites using both experimental and predicted V m ( Table 3). The results are presented in Table 3 (calcU POT ) and in Fig. 3. The values calculated based on Eq. (2) differ both from exptlU POT and from intuitively expected numbers. The U POT value depends not only on the morphology and distribution of the individual atoms  in the structure but also to a large extent on the chemical nature of these atoms, which is not included in the calculations. For example, for the apatite pair Ca 10 (PO 4 ) 6 F 2 and Cd 10 (PO 4 ) 6 Cl 2 , the experimentally determined exptlU POT values are 17,124 and 18,063 kJ mol −1 , respectively. However, since the difference in exptlV m for these end-members is small (on the order of 0.4%), the calcU POT values determined for these apatites from Eq. (2) are 16,554 and 16,577 kJ mol −1 , respectively. Not only do these values deviate significantly from experimental determinations, but they are also almost indistinguishable from one another. This is, among other things, an artifact of using the molar volume V m as the only variable in Eq.
(2). In contrast, the plot of exptlU POT against V m shows that there is a linear relationship between them within the distinct apatite subgroups (Fig. 3). The different slopes of the trend lines show the varying effect of the halogen on the thermochemical behavior for apatite subgroups. Some apatites have Note: exptlV m -experimental data extracted from Table 1; calcV m -based on Glasser and Jenkins (2008); predV m -based on Eq. 1; %diff 1 = 100 · (exptlV m -calcV m ) / exptlV m ; %diff 2 = 100 · (exptlV m -predV m ) / exptlV m ; the data in the first column will be used in further calculations  Fig. 3 can be used for interpolation and extrapolation to predict missing U POT values. The steps in determining U POT and the prediction process are similar to the prediction of V m . The exptlU POT data of apatites and their synthetic analogs should be divided into apatite subgroups. The subgroups should be considered separately within the X substitutions excluding OH (X = F, Cl, Br, I), e.g., Ca 10 (PO 4 ) 6 X 2 , Pb 10 (PO 4 ) 6 X 2 , Ca 10 (AsO 4 ) 6 X 2 , Pb 10 (AsO 4 ) 6 X 2 , Ca 10 (VO 4 ) 6 X 2 , Pb 10 (VO 4 ) 6 X 2 , etc. If there are at least two known values of exptlU POT within a subgroup, the parameters a and b of the linear regression between exptlU POT and the molar volume V m are calculated. Lattice energy predU POT is predicted from the equation: The predU POT values obtained by this method are plotted in Fig. 3 as empty marks. Linear regression coefficients a and b along with Pearson coefficient R and R 2 are listed in Table SI 4. A comparison of the predU POT values and calcU POT obtained using Eq. (2) with the experimental values shows that greater precision in predicting U POT values was achieved (as assessed by the relative percentage deviation from experimental data). The difference between exptlU POT and the values calculated from the regression does not exceed 0.05% for any apatite considered. In contrast, the differences calculated using the values computed by the Glasser-Jenkins (2000) method are up to 4% for calcium phosphate apatites, 8% for lead phosphate apatites, or 9% for cadmium phosphate apatites. All the predU POT values summarized in Table 3 will be used in further calculations below on par with the experimental data. Figure 4 shows examples of the linear correlation of ΔH°f ,el of apatites as a function of U POT for selected phosphate

Prediction of ΔH°f ,el using U POT
apatites. The linearity of these correlations is enforced by the Born-Haber cycle. Phosphate apatites were chosen to present these correlations. This is currently impossible for As-and V-apatites due to the lack of data. Using all the exptlU POT and the predU POT calculated from Eq. 3, the predΔH°f ,el can be determined by extrapolating the linear relationships shown in Fig. 4: The linear regression coefficients given in Table SI 5 were used for the calculations according to Eq. 4. The values obtained by this method are plotted in Fig. 4 as empty marks. A comparison of the predΔH°f ,el with the experimental ones is shown in Table 4. The discrepancies do not exceed 0.1% relative error. This correlation allowed the prediction of eight, so far unknown, ΔH°f ,el values for the following end-members: Ca 10 (PO 4 ) 6 I 2 , Sr 10 (PO 4 ) 6 Br 2 , Sr 10 (PO 4 ) 6 I 2 , Ba 10 (PO 4 ) 6 Br 2 , Ba 10 (PO 4 ) 6 I 2 , Cd 10 (PO 4 ) 6 Br 2 , Cd 10 (PO 4 ) 6 I 2 , and Pb 10 (PO 4 ) 6 I 2 .

Prediction of ΔH°f ,el of apatites using ΔH°f ,el of X −
The correlations of ΔH°f ,el with exptlU POT do not allow the prediction of missing ΔH°f ,el for As-and V-apatites even in the case when ΔH°f ,el is available: the enthalpy of formation of the gaseous AsO 4 3− and VO 4 3− ions is still unknown, making it impossible to determine exptlU POT using Born-Haber cycles. Moreover, "ab initio " calculations are also not feasible due to the structural complexity of these particular apatites. To address this issue, we explored a linear relationship between ΔH°f ,el of apatite and ΔH°f ,el of monovalent anion X − .
The experimental ΔH°f ,el from Table 1 and the predictions  from Table 4 were used to plot these relationships (Fig. 5). In addition to halides, the OH − anion and OH-apatites were used because they fit the linear trends with R 2 > 0.99 (except Pb 10 (PO 4 ) 6 X 2 Cd 10 (PO 4 ) 6 X 2 Ca 10 (PO 4 ) 6 X 2 Ba 10 (PO 4 ) 6 X 2 Sr 10 (PO 4 ) 6 X 2 F Cl I Br for Ba 10 (PO 4 ) 6 X 2 where R 2 = 0.97; Table SI 6). The extrapolation of the regression lines allowed to obtain a prediction of ΔH°f ,el for calcium and lead As-apatites. For calculation of predicted ΔH°f ,el from the equation: The ΔH°f ,el of X − from Table SI 1 and linear correlation  coefficients from Table SI 6 were used. The existing and predicted ΔH°f ,el data are compared in Table 5. The difference between exptlΔH°f ,el and the values calculated from the regression does not exceed 0.27% for any apatite considered. The ΔH°f ,el values were predicted for the following apatites: Ca 10 (AsO 4 ) 6 Cl 2 , Ca 10 (AsO 4 ) 6 Br 2 , Ca 10 (AsO 4 ) 6 I 2 , Pb 10 (AsO 4 ) 6 Br 2 , and Pb 10 (AsO 4 ) 6 I 2 . Linear extrapolation from only two points was used for Ca 10 (AsO 4 ) 6 X 2 . The linear correlation was assumed based on the linearity within other apatite subgroups.
Prediction of ΔH°f ,el of apatites using ΔH°f ,el of AO 4

3−
Due to lack of data, the prediction methods presented above do not allow estimation of ΔH°f ,el for V-apatites. Only two experimental ΔH°f ,el for the synthetic vanadinite analog Pb 10 (VO 4 ) 6 Cl 2 are known. Therefore, an attempt was made to use the relationship between ΔH°f ,el of apatite and ΔH°f ,el of the AO 4 3− anion. The availability of experimental data allows to plot such a dependence only for lead apatites Pb 10 (AO 4 ) 6 Cl 2 , where A = P, V, or As (Fig. 6). Since ideal linear fit is apparent (R 2 = 1.00), we hypothesize that linear correlation also exists for other apatite subgroups, with the same Me and X but different A. The lines drawn for the various P-and As-apatites (Fig. 7) allow to Fig. 4 Correlation of ΔH°f ,el with U POT (from the first column in Table 3 Cd 10 (PO 4 ) 6 X 2 Ca 10 (PO 4 ) 6 X 2 Ba 10 (PO 4 ) 6 X 2 Sr 10 (PO 4 ) 6 X 2 F Ca 10 (PO 4 ) 6 X 2 Sr 10 (PO 4 ) 6 X 2 Ba 10 (PO 4 ) 6 X 2 Pb 10 (PO 4 ) 6 X 2 Cd 10 (PO 4 ) 6 X 2 Ca 10 (AsO 4 ) 6 X 2 Pb 10 (AsO 4 ) 6 X 2 Fig. 5 Correlation of the ΔH°f ,el of apatites vs. the ΔH°f ,el of anions X − . Experimental data and values predicted using Eq. 4 (Table 4) were used to plot regression lines (full symbols). Empty marks indicate values calculated from Eq. 5 determine the ΔH°f ,el of their vanadate counterparts. Linear regression coefficient a and b given in Table SI 7 were used to calculate predΔH°f ,el of V-apatites using Eq. 6: The ΔH°f ,el of AO 4 3− were extracted from Table SI 3. The results are summarized in Table 6. It is important to note that prediction of ΔH°f ,el for V-apatites would not have been possible without firstly estimating the values of ΔH°f ,el by the predictive methods described above.

Discussion
Experimental data selected from Table 1 and predicted values recommended in this work (Table 7) allow for comparison of ΔH°f ,el and presentation of the linear relationships observed within apatite subgroups (Fig. 8). The dependence of ΔH°f ,el on the molecular weight is apparent. The heavier halide substituted within any of the apatite subgroups the less negative ΔH°f ,el (apatite is less stable). This relationship is identical within all apatite subgroups studied but the intensity of this effect varies as evidenced by different slope coefficients of the trend lines. This observation also applies to the molecular weight of whole apatite. The lightest phosphate apatites have the most negative ΔH°f ,el and the heaviest lead arsenate apatites have the least negative ΔH°f ,el . Therefore, Sr 10 (PO 4 ) 6 F 2 is enthalpically the most stable of all the apatites studied while Pb 10 (AsO 4 ) 6 I 2 is the least stable one.
The mass of the tetrahedral anion AO 4 3− and the mass of the anion at the X position strongly and equally affect the ΔH°f ,el but the mass of the metal cation Me 2+ does not influence ΔH°f ,el unambiguously. Apatites containing alkaline earth metal cations (Ca 2+ , Sr 2+ , Ba 2+ ) are more enthalpically stable than apatites of other metals, e.g., Pb and Cd (but also Zn, Cu, Fe, see Drouet 2015Drouet , 2019, regardless of Table 5 Comparison of the experimental enthalpies of formation and selected predΔH°f ,el from Table 4 with values predicted using ΔH°f ,el of monovalent anion X − Note: exptlΔH°f ,el -experimental data extracted from Table 1; a predΔH°f ,el -calculated based on Eq. 4; b predΔH°f ,el -calculated based on Eq. 5; %diff = 100 · (exptlΔH°f ,el -b predΔH°f ,el ) / exptlΔH°f ,el or %diff = 100 · ( a predΔH°f ,el -b predΔH°f ,el ) / a predΔH°f ,el Pb 10 (AO 4 ) 6 I 2 Pb 10 (AO 4 ) 6 F 2 Pb 10 (AO 4 ) 6 Cl 2 Pb 10 (AO 4 ) 6 OH 2  or Ba 2+ is minimal compared to the differences with other apatites (even though the difference in the molecular mass of these cations is very pronounced, and the contribution of the cation to the formula is the largest). This may indicate that it is the chemical character of the Me 2+ bond in the apatite structure that has also a strong effect on the ΔH°f ,el . The chemical character of the bonds is similar within alkali earth elements (Ca 2+ , Sr 2+ , and Ba 2+ ) and different for heavy metals (Pb 2+ , Cd 2+ , etc.). Hydroxylapatites fit well into linear regression line in the relationship between ΔH°f ,el of apatite and ΔH°f ,el of X − (Fig. 8A). This is somewhat obvious since the enthalpy of X − is the component for the calculation of the enthalpy of apatite formation. Such complete linear dependencies are rare. Deviations of hydroxylapatites from the trend line are more often observed as in the case of ΔH°f ,el of apatite versus V m (Fig. 8B). This reflects the dissimilarity of the OH − anion from the halide anion, due, among other things, to the anisotropy of the charge distribution. The presence of such an anion in the hexagonal tunnel of the apatite crystalline structure imposes a higher energy penalty resulting in higher ΔH°f ,el (more endothermic). The presence of H + on the OH − group can also lead to hydrogen bonding effects within apatite channels, probably causing a stabilizing effect. Additionally, the OH − ions in the X-position ions in the channels within the apatite structure do not occupy the same positions in the z value along the c-axis. A larger X-site anion results in more separation from the mirror plane of the triangular cationic II sites.
The recommended ΔH°f ,el (Table 8) show linear correlation also with the electronegativity of the halide X, the ionization energy of the halide X and the ΔH°f ,el of MeX 2 (Figs. SI 1, 2, 3). All relationships give a very good or good linear fit. These correlations have been reported before but referred only to experimental data (Cruz et al. 2005b;Drouet 2015;Puzio et al. 2022). The fact that the experimental and predicted values match these lines equally well can be taken as evidence of their reliability.
Linear relationships between selected parameters within apatite subgroups are used to predict missing thermodynamic data by regression analysis. The proposed complete procedure consists of 5 steps and is shown in Table 9. The  (Table 8)  Cd 10 (PO 4 ) 6 X 2 Ca 10 (AsO 4 ) 6 X 2 Pb 10 (AsO 4 ) 6 X 2 Pb 10 (VO 4 ) 6 X 2 Ca 10 (VO 4 ) 6 X 2 order in which the calculations are performed is crucial because only by supplementing the database with the values obtained from one prediction could the calculations for obtaining subsequent prediction values be performed. This procedure allowed for the prediction of 22 thus far experimentally unknown ΔH°f ,el values for apatite end-members. This includes 9 values for iodoapatites which are the least characterized apatites. The percentage relative difference which is a measure of precision is in most cases less than 1%. The prediction precision is due to the high regression coefficients (above R 2 = 0.98). Such precision is comparable to the experimental uncertainty obtained when reproducing experimental data using calorimetric measurements or dissolution experiments. It is also higher than in other prediction methods proposed so far. Using the ΔH°f ,el recommended in this work the solubility constants K sp,298.15K can be calculated and compared where available with experimental data. It is based on dissolution reaction: LogK sp, 298.15K is calculated from the equation: where ΔG°r is the free Gibbs energy of the dissolution reaction (7), T is temperature (in K), R is the gas constant (8.31447 J mol −1 K −1 ) and superscript "°" denotes normal conditions. The thermodynamic data used in calculations are provided in Tables SI 3 and SI 9. Comparison of the calculated K sp,298.15 K with previously reported values indicates very good or good agreement within the experimental error (Tab. SI 9). This confirms the usefulness and reliability of the ΔH°f ,el predicted here for thermodynamic calculations.

Conclusions
A method for predicting the ΔH°f ,el of apatites using molar volume, lattice energy, and ΔH°f ,el of anions AO 4 3− or X − was proposed and demonstrated on phosphate, arsenate, and vanadate apatites containing Ca, Sr, Ba, Pb, and Cd at the cationic positions and F, OH, Cl, Br, and I at the halide position. The approach is based on regression analysis of the correlations occurring within apatite subgroups. These subgroups are formed by Me 10 (AO 4 ) 6 X 2 apatites with the same Me 2+ cations and tetrahedral AO 4 3− anions and with different halides in the X position (or a complex monovalent OH − anion). This approach not only leads to more accurate predictions (with precision comparable with the experimental uncertainty) but allows to see important relationships between apatites and should also be used when analyzing other properties of apatite end-members. The proposed prediction procedure allowed for the prediction of 22 so far unknown ΔH°f ,el and can be applied to a wider range of apatites than other methods. Due to lack of experimental data, it is still not possible to predict the ΔH°f ,el for Sr 10 (VO 4 ) 6 X 2 , Ba 10 (VO 4 ) 6 X 2 , Cd 10 (AsO 4 ) 6 X 2 , Sr 10 (AsO 4 ) 6 X 2 or Ba 10 (AsO 4 ) 6 X 2 . The new prediction method for ΔH°f ,el of apatites could provide important insights, e.g., allowing optimization of the chemical composition and properties of apatite-based materials for their suitability to various forms of nuclear waste deposited in geological repositories.

Acknowledgements
The authors would like to thank the two anonymous reviewers for their diligent and very thorough work on the manuscript.
Authors contributions Conceptualization; Methodology; Formal analysis; Investigation resources; Data curation; Writing-original draft Table 8 The prediction method for standard enthalpies of apatites using the molar volume, lattice energy, ΔH°f ,el of anions AO 4 3− or X − and linear regression Step Procedure 1 Compilation of existing experimental molar volume V m data for apatites and estimation of missing data (where possible) based on the linear relationship of V m with halide ionic radius R i (X − ) plotted separately for the halide apatite subgroups 2 Compilation of existing lattice energy U POT experimental data and estimation of missing data (where possible) based on the linear dependence of U POT on V m plotted separately for the halide apatite subgroups (utilizing both experimental V m and values predicted in step 1) 3 Compilation of existing experimental enthalpies of formation from elements ΔH°f ,el for apatites and estimation of the missing data (where possible) based on linear relationship of ΔH°f ,el of apatites with U POT plotted separately for the halide apatite subgroups 4 Using the experimental ΔH°f ,el and values predicted for apatites in step 3, estimation of missing values (where possible) from the linear relationship of ΔH°f ,el of apatites with ΔH°f ,el of halide anions X − plotted separately for the halide subgroups 5 Using the experimental ΔH°f ,el and values predicted in the steps above, estimation of missing values (where possible) based on the linear relationship of ΔH°f ,el of apatites with ΔH°f ,el of tetrahedral anions AO 4 3− plotted separately for the apatite subgroups containing the same Me 2+ cations and the same X − anions preparation; Visualization; Project administration; Funding acquisition: Bartosz Puzio; Writing-review and editing; supervision: Maciej Manecki. All authors have read and agreed to the published version of the manuscript.

Funding
The authors have no relevant financial or non-financial interests to disclose. The research leading to these results received funding from the Polish National Science Center (NCN) under Grant Agreement [Grant No. 2017/27/N/ST10/00776].
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