# Linear Correlations of Gibbs Free Energy of REE Phosphates (Monazite, Xenotime, and Rhabdophane) and Internally Consistent Binary Mixing Properties

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{f}) values reported for these minerals in the literature vary by up to 25 kJ mol

^{−1}. Here, we present linear free energy relationships that allow the evaluation and estimation of the ∆G°

_{f}values at 25 °C and 1 bar for the three minerals from the ionic radius (r

_{REE}

^{3+}) and the non-solvation Gibbs free energy contribution to the REE

^{3+}aqua ion (∆G°

_{n}

_{, REE}

^{3+}): ∆G°

_{f}

_{,monazite}− 399.71 r

_{REE}

^{3+}= 1.0059 ∆G°

_{n}

_{,REE}

^{3+}− 2522.51; ∆G°

_{f}

_{,xenotime}− 344.08 r

_{REE}

^{3+}= 0.9909 ∆G°

_{n}

_{,REE}

^{3+}− 2451.53; and ∆G°

_{f}

_{,rhabdophane}− 416.17 r

_{REE}

^{3+}= 1.0067 ∆G°

_{n}

_{, REE}

^{3+}− 2688.86. Moreover, based on the new dataset derived for REE end-members, we re-fitted the binary Margules parameter (W) from previous theoretical calculations into linear correlations: W + 0.00204 ∆G°

^{’}

_{n}

_{,monazite}= 39.3549 ∆V + 0.0641; W + 0.00255 ∆G°

^{’}

_{n}

_{,xenotime}= 25.4885 ∆V − 0.0062. The internally consistent thermodynamic properties of these REE phosphates are incorporated into the computer program Supcrtbl, which is available online at Zhu’s research website.

## 1. Introduction

_{f}) are key thermodynamic parameters for REE mineral end-members. These parameters determine the REE species solubilities, transfer mechanism and precipitation in hydrothermal fluids, fluid–rock interaction process, and mineral chemical processing [7,8,14]. However, ∆G°

_{f}values retrieved from different sources vary greatly, with the ranges typically > ±20–40 kJ mol

^{−1}[8,11,13,15,16,17,18,19,20]. Figure 1 shows the ranges of standard-state ∆G°

_{f}for monazite–(La) and xenotime–(Er) end-members, which vary by up to 24 kJ mol

^{−1}.

_{f}values of minerals in an isostructural family may be a way of evaluating the internal consistency of ∆G°

_{f}values of the end-members. Sverjensky and Molling [21] pioneered an empirical linear free energy correlation for crystalline solids within the same structure families as follows:

_{f}

_{,MX}refers to the standard-state Gibbs free energy of formation for the solid MX. r

_{M2+}represents the Shannon–Prewitt ionic radii [22] of M

^{2+}in a given coordination state, and ∆G°

_{n}

_{,M}

^{2+}represents the non-solvation contribution to the Gibbs free energy of formation of the aqueous M

^{2+}ion. The parameters a

_{MX,}b

_{MX}, and β

_{MX}are regression parameters for the isostructural family of minerals. A list of symbols and definitions is provided in Table 1. This correlation equation is similar to the Hammett relationship for reactions in organic compounds [23]. This linear correlation has been successfully applied to other isostructural families, including carbonate, pyrochlore, zirconolite, and uranate (MUO

_{4}) minerals [24,25,26,27,28]. Wang and Xu [27] applied this correlation equation to study the metal partitioning between carbonate minerals and aqueous solutions. Zhu [29] used a similar linear correlation to estimate the surface precipitation constants for the sorption of divalent metals onto hydrous ferric oxide and calcite.

_{n}denotes the non-solvation contribution to the standard partial molal Gibbs free energy of formation for the aqueous ions [33]. The parameters a, b, and β are regressed from the experimental Margules parameters. It provides a method to estimate or evaluate the binary mixing properties between the monazite and xenotime end-members for those end-members for which there are no or limited experimental data.

_{4}; LREE: La to Gd), xenotime (HREEPO

_{4}; Tb to Lu, plus Y), and rhabdophane (LREEPO

_{4}·0.667H

_{2}O; LREE: La to Gd) are major REE phosphates, mainly formed in hydrothermal fluids at crustal conditions (Figure 2). Each of the three solid phases has its isostructural family with a very similar REE

^{3+}crystallographic radius decreasing from La

^{3+}to Gd

^{3+}for monazite and rhabdophane end-members and from Tb

^{3+}to Lu

^{3+}for xenotime end-members. However, even though the thermodynamic properties (e.g., ∆G°

_{f}, ∆H°

_{f}, and S°) of REE minerals have been experimentally determined (e.g., via calorimetry, solubility experiments, etc.), the reported thermodynamic properties vary greatly [8,11,13,15,16,18]. For this study, our objectives are to test if the linear relationships in Sverjensky and Molling [21] and Zhu [32] are applicable to the experimentally derived ∆G°

_{f}values and W interaction parameters of the trivalent REE phosphates (monazite, xenotime, and rhabdophane) end-members and their binaries, as well as to recommend a set of internally consistent thermodynamic databases for REE phosphate end-members and end-member binaries after the careful evaluation of their internal consistency.

## 2. Mathematical Formulation

#### 2.1. Linear Free Energy Correlation

_{4}or REEPO

_{4}·0.667H

_{2}O, M

^{2+}with REE

^{3+}, and X

^{2−}with PO

_{4}

^{3−}or PO

_{4}

^{3−}·0.667H

_{2}O, respectively, to examine its application in trivalent isostructural families of solids. We followed the procedure in Sverjensky and Molling [21] to derive the linear correlations of the ∆G°

_{f}values of the isostructural REE phosphate families.

^{3+}ions (∆G°

_{n}

_{,REE}

^{3+}) was calculated as follows:

_{s}

_{,REE}

^{3+}refers to the solvation contribution to the Gibbs free energy of formation of the aqueous REE

^{3+}ions (∆G°

_{f}

_{,REE}

^{3+}). ∆G°

_{s}

_{,REE}

^{3+}can be calculated using the following equation:

_{REE}

^{3+}denotes the conventional Born solvation coefficient for the aqueous ion REE

^{3+}and ε refers to the dielectric constant of water, which is 78.47 at 25 °C and 1 bar [37]. The value of ω

_{REE}

^{3+}can be calculated using the following equation:

^{abs}

_{H}

^{+}equals 2.254 × 10

^{5}J mol

^{−1}[38]; ω

^{abs}

_{REE}

^{3+}refers to the absolute Born coefficient of the aqueous REE

^{3+}ion, which can be further derived using the following equation:

_{e,REE}

^{3+}denotes the effective electrostatic radius of the aqueous REE

^{3+}ion, which can be obtained using the following equation:

_{REE}

^{3+}refers to the crystallographic radius of the REE

^{3+}ion, which represents its Shannon–Prewitt radius (Å) [39] (Table 2). Z in Equations (5)–(7) represents the charge of the REE

^{3+}ion, which is +3 for trivalent ions.

_{n}(Table 2) for the 16 trivalent aqueous REE

^{3+}cations. To present more clearly its linear correlations for ∆G°

_{f}

_{,REEPO4}, ∆G°

_{n}

_{,REE}

^{3+}, and r

_{REE}

^{3+}, we expressed the equation as shown below and plotted the left-hand sides of Equations (8) and (9) against the aqueous cation parameter ∆G°

_{n}

_{,REE}

^{3+}as follows:

_{4}refers to the isostructural family of monazite, xenotime, and rhabdophane solids. The other parameters in the equations have been introduced in Equation (1).

#### 2.2. Linear Correlation between Mixing Binaries

_{Cα}is defined as follows:

_{n}denotes the non-solvation contribution to the ∆G°

_{f}values for the aqueous ions [33]. β is an empirically derived parameter.

## 3. Data Availability and Formatting

^{3+}crystallographic radii r, ∆G°

_{n}non-solvation contributions to the ∆G°

_{f}values of the aqueous REE

^{3+}ions, and ∆G°

_{f}values of the monazite, xenotime, and rhabdophane end-members are needed to conduct the regression of the Gibbs free energy linear correlation. ∆G°

_{n}values were derived from Equations (3)–(7), and their values are provided in Table 2.

#### 3.1. Aqueous REE^{3+} Ion Parameters

^{3+}ions must be accounted for when dealing with the regression of the linear free energy correlations. The REE

^{3+}ions in monazite, xenotime, and rhabdophane have 9-, 8-, and 9-fold coordination numbers, respectively [34,43], and their REE

^{3+}crystallographic radii vary with the numbers of their coordination state. In this study, the REE

^{3+}crystallographic radii (r

_{REE}

^{3+}) at different coordination numbers are derived from Shannon [22]. The ∆G°

_{f}values of the aqueous REE

^{3+}ions are derived directly from Shock and Helgeson [38].

#### 3.2. REE Phosphate (Monazite, Xenotime, and Rhabdophane) Parameters

_{4}and TbPO

_{4}, xenotime GdPO

_{4}) were calculated from the correlation of the monazite and xenotime isostructural families, derived from the regressed correlations in Ni et al. [34] by adding the volume data of monazite–(Tb) from Ushakov et al. [41]. The molar volume of the hypothetical end-members will be used to calculate the Margules parameters between the LREE and HREE phosphate end-members. The calculated correlations are as follows:

^{1/3}) are excellent (R

^{2}> 0.995).

_{f}values of the monazite and xenotime end-members for regression analysis were sourced from Migdisov et al. [8], who derived these values from the solubility data (log K

_{sp}at 25 °C) of REE phosphates reported in Liu and Byrne [40]. We adopted the ∆G°

_{f}values of the rhabdophane end-members from Gausse et al. [9], which were calculated via the ideal formula of LREEPO

_{4}·0.667H

_{2}O. The ∆G°

_{f}values from other review studies [13,15,16] were also included in the discussion for comparison with the results obtained via the linear correlation study.

#### 3.3. Available Margules Parameters for REE Phosphate End-Member Binaries

_{4}paired with NdPO

_{4}, EuPO

_{4}, and GdPO

_{4}[44]; LaPO

_{4}paired with EuPO

_{4}and GdPO

_{4}from Neumeier et al. [45], and the binary of ErPO

_{4}-YbPO

_{4}from Strzelecki et al. [46]. The values of those Margules parameters of end-member binaries measured via calorimetric experiments differ greatly, and currently, there are only three existing end-member binaries in the literature for the monazite family [44]. Therefore, there are not enough calorimetric experimental data to build the linear correlations.

_{x}) and ionic radii r

_{x}in the xenotime family using available elastic constants data from the literature and generated an E

_{x}− r

_{x}function for xenotime: E

_{x}= 1320.4 (±179.6) − −1142.8 (±178.0)·r

_{x}. Furthermore, the study determined the W

_{x}values (Table 3 in Migdisov et al. [7]) of all xenotime end-member binaries using the equation of W = E/(dV

^{2}/6V) derived by Kowalski and Li [48], where E is the average Young’s modulus and dV is the mismatch of the cell volume values (namely V°

_{TPO4}− V°

_{CPO4}). Since the parameters E and V represent the average values of the Young’s modulus and molar volume of monazite and xenotime, and their values are constant, the W equation can be further simplified as follows:

## 4. Results

#### 4.1. Linear Free Energy Relationships for REE Phosphates and Aqueous Ions

^{3+}ionic radii and ∆G°

_{n}from Table 2. The regression results for the three isostructural families are summarized in Table 3 and plotted in Figure 3. The differences between the values retrieved from solubility experiments and calculated values from linear correlations are shown in Figure 4. The results show that the linear correlation lines of monazite, xenotime, and rhabdophane have R

^{2}values of 0.995, 0.996, and 0.998, respectively, indicating the strong linear relationship for those selected thermodynamic parameters. The linear free energy relationships are expressed as follows:

_{f}values of monazite and rhabdophane–(Pu, Tb, and Dy) and the fictive end-members of monazite and xenotime were also calculated to study the substitution of LREE in a xenotime structural lattice and HREE in a monazite lattice because the latter are difficult to synthesize in the corresponding crystal structures of the host minerals.

^{−1}. The largest discrepancies are 4.1 kJ mol

^{−1}for LaPO

_{4}and 3.2 kJ mol

^{−1}for PrPO

_{4}; these uncertainties are within those reported in calorimetric experiments. Figure 3 and the regressed coefficients listed in Table 3 indicate that the REE phosphates with the monazite, xenotime, and rhabdophane structures all have essentially the same slopes a

_{REEPO4}, implying that the parameter a

_{REEPO4}in Equation (8) is constant for all polymorphs of the composition REEPO

_{4}and its hydrous phases. The linear correlations for monazite and xenotime are generally parallel (Figure 3), which agrees with the correlations for different isostructural families with the same chemical formula [21,27,28]. The coefficient b in Equation (8) seems to only be related to the stoichiometry of solids [28]. These results indicate that the regressed linear correlations closely fit the experimentally derived ∆G°

_{f}values of isostructural families of monazite, xenotime, and rhabdophane.

_{f}values retrieved for monazite based on the solubility data from Van Hoozen et al. [18] also generated a linear correlation. The latter is generally parallel to the linear correlation line derived from the data compiled by Migdisov et al. [8], which is based on the low-temperature solubility experiments conducted by Liu and Byrne [40]. The resulting y-axis intercept displays an average shift that is 7.2 kJ mol

^{−1}lower than that derived from the regression of the data conducted by Migdisov et al. [8]. As the LREE phosphates form a more soluble hydrous phase (i.e., metastable rhabdophane) at room temperature, we recommend the correlations corrected based on the solubility data provided by Van Hoozen et al. [18]. In their study, the ∆G°

_{f}for monazite was extrapolated to 25 °C based on experiments conducted at elevated temperatures (100–250 °C) at which the monazite phase was stable and controlled solubility. Therefore, we adjusted the linear correlation in Equation (19) by subtracting 7.2 kJ mol

^{−1}from the parameter b to obtain Equation (20) to represent the linear correlation of monazite.

#### 4.2. A Semi-Empirical Correlation for Margules Parameters of REE Phosphate End-Members

_{n}values calculated in the first part of the mathematical formulation section. Multiple linear regression analysis following Equations (13)–(15) resulted in the following correlations. Other results for the correlation for Margules parameters can be found in Table 4.

^{2}is > 0.99 and the intercept is close to zero (Figure 5). Theoretically, if both the ionic radii and ionicity are adequately taken into account in the regression, the intercept should be zero. This is consistent with our regression results showing that the intercepts are 0.0641 and −0.0062 for monazite and xenotime, respectively. The β values in Equations (23) and (24) are extremely small, close to zero, indicating that the contribution of ionicity properties to the whole W is very small or can even be ignored. The differences between the theoretical calculations from the literature and the results based on the correlation are within 1.0 kJ mol

^{−1}and 0.3 kJ mol

^{−1}for monazite and xenotime end-member binaries, respectively.

_{n}value difference, or the ionic properties, to the binary mixing interaction can be considered to be a minor factor or even be omitted.

_{x}HREE

_{1−x})PO

_{4}and (Y

_{x}LREE

_{1−x})PO

_{4}, respectively [61,62]. Table 5 presents a comprehensive dataset of the Margules parameters W for modeling the monazite and xenotime binary mixing solutions in natural or laboratory conditions. It is also common for REE phosphates with a monazite structure to bond certain HREE (e.g., Y) and xenotime bonding LREE (e.g., La, Ce, Pr, Nd, Sm) in a natural environment [61,63]. Therefore, in our study, we also accounted for two types of fictive phosphate end-members to model the natural occurrence. The Margules parameters W for those fictive phosphate end-member binaries are provided in Table 5.

## 5. Discussion

#### 5.1. Comparisons with Previous Studies and Data Evaluation

#### 5.1.1. Linear Correlation Calculated by Other ∆G°_{f} Choice in the Literature

_{f}values for monazite and xenotime isostructural families based on calorimetric measurements. These data also show linear correlations (Figure 6a). However, the discrepancy between the experimental ∆G°

_{f}values and correlations can reach up to 10 kJ mol

^{−1}for monazite and up to 5 kJ mol

^{−1}for xenotime (Figure 6b,c). In contrast to the solubility-based ∆G°

_{f}linear correlations, the linear correlations for calorimetry-based ∆G°

_{f}values are not as predictive as linear correlation based on solubility measurements.

#### 5.1.2. Comparisons of the Calculated ∆G°_{f} Values with Other Data in the Literature

_{f}values used for our linear correlation calculations for monazite are sourced from Migdisov et al. [8], who derived the ∆G°

_{f}values from the solubility data reported in Liu and Byrne [40]. The LREE phosphates used in the experiments in Liu and Byrne [40] had rhabdophane components under ambient conditions. Moreover, the ∆G°

_{f}values of those LREE phosphate end-members are slightly too positive due to the higher solubility of rhabdophane, which can control the solubility of LREE phosphates at temperatures <100 °C. The ∆G°

_{f}values of monazite in Navrotsky et al. [16] were derived from the calorimetric study of Popa and Konings [15], in which the ∆H°

_{f}values were recalculated from the experimental study of Ushakov et al. [41]. Pan et al. [13] provides a set of recommended thermodynamic properties for REE phosphates and REE aqueous species, where the ∆G°

_{f}values for monazite are mainly derived from previous calorimetric studies. Van Hoozen et al. [18] provides ∆G°

_{f}values for monazite end-members at 25 °C and 1 bar by regressing solubility data derived at higher temperatures. A correction by 7.2 kJ mol

^{−1}indicated for the regression in Figure 3a presents an option for an adjusted linear correlation that fits most of the high temperature solubility- and calorimetry-based ∆G°

_{f}values.

_{f}values for DyPO

_{4}, YPO

_{4}, ErPO

_{4}, and YbPO

_{4}based on hydrothermal solubility experiments. These values agree with the regression (Equation (21)) based on the data by Migdisov et al. [8] for Y, Dy, and Er but display a significant deviation for Yb (Figure 3b). The latter could be explained by a need to revise the properties of the aqueous REE

^{3+}species instead of the mineral properties [13]. The calorimetry-based ∆G°

_{f}values from Navrotsky et al. [16] are generally parallel with the regression from Equation (21) but display discrepancies of up to 25 kJ mol

^{−1}. The ∆G°

_{f}values from Navrotsky et al. [16] were calculated from the ∆H°

_{f}values from the calorimetric data in Ushakov et al. [41], as well as the S°

_{f}(ErPO

_{4}) values from Gavrichev et al. [20] and other S°

_{f}(HREE) values from Tananaev et al. [64]. The ∆G°

_{f}values for xenotime used in the thermodynamic optimization study by Pan et al. [13] were mainly derived from the ∆H°

_{f}values from Ushakov et al. [41] and S°

_{f}(HREE) values from Gysi et al. [11], Gavrichev et al. [20], Gavrichev et al. [51], Gavrichev et al. [53], Gavrichev et al. [55], Ji et al. [58], and Tyurin et al. [59].

_{f}values of rhabdophane end-members from Gausse et al. [9] are the only available data.

#### 5.1.3. Comparisons of the Solubility Product (log K_{sp}) Performed in This Study using the Data in the Literature

_{sp}), on the other hand, give a better indicator of whether the correlations are a good representation of the experimental data. For this purpose, we calculated the log K

_{sp}values for REE phosphate (monazite, xenotime, and rhabdophane) end-members using the ∆G°

_{f}values derived from the linear correlations and the following sets of reactions:

_{r}is the standard-state Gibbs free energy of reaction in Equations (25) or (26), T is the temperature in Kelvin, and R is the ideal gas constant, 8.314 J mol

^{−1}K

^{−1}. The ∆G°

_{f}values for REE

^{3+}, PO

_{4}

^{3−}, and H

_{2}O were taken from Shock et al. [65].

#### 5.2. Evaluation of Margules Parameter W and Comparisons with Previous Studies

_{n}, as shown in Equations (13)−(15). The linear correlations found in this study are not new; they are similar relationships to those reported by Kowalski and Li [48] and Li et al. [49]. As different molar volumes were used in these studies, the slopes received (a in Equation (18)) are different. The W parameters used for the regression of xenotime only considered the contribution of the elastic effects [7], while our regression also considered ionic properties using the method outlined in Zhu [32]. However, the results are essentially the same, indicating that elasticity, not ionic properties, is the dominant factor. However, our W parameters are consistent with the molar volume data in Table 2.

^{−1}under ambient conditions. This is consistent with the calculated W parameter dataset for xenotime, with all W values being well below 5.0 kJ mol

^{−1}(Table 5). This indicates the absence of any miscibility gaps for the HREE solid solutions in the xenotime group structure.

_{4}paired with NdPO

_{4}, EuPO

_{4}, and GdPO

_{4}were measured via drop calorimetry at a temperature of 727 °C in Popa et al. [44], indicating that excess enthalpy decreases with increasing temperature. The ∆H°

_{f}values used for retrieving the Margules interaction parameters of the LaPO

_{4}paired with EuPO

_{4}and GdPO

_{4}were determined via high-temperature oxide melt solution calorimetry at 700 °C [45]. The only available W value for the xenotime end-member binary measured via calorimetry experiments at up to 727 °C is the ErPO

_{4}-YbPO

_{4}binary xenotime from Strzelecki et al. [46].

## 6. Conclusions and Recommendations

_{f}values for monazite–(La, Pr, and Nd) are from Van Hoozen et al. [18]. The ∆G°

_{f}values of monazite–(Ce, Sm, Eu, and Gd) were predicted from the linear correlations. The ∆G°

_{f}value for monazite–(Eu) extrapolated to room temperature from the hydrothermal solubility data by Van Hoozen et al. [18] is not used. Using the linear correlation, we predicted the ∆G°

_{f}values for fictive heavy REE end-members for the study of impurities in monazite.

_{f}values of xenotime end-members from Migdisov et al. [8] due to the excellent linear correlations and alignment with the values extrapolated to 25 °C from the solubility study by Gysi et al. [11] (Figure 3b). The ∆G°

_{f}values for xenotime–(Y) from Gysi et al. [11] are not used. These recommended ∆G°

_{f}values are significantly different from the calorimetry values. The reconciliation of these values will be addressed in future studies. For rhabdophane, there are not many measurements. We adopt the ∆G°

_{f}values from Gausse et al. [9] and predicted the ∆G°

_{f}values of rhabdophane–(Pu, Tb, and Dy).

_{f}values of monazite and xenotime end-members, the recommended ΔS°

_{f}values from Navrotsky et al. [16] are used to recalculate the ΔH°

_{f}values using the Benson–Helgeson convention as follows: ΔG

_{f}= ΔH

_{f}− TΔS

_{f}. The molar volume data (V°

_{m}) for monazite and xenotime are taken from Ni et al. [34]. The heat capacity function for monazite and xenotime are based on the equation below, with T in Kelvin:

## Supplementary Materials

_{f}) of monazite and xenotime at 25 °C and 1 bar sourced from the literature and reviewed by Pan et al. [13].

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The ranges of ∆G°

_{f}values for (

**a**) monazite–(La) and (

**b**) xenotime–(Er) end-members reported in the literature [8,11,13,15,16,17,18,19,20]. Data retrieved or compiled from calorimetric experiments are plotted as blue symbols, and data retrieved from mineral solubility experiments are plotted as green symbols.

**Figure 2.**Representative REE phosphate mineral structures of isostructural families of (

**a**) monazite –(Ce), (

**b**) xenotime–(Y), and (

**c**) rhabdophane–(Sm). Crystal structure data of monazite and xenotime are derived from Ni et al. [34], and the data of rhabdophane are derived from Mesbah et al. [35]. The virtualizations of the crystal structures of the three minerals were performed using the program VESTA [36].

**Figure 3.**Graphical representations of linear correlations of isostructural (

**a**) monazite, (

**b**) xenotime, and (

**c**) rhabdophane mineral families and comparisons with other experimental studies [8,9,11,15,16,17,18,20,51,52,53,54,55,56,57,58,59], as shown in Table S1. In (

**a**), the dashed line represents the linear correlation line based on the regression of the data sourced from Migdisov et al. [8], and the solid line represents a modified parallel line by subtracting 7.2 kJ mol

^{−1}for its y-axis intercept to fit the solubility data provided by Van Hoozen et al. [19].

**Figure 5.**Linear correlations between the modified Margules parameters W with a term of volume mismatch (∆V) for binary mixing solutions in the (

**a**) monazite and (

**c**) xenotime isostructural families and comparisons with other experimental studies [7,44,45,47,48,50,60]. Residuals between the regressed and theoretically calculated (

**b**) monazite and (

**d**) xenotime binary excess properties.

**Figure 6.**(

**a**) Graphical representation of linear correlations of isostructural monazite and xenotime groups using the ∆G°

_{f}data of monazite and xenotime end-members from Navrotsky et al. [16] and Migdisov et al. [8], Gausse et al. [9]. (

**b**,

**c**) Residuals between the experimentally measured and recalculated ∆G°

_{f}values of (

**b**) monazite and (

**c**) xenotime based on the calculation of linear correlations.

**Figure 7.**Linear correlation for solubility products for xenotime end-members with REE aqueous ion properties. Experimental data are taken from Liu and Byrne [40].

Symbol | Definition |
---|---|

∆G°_{f} | Gibbs free energy of formation |

∆G°_{n} | Non-solvation contribution to the Gibbs free energy of formation |

∆G°_{s} | Solvation contribution to the Gibbs free energy of formation |

∆G’_{f} | Adjusted Gibbs free energy of formation (∆G°_{f}_{,REEPO4} − β r_{REE}^{3+}) |

ΔS°_{f} | Entropy of formation from the elements |

S° | Absolute entropy |

ΔH_{f} | Enthalpy of formation from the elements |

Cp | Heat capacity |

V° | Molar volume |

∆V | Defined volume mismatch term in this study |

dV | Mismatch of the cell volume values under the definition of Young’s moduli |

R | Gas constant (8.314 J mol^{−1} K^{−1}) |

K | Equilibrium constant |

K_{sp} | Solubility constants |

W’ | Adjusted Margules parameter |

W | Margules parameter |

ω | Born coefficient of an ion |

ω^{abs} | Absolute Born coefficient of an ion |

r_{REE}^{3+} | Crystallographic radius of the aqueous REE^{3+} ion |

Z | Charge number of an ion |

α | Anion part of a mineral, such as PO_{4} |

ψ | Effective bulk modulus |

MX | Chemical formula of a solid |

T^{3+} | Trace cation of REEPO_{4} with charge of 3+ |

C^{3+} | Carrier cation of REEPO_{4} with charge of 3+ |

REE | Rare Earth Elements |

LREE | Light Rare Earth Elements |

HREE | Heavy Rare Earth Elements |

REE | r(REE^{3+}) | r(REE^{3+}) | V° (cm^{3} mol^{−1}) | ∆G°_{s} (kJ mol^{−1}) | ∆G°_{f} (kJ mol^{−1}) | ∆G°_{n} (kJ mol^{−1}) | ∆G°_{f} (kJ mol^{−1}) ^{2} | ∆G°_{f} (kJ mol^{−1}) ^{3} |
---|---|---|---|---|---|---|---|---|

Å (CN9) ^{1} | Å (CN8) ^{1} | REEPO_{4(s)} | REE^{3+}_{(aq)} | REE^{3+}_{(aq)} | REE^{3+}_{(aq)} | REEPO_{4(s)} | REEPO_{4}·0.667H_{2}O_{(s)} | |

La | 1.216 | 1.160 | 46.03 | −861.74 | −686.18 | 175.56 | −1848.53 | −2004.00 |

Ce | 1.196 | 1.143 | 45.16 | −869.35 | −676.13 | 193.22 | −1844.48 | −1997.00 |

Pr | 1.179 | 1.126 | 44.45 | −875.89 | −680.32 | 195.57 | −1850.50 | −2003.00 |

Nd | 1.163 | 1.109 | 43.86 | −882.09 | −671.95 | 210.14 | −1840.30 | −1994.00 |

Pm | 1.144 | 1.093 | 43.26 | −889.51 | −661.07 | 228.44 | ||

Sm | 1.132 | 1.079 | 42.81 | −894.24 | −665.67 | 228.57 | −1833.45 | −1989.00 |

Eu | 1.120 | 1.066 | 42.40 | −899.00 | −574.46 | 324.54 | −1741.10 | −1896.00 |

Gd | 1.107 | 1.053 | 42.01 | −904.19 | −663.58 | 240.60 | −1828.50 | −1984.00 |

Tb | 1.095 | 1.040 | 41.53 ^{4} | −909.00 | −667.35 | 241.65 | ||

Dy | 1.083 | 1.027 | −913.85 | −664.00 | 249.85 | |||

Tb | 1.095 | 1.040 | 43.90 | −931.47 | −667.35 | 264.12 | −1831.07 | |

Dy | 1.083 | 1.027 | 43.35 | −936.87 | −664.00 | 272.87 | −1829.10 | |

Y | 1.075 | 1.019 | 43.14 | −940.21 | −685.34 | 254.87 | −1849.12 | |

Ho | 1.072 | 1.015 | 42.90 | −941.89 | −675.30 | 266.59 | −1836.79 | |

Er | 1.062 | 1.004 | 42.37 | −946.52 | −669.02 | 277.50 | −1831.26 | |

Tm | 1.052 | 0.994 | 42.00 | −950.75 | −669.02 | 281.73 | −1830.52 | |

Yb | 1.042 | 0.985 | 41.64 | −954.58 | −640.15 | 314.43 | −1801.02 | |

Lu | 1.032 | 0.977 | 41.22 | −958.00 | −666.93 | 291.07 | −1826.71 |

^{1}CN: Coordination numbers: 9 for monazite–(La-Dy) and 8 for xenotime–(Tb-Lu, and Y).

^{2}Monazite–(La to Gd) and xenotime–(Tb to Lu) ∆G°

_{f}values are derived from Migdisov et al. [8], which were in turn derived from the solubility data of Liu and Byrne [40].

^{3}Rhabdophane–(La to Gd) ∆G°

_{f}values are derived from Gausse et al. [9].

^{4}Molar volume (V°) of monazite TbPO

_{4}is derived from Ushakov et al. [41]. Note that all V°, ∆G°

_{s}, ∆G°

_{n}, and ∆G°

_{f}data included here are actual figures, meaning that those data are directly measured or calculated by 9 C. N. for monazite–(La-Dy) and 8 C. N. for xenotime–(Tb-Lu, and Y). The references of the data sources in this table have been introduced in Section 3 if they are not mentioned here.

REE Phosphate Lsostructural Type | Parameters | |||
---|---|---|---|---|

a | b (kJ mol^{−1}) | β (kJ Å^{−1}) | R^{2} | |

Monazite | 1.0059 (0.0398) | −2515.31 (63.23) | 399.71 (48.33) | 0.9975 |

Xenotime | 0.9909 (0.0325) | −2451.53 (35.58) | 344.08 (27.89) | 0.9979 |

Rhabdophane | 1.0067 (0.0240) | −2688.86 (38.02) | 416.17 (29.06) | 0.9991 |

REE Phosphate Lsostructural Type | Parameters | |||
---|---|---|---|---|

a (kJ V°^{−1}) | b (kJ mol^{−1}) | β | R^{2} | |

Monazite | 39.3549 (0.5659) | 0.0641 (0.0958) | −0.0020 (0.0017) | 0.9939 |

Xenotime | 25.4885 (0.5589) | −0.0062 (0.0269) | −0.0025 (0.0007) | 0.9931 |

**Table 5.**Calculated Margules interaction parameters W in kJ/mol (upper of diagonal) for the monazite and xenotime solid solutions.

Monazite | Cation | La | Ce | Pr | Nd | Pm | Sm | Eu | Gd | Tb | Dy | Y^{1} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

W (kJ/mol) | 0.76 | 2.32 | 4.36 | 7.14 | 9.70 | 12.6 | 15.3 | 19.4 | 23.6 | 26.4 | La | ||

0.52 | 1.61 | 3.41 | 5.21 | 7.40 | 9.46 | 12.7 | 16.0 | 18.3 | Ce | ||||

0.41 | 1.41 | 2.60 | 4.23 | 5.73 | 8.28 | 11.0 | 12.9 | Pr | |||||

0.43 | 1.12 | 2.28 | 3.33 | 5.32 | 7.50 | 9.02 | Nd | ||||||

0.25 | 0.95 | 1.56 | 2.98 | 4.64 | 5.83 | Pm | |||||||

0.42 | 0.69 | 1.69 | 2.96 | 3.89 | Sm | ||||||||

0.04 | 0.66 | 1.58 | 2.28 | Eu | |||||||||

0.33 | 0.93 | 1.41 | Gd | ||||||||||

0.26 | 0.49 | Tb | |||||||||||

0.06 | Dy | ||||||||||||

Y | |||||||||||||

Xenotime | Cation | Tb | Dy | Y | Ho | Er | Tm | Yb | Lu | ||||

W (kJ/mol) | 0.19 | 0.46 | 0.59 | 1.44 | 2.23 | 3.25 | 4.50 | Tb | |||||

0.13 | 0.10 | 0.58 | 1.12 | 1.89 | 2.85 | Dy | |||||||

−0.09 | 0.26 | 0.70 | 1.37 | 2.21 | Y | ||||||||

0.19 | 0.52 | 1.09 | 1.80 | Ho | |||||||||

0.09 | 0.41 | 0.85 | Er | ||||||||||

0.16 | 0.39 | Tm | |||||||||||

0.04 | Yb | ||||||||||||

Lu | |||||||||||||

Xenotime | Cation | La | Ce | Pr | Nd | Pm | Sm | Eu | Gd | ||||

Fictive | W (kJ/mol) | 22.9 | 17.4 | 12.8 | 8.86 | 5.89 | 3.74 | 1.98 | 1.08 | Y^{1} |

^{1}W parameters for fictive REE phosphate end-members, as marked in italic.

**Table 6.**Recommended standard thermodynamic properties of monazite, xenotime, and rhabdophane at a temperature of 298.15 K and pressure of 1 bar, as well as the heat capacity function, with T in Kelvin.

ΔG°_{f} | ΔH°_{f} | ΔS°_{f} [16] | S° [16] | V°_{m} [34] | Cp = a + bT + c/T^{2}
+ d/T^{0.5} | |||||
---|---|---|---|---|---|---|---|---|---|---|

kJ mol^{−1} | kJ mol^{−1} | J mol^{−1} K^{−1} | J mol^{−1} K^{−1} | J mol^{−1} bar^{−1} | a | b*100 | c | d | Reference | |

Monazite | ||||||||||

LaPO_{4} | −1861.2 | −1980.46 | −400.0 | 108.3 | 46.03 | 121.13 | 3.0116 | −2,562,500 | - | [15] |

CePO_{4} | −1851.7 | −1971.33 | −401.3 | 120.0 | 45.16 | 125.21 | 2.7894 | −2,408,500 | - | [15] |

PrPO_{4} | −1855.6 | −1975.55 | −402.3 | 123.2 | 44.45 | 124.50 | 3.0374 | −2,449,500 | - | [15] |

NdPO_{4} | −1846.2 | −1965.76 | −401.0 | 122.9 | 43.86 | 132.96 | 2.2541 | −3,100,900 | - | [15] |

SmPO_{4} | −1840.7 | −1959.52 | −398.7 | 122.5 | 42.81 | 133.13 | 2.3468 | −3,068,700 | - | [15] |

EuPO_{4} | −1748.3 | −1871.17 | −412.1 | 117.2 | 42.40 | 137.56 | 1.7693 | −2,785,400 | - | [15] |

GdPO_{4} | −1835.7 | −1953.47 | −395.0 | 124.6 | 42.01 | 133.24 | 1.2793 | −3,097,200 | - | [15] |

TbPO_{4} | −1841.7 | - | - | - | 41.53 ^{1} | - | - | - | - | This study |

DyPO_{4} | −1838.3 | - | - | - | 41.10 ^{2} | - | - | - | - | This study |

(YPO_{4}) | −1859.7 | - | - | - | 40.82 ^{2} | - | - | - | - | This study |

(HoPO_{4}) | −1849.6 | - | - | - | 40.71 ^{2} | - | - | - | - | This study |

(ErPO_{4}) | −1843.1 | - | - | - | 40.37 ^{2} | - | - | - | - | This study |

(TmPO_{4}) | −1843.0 | - | - | - | 40.03 ^{2} | - | - | - | - | This study |

(YbPO_{4}) | −1813.8 | - | - | - | 39.68 ^{2} | - | - | - | - | This study |

(LuPO_{4}) | −1859.7 | - | - | - | 39.34 ^{2} | - | - | - | - | This study |

Xenotime | ||||||||||

(LaPO_{4}) | −1857.1 | - | - | - | 49.35 ^{2} | - | - | - | - | This study |

(CePO_{4}) | −1846.4 | - | - | - | 48.56 ^{2} | - | - | - | - | This study |

(PrPO_{4}) | −1849.8 | - | - | - | 47.77 ^{2} | - | - | - | - | This study |

(NdPO_{4} | −1840.6 | - | - | - | 47.00 ^{2} | - | - | - | - | This study |

(SmPO_{4}) | −1832.7 | - | - | - | 45.66 ^{2} | - | - | - | - | This study |

(EuPO_{4}) | −1741.6 | - | - | - | 45.08 ^{2} | - | - | - | - | This study |

(GdPO_{4}) | −1829.1 | - | - | - | 44.51 ^{2} | - | - | - | - | This study |

TbPO_{4} | −1831.1 | −1946.3 | −386.6 | 138.1 | 43.90 | 116.4 | 4.55 | −2,190,000 | - | [67] |

DyPO_{4} | −1829.1 | −1945.0 | −388.6 | 138.1 | 43.35 | 185.5 | 0.00 | −3,261,000 | −751.900 | [68] |

YPO_{4} | −1849.1 | −1964.4 | −386.8 | 108.8 | 43.14 | 131.3 | 1.992 | −3,563,700 | - | [54] |

HoPO_{4} | −1836.8 | −1951.4 | −384.4 | 142.3 | 42.90 | 124.4 | 2.658 | −2,690,000 | - | [59] |

ErPO_{4} | −1831.3 | −1952.8 | −407.7 | 116.6 | 42.37 | 205.5 | −0.076 | −859,073 | −1651.88 | [20] |

TmPO_{4} | −1830.5 | −1945.9 | −387.1 | 138.1 | 42.00 | 128.8 | 1.904 | −3,090,000 | - | [12] |

YbPO_{4} | −1801.0 | −1913.5 | −377.1 | 133.9 | 41.64 | 198.0 | 0.448 | −991,250 | −1506.38 | [55] |

LuPO_{4} | −1826.7 | −1941.6 | −385.2 | 117.2 | 41.22 | 130.7 | 1.85 | −3,330,000 | - | [53,69] |

Rhabdophane | ||||||||||

LaPO_{4}·0.667H_{2}O | −2004.0 | −2151.3 | −494.0 | 170.0 | - | - | - | - | - | [9] |

CePO_{4}·0.667H_{2}O | −1997.0 | −2147.3 | −504.0 | 175.0 | - | - | - | - | - | [9] |

PrPO_{4}·0.667H_{2}O | −2003.0 | −2144.0 | −473.0 | 210.0 | - | - | - | - | - | [9] |

NdPO_{4}·0.667H_{2}O | −1994.0 | −2142.8 | −499.0 | 180.0 | - | - | - | - | - | [9] |

SmPO_{4}·0.667H_{2}O | −1989.0 | −2137.8 | −499.0 | 177.0 | - | - | - | - | - | [9] |

EuPO_{4}·0.667H_{2}O | −1896.0 | −2056.4 | −538.0 | 149.0 | - | - | - | - | - | [9] |

GdPO_{4}·0.667H_{2}O | −1984.0 | −2130.7 | −492.0 | 182.0 | - | - | - | - | - | [9] |

TbPO_{4}·0.667H_{2}O | −1989.9 | - | - | - | - | - | - | - | - | This study |

DyPO_{4}·0.667H_{2}O | −1986.6 | - | - | - | - | - | - | - | - | This study |

^{1}V

_{m}of monazite TbPO

_{4}is calculated from Ushakov et al. [41].

^{2}V

_{m}of monazite DyPO

_{4}and other monazite fictive end-members are calculated from Equation (16); V

_{m}of xenotime fictive end-members are calculated from Equation (17). Please note that in the ΔG

^{°}

_{f}and V

^{°}

_{m}columns, the data in regular font and red color are retrieved from experiments, and the data in italic font are calculated in this study. The REE phosphates in parentheses are fictive end-members. The references for the ΔG

^{°}

_{f}values retrieved from experiments can be found in the text.

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## Share and Cite

**MDPI and ACS Style**

Pan, R.; Gysi, A.P.; Migdisov, A.; Gong, L.; Lu, P.; Zhu, C.
Linear Correlations of Gibbs Free Energy of REE Phosphates (Monazite, Xenotime, and Rhabdophane) and Internally Consistent Binary Mixing Properties. *Minerals* **2024**, *14*, 305.
https://doi.org/10.3390/min14030305

**AMA Style**

Pan R, Gysi AP, Migdisov A, Gong L, Lu P, Zhu C.
Linear Correlations of Gibbs Free Energy of REE Phosphates (Monazite, Xenotime, and Rhabdophane) and Internally Consistent Binary Mixing Properties. *Minerals*. 2024; 14(3):305.
https://doi.org/10.3390/min14030305

**Chicago/Turabian Style**

Pan, Ruiguang, Alexander P. Gysi, Artas Migdisov, Lei Gong, Peng Lu, and Chen Zhu.
2024. "Linear Correlations of Gibbs Free Energy of REE Phosphates (Monazite, Xenotime, and Rhabdophane) and Internally Consistent Binary Mixing Properties" *Minerals* 14, no. 3: 305.
https://doi.org/10.3390/min14030305