A comparison of methods for the estimation of the enthalpy of formation of rare earth compounds

Abstract

Rare earth elements are helping drive the global transition towards a greener economy. However, the way in which they are produced is far from being considered green. One of the major obstacles to developing greener production methods and the design of novel processes and materials involving rare earth elements is the limited thermodynamic data available. In the present work, we apply a suite of methods to estimate the enthalpy of formation of several rare earth compounds, including a new method based on a linear relationship, established by the authors. Experimental values of the enthalpy of formation of LnCl₃, LnOCl, LnPO₄, Ln₂O₂S, Ln₂O₂CO₃ and NaLnO₂ were collated and used to assess the accuracy of the different methods, which were then used to predict values for compounds for which no data exists. It is shown that Mostafa et al.'s group contribution method and the linear relationship proposed by the authors give the lowest mean absolute error (<9%). The volume based thermodynamics (VBT) method yields estimates with absolute mean errors below 16.0% for LnPO₄ and Ln₂O₂S, but above 26.0% for other compounds. Correction of the VBT method using an improved estimate of the Madelung energy for the calculation of the lattice enthalpy decreases the absolute mean error below 12.0% for all compounds except LnPO₄. These complementary methods provide options for calculating the enthalpy of formation of rare earth compounds, depending on the experimental data available and desired accuracy.

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Introduction

Rare earth elements, commonly called the lanthanides (Ln), are playing a major role in the move towards a more sustainable society, due to their use in green technologies, including electric vehicles and wind turbines.^4,5 Rare earth oxycarbonates (Ln₂O₂CO₃) are important for manufacturing high-temperature superconductors, solid oxide fuel cells and permeable membranes; and as alternative catalysts for biodiesel production and dehydrogenation of primary aliphatic alcohols, due to their environmental benefits and process efficiency.^6–8 In addition, rare earth oxysulfides (Ln₂O₂S) are becoming important in the manufacture of red phosphors and oxygen storage, based on the reversible oxidation of sulphur to produce Ln₂O₂SO₄.^9–11 The use of rare earth elements in almost all of these applications is linked to their characteristic chemical, magnetic and electronic properties, making their substitution difficult.¹²

The applications discussed above, are pushing the global demand for rare earth compounds, increasing their beneficiation and extraction from ores and recycling from end-of-life products. The current chemical technologies used for extraction of rare earth elements have serious environmental impacts and are energy inefficienct.^13,14 In view of this, industry is looking to design novel processes for their extraction and recovery.^15–19 For instance, in the early 80s high temperature reactions between Ln₂O₃ with different sodium compounds was studied, due to the formation of NaLnO₂ compounds affecting the swelling behavior of nuclear reactors during the cooling of liquid sodium;²⁰ a similar approach has been applied to recover REEs from waste fluorescent phosphors.^21,22 One of the main obstacles inhibiting the design of novel processing routes and synthesis of new rare earth element compounds is the lack of thermodynamic data available, because experimental measurements of their thermodynamic properties are extremely challenging.²³

Knowledge of the enthalpies of formation of rare earth element compounds is crucial for calculations of Gibbs free energies and chemical equilibria. Methods exist for predicting these values, but some are only parameterised for a particular set of chemical compounds, while more general methods, based on group contribution, fail to provide a user-friendly mathematical model or the required data is only available for particular compounds of interest.^23,24 Other methods rely on the lattice potential energy, from the Born–Haber–Fajans cycle, to calculate enthalpies formation. Born-Lande developed in 1918 an equation to estimate lattice potential energies of binary ionic crystals which was further improved on by Born and Mayer. Kapustinskii developed the previous equations to allow calculation of the lattice potential energy of any simple ionic crystal.²⁵ Jenkins et al.^26,27 extended Kapustinskii's equation to link the lattice potential energy with the molecular volume of the crystal unit cell.

In view of this, in the present work, we evaluate a suite of methods for estimating the enthalpy of formation of rare earth compounds, using the rare earth oxides, phosphates, chlorides, oxychlorides, oxysulfides, oxycarbonates and sodium lanthanide oxides as a test set. In particular, we assess the performance of the volume based thermodynamic (VBT) method developed by Glasser and Jenkins^2,3 and group contribution method reported by Mostafa and Eakman,¹ which were chosen for their ease of use. In addition, we show, for the first time, that a linear relationship exists between ΔH⁰_f(Ln₂O₃) and ΔH⁰_f(LnSalt), where ΔH⁰_f(Ln₂O₃) is the enthalpy of formation of the rare earth oxides (Ln₂O₃) and ΔH⁰_f(LnSalt) is the enthalpy of formation of the rare earth salts under study. If experimental data is available for some rare earth compounds in the series, these can be used to easily estimate values for others.

Crystal structures

Before describing the methodology employed in this work, it is useful to provide a brief description of the crystal structures of the compounds considered.

Rare earth sesquioxides (Ln₂O₃)

Rare earth sesquioxides are polymorphic. At ambient conditions the cubic structure (form C, space group Ia3) is the most thermodynamically stable for all rare earth sesquioxides, with the exception of La₂O₃, Ce₂O₃ and Nd₂O₃.²⁸ In the cubic phase, the rare earth ions are six-coordinate and there are sixteen formula units per unit cell.^29,30 For La₂O₃, Ce₂O₃ and Nd₂O₃ a hexagonal structure (form A, space group P32/m) is observed.²⁹ In the hexagonal phase, the rare earth ions are seven-coordinate and there is one formula unit per unit cell. For the light rare earth elements, the thermochemical stability of forms A and C are similar, such that lanthanide sesquioxides with a mixture of both phases can be found at room temperature.³¹

Anhydrous rare earth trichlorides (LnCl₃)

Rare earth trichlorides are also polymorphic, presenting three structures at room temperature, depending on atomic number.³² The trichlorides from LaCl₃ to GdCl₃ are isostructural, having a hexagonal UCl₃ structure (space group P63/m) in which rare earth ions are symmetrically coordinated to nine chloride ions.³³ The rare earth chlorides from DyCl₃ to LuCl₃ present a monoclinic YCl₃ structure (space group C2/m) in which the chloride ions are arranged in a nearly cubic close packed arrangement, with the rare earth ions located in the octahedral holes formed by alternate pairs of close packed layers of chloride ions.³⁴ TbCl₃ has the orthorhombic PuBr₃ structure (space group Cmcm), which can be described as triangular prisms in which Tb³⁺ is eight coordinated, with Tb³⁺ located at the center and Cl⁻ situated at the corners.³⁵ Rycerz and Gaune-Escard³⁶ investigated the correlation between the thermodynamic properties and crystal structure of rare earth trihalides, using differential scanning calorimetry. They found that TbCl₃ was the first compound of the series to present a solid–solid phase transition, pointing out that the structure of TbCl₃ depends on the methodology employed for its synthesis, being possible to obtain either the hexagonal (UCl₃) or orthorhombic (PuBr₃) structure. It is only possible to obtain the hexagonal form of TbCl₃, when it is prepared below 640 K.³⁶

Rare earth oxychlorides (LnOCl)

Rare earth oxychlorides from LaOCl–HoOCl crystallize in the PbFCl tetragonal crystal structure (space group P4/nmm), meanwhile TmOCl–LuOCl adopt the SmSI hexagonal structure (space group R32/m). ErOCl is dimorphic and can present both structures. The change from tetragonal to hexagonal structure is due to the rearrangement of the chloride ions to reduce the strain present in the tetragonal form, when the ionic radius of the rare earth cation is small enough to allow it.^37,38

Rare earth phosphates (LnPO₄)

Rare earth phosphates change crystal structure with increasing ionic radius. From LaPO₄ to GdPO₄ they adopt the monoclinic monazite structure (space group P21/n), while from TbPO₄ to LuPO₄ they adopt the tetragonal xenotime structure (space group I41/amd). Depending on preparation conditions, GdPO₄, TbPO₄ and DyPO₄ can adopt either structure.³⁹ Both structures contain isolated PO₄³⁻ tetrahedral separated by LnO₉ polyhedra for monazite or LnO₈ polyhedra for xenotime structure.⁴⁰

Ternary sodium lanthanide oxides (NaLnO₂)

All the sodium rare earth oxides have an ordered rocksalt lattice and can present three different crystal structures, depending on the relative size of the Na⁺ and Ln³⁺. If the Ln³⁺ ion is bigger than Na⁺ (i.e. from La to Gd) the tetragonal α-LiFeO₂ structure (space group I4₁/amd) is found. If the Ln³⁺ is smaller than Na⁺, two different structures are possible. The monoclinic β-LiFeO₂ structure (space group C2/c) is found from Tb to Er and the hexagonal α-NaFeO₂ structure (space group R3m) from Tm to Lu.^41,42 Impurities from other Ln³⁺ ions may alter the structure suggesting polymorphisms in the NaLnO₂ series.²⁰

Lanthanide oxycarbonates (Ln₂O₂CO₃)

Rare earth oxycarbonates exists in three polymorphic crystalline structures. The tetragonal Type I structure has square Ln₂O₂²⁺ layers separated by CO₃²⁻ ions. The type IA is a monoclinic distortion of the type I. The type II structure has hexagonal Ln₂O₂²⁺ layers separated by CO₃²⁻ ions.⁷ The transformation of the type I to the type II structure has been observed for the light lanthanides at temperatures higher than 400 °C, whereas no type II structure has been reported for members beyond Gd.⁴³

Lanthanide oxysulfides (Ln₂O₂S)

Rare earth oxysulfides have a trigonal crystal structure (space group P3m1), with one formula per unit cell. This structure is closely related with the A-type structure of the rare earth sesquioxides and can be described as an alternative stacking of Ln₂O₂²⁺ and S²⁻ layers.^10,53,54

Methodology

In this section we outline the various methods used for calculating enthalpies of formation. Example calculations, for LaPO₄, are given in the ESI.†

Experimental enthalpies of formation

A literature review of the experimental enthalpy of formation values of the different rare earth compounds considered in this work was carried out and their average values are presented in Table 1.

Table 1 Average experimental value of the enthalpy of formation (ΔH⁰_f(LnSalt)) (kJ mol⁻¹) for the rare earth compounds considered in this study^39,44–52

Ln	Ln2O3	LnCl3	LnOCl	LnPO4	Ln2O2CO3^a	Ln2O2S	NaLnO2
u.k. = unknown.a Type II oxycarbonates.
La	−1793.8 ± 1.3	−1072.1 ± 0.9	−1013.8 ± 4.4	−1937.2 ± 47.4	−2388.4 ± 3.5	−1723.8 ± 9.3	u.k.
Ce	−1799.7 ± 7.1	−1056.0 ± 3.3	−1002.3 ± 2.4	−1949.8 ± 25.5	u.k.	−1743.0	−1127.0
Pr	−1812.6 ± 6.7	−1057.6 ± 1.0	−1013.6 ± 0.6	−1969.5	u.k.	−1742.4 ± 6.8	u.k.
Nd	−1808.2 ± 0.6	−1041.5 ± 0.4	−1000.5 ± 0.5	−1931.9 ± 51.7	−2382.1 ± 9.4	−1733.7 ± 19.1	u.k.
Pm	−1811.0	−1029.3	u.k.	u.k.	u.k.	−1751.4	u.k.
Sm	−1824.0 ± 1.7	−1026.4 ± 1.2	−994.9 ± 4.8	−1922.2 ± 61.6	u.k.	−1747.1 ± 18.0	u.k.
Eu	−1658.3 ± 6.0	−936.8 ± 1.5	u.k.	−1880.9 ± 14.5	−2202.4 ± 12.0	−1578.0 ± 19.5	u.k.
Gd	−1824.5 ± 10.6	−1007.3 ± 1.9	−980.8 ± 2.5	−1924.6 ± 53.2	u.k.	−1744.2 ± 28.0	u.k.
Tb	−1865.4 ± 0.0	−1000.1 ± 4.6	−981.8 ± 5.5	−1947.8 ± 38.2	u.k.	−1762.9 ± 13.5	u.k.
Dy	−1863.3 ± 0.6	−996.6 ± 4.9	−986.3 ± 2.5	−1937.9 ± 42.4	u.k.	−1762.5 ± 20.0	u.k.
Ho	−1881.0 ± 0.5	−1003.2 ± 5.5	−996.2 ± 7.4	−1948.1 ± 33.2	u.k.	−1774.1 ± 9.5	u.k.
Er	−1898.1 ± 0.6	−996.8 ± 2.6	−993.9 ± 5.5	−1957.0 ± 28.1	u.k.	−1782.5 ± 2.4	u.k.
Tm	−1888.9 ± 0.6	−989.8 ± 4.0	−990.0 ± 1.7	−1946.8 ± 25.4	u.k.	−1777.7 ± 10.4	u.k.
Yb	−1821.7 ± 19.3	−960.3 ± 0.7	−962.2 ± 0.3	−1906.1 ± 33.0	u.k.	−1692.6	u.k.
Lu	−1878.4 ± 0.6	−966.2 ± 23.4	−976.8 ± 22.9	−1948.4 ± 9.9	u.k.	−1789.2	u.k.

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Group contribution method

Mostafa et al.¹ developed a group contribution (GC) method to estimate the enthalpy of formation (ΔH⁰_f) of solid inorganic salts. It is based on a multiple linear regression analysis, which calculates the contribution of each cation, anion and ligand molecule to the total heat of formation of the inorganic solid. We used this method to estimate the enthalpy of formation of rare earth compounds using the values of the different ionic contributions in ref. 1. The contribution of the ion S²⁻ was estimated as the difference between the contribution of the ligand H₂S and two times the contribution of the ion H⁺ giving a value of 108.8 kJ mol⁻¹.

Volume based thermodynamics

The volume based thermodynamics method (VBT), is a range of correlation methods that rely on the volume of the condensed phases to estimate thermodynamic properties.^2,3 This method was used to predict the lattice potential energy of rare earth compounds, which was input into the Born–Haber–Fajans cycle (BHFC) to calculate ΔH⁰_f from their individual elements. The lattice potential energy was calculated using the equations proposed by Jenkins and Glasser^55,56where V_m is the molecular volume in nm³ calculated from experimental X-ray diffraction data, α = 139.0 kJ mol⁻¹ and β = 28.0 kJ mol⁻¹ are empirical constants determined by statistical analysis of experimental data;⁵⁷I = 1/2∑n_iz_i² is the ionic strength factor, which is a summation over the product of the n_i ions with a charge of z_i for the different ionic constituents of the chemical formula; A = ½N_AMe²/4πε₀ is a standard electrostatic constant with the value of 121.4 kJ mol⁻¹; N_A is Avogadro's number, M is the Madelung constant of sodium chloride; e the charge of an electron and 4πε₀ the vacuum permittivity. As recommended by Jenkins and Glasser,^55,56 for lattice energies below 5000.0 kJ mol⁻¹, eqn (1) was used, otherwise eqn (2) was used.

Since the BHFC is an enthalpy cycle, the lattice energy (U_POT) calculated using eqn (1) and (2) was transformed into lattice enthalpy⁵⁵ using the following expression

where s_i is the number of ions of type i in the formula unit; c_i is a constant that takes a value of 3, 5 and 6 depending on whether the ion i is monoatomic, linear polyatomic or polyatomic, respectively, and RT is approximately 2.5 kJ mol⁻¹.

The thermochemical cycles considered in each BHFC are presented in Fig. S1a–f in ESI.† The values of the lattice enthalpy and the standard enthalpies of sublimation, ionization, dissociation, and electronic affinity used in this work can also be found in the ESI.†

Predictions based on lattice energy calculations using EUGEN

EUGEN is a code for calculating Madelung constants of salts,⁵⁸ and outputs Madelung energies (E_Madelung). Madelung energies output from the EUGEN code were used to estimate lattice potential energies (U_POT). Based on the observation of Glasser,⁵⁹ lattice potential energies were computed using the relation: U_POT = 0.85E_Madelung. The factor of 0.85 accounts for the fact that Madelung energies are based on coloumbic interactions alone and neglect other types of interactions, such as van der Waals repulsions. Note that, this relation was used to calculate lattice potential energies for all compounds, with the exception of the trichlorides. For the trichlorides it was found that scaling Madelung energies by a factor of 0.85 led to poor corresponding enthalpies of formation, and so for these compounds lattice potential energies were taken to be Madelung energies, without scaling. This is in keeping with the work of Glasser,⁵⁹ who reported the Madelung energies and lattice potential energies of LaCl₃, and YCl₃ to be within 1–2%. Calculated lattice potential energies were then converted to enthalpies using eqn (3) and input into the BHFC to calculate enthalpies of formation.

The EUGEN code requires the positions and charges of anions and cations in order to calculate Madelung energies. In our calculations compounds were treated as ionic and charges on anions and cations were taken to be their oxidation number. Of course, the validity this is an approximation decreases with the covalent character of the compound, but calculation of the fractional charges on ions is out of the scope of this study. Using the oxidation number means that the method is kept simple and, as we will show, leads to reasonable results.

The crystal structures of the rare earth oxysulfide and sodium lanthanide oxides were taken from The Materials Project (https://materialsproject.org),⁶⁰ which contains a database of structures calculated from density functional theory calculations. Due to the absence of some structure for the phosphates, chlorides, and oxychlorides in The Materials Project database, we calculated them ourselves using the density functional theory based VASP code,^61,62 with the projector augmented wave method.^62–64 The PBEsol exchange–correlation functional⁶⁵ and an energy cut-off of 1000 eV were used for all calculations. The valence electron configurations for the potentials were: 5s² 5p⁶ 5d¹ 6s² for all rare earth elements from La to Sm and 5p⁶ 5d¹ 6s² for all rare earth elements from Eu to La (with the 4f electrons frozen in the core); 3s² 3p³ for P; s² p⁵ for Cl; and 2s² 2p⁴ for O. Brillouin zone sampling was carried out using the following Monkhorst–Pack grids:⁶⁶ phosphates (monazite) 4 × 4 × 4; phosphates (xenotime) 4 × 4 × 4; chlorides (hexagonal) 6 × 6 × 8; chlorides (monoclinic) 4 × 3 × 4; oxychlorides 16 × 16 × 16.

In order to validate the use of the EUGEN code for compounds in our test set, we calculated the lattice potential enthalpies for some similar compounds, e.g. sulfides, chloride and oxides, for which values determined from experimental data exist.⁶⁷ The crystal structures were taken from The Materials Project (https://materialsproject.org)⁶⁰ and the charges on ions were taken to be their oxidation number. The results of these calculations are in Table S1 of the ESI.† In general, good agreement is found.

Linear relationship

The correlation between ΔH⁰_f(Ln₂O₃) and ΔH⁰_f(LnSalt) for our test set of rare earth compounds is presented in Fig. 1a–c using the values given in Table 1. Promethium was excluded from the analysis due to the lack of data for most of the compounds under study. Moreover, from Table 1 the values for europium and ytterbium compounds are anomalous and so were also excluded from our analysis. Glasser⁶⁸ explained that the anomalous behaviour of europium arises from the ground state of Eu³⁺ having zero magnetic spin entropy, compared to other trivalent lanthanide ions, which have finite values. The discrepancy between the enthalpies of formation of Lu and Yb compounds are attributed to a fall in the ionization energy of the ion Lu³⁺ with respect to the Yb³⁺ due to the fully shielded outer f-electron shell of Yb. To our best knowledge, this is the first time these linear relationships have been reported. They offer a new way to predict the enthalpy of formation of rare earth compounds for which values are known for other rare earth elements in the series.

Fig. 1 Relationship between ΔH⁰_f(Ln₂O₃) and ΔH⁰_f(LnSalt) for (a) LnCl₃ and LnOCl, (b) LnPO₄ and Ln₂O₂S and (c) Ln₂O₂CO₃. Pm, Eu and Yb were excluded from all analysis and for Ln₂O₂CO₃ only La, Nd and Eu were considered.

Table 2 shows the statistical descriptors for each regression. As can be observed in Table 2, the r² values obtained for LnCl₃, Ln₂O₂S and Ln₂O₂CO₃ show a linear dependence between ΔH⁰_f(Ln₂O₃) and ΔH⁰_f(LnSalt). A poor dependence is observed for LnOCl and LnPO₄. However, the F-test carried out at a 99% confidence level, shows that F calc > F critical in all cases indicating that there is a relationship between ΔH⁰_f(Ln₂O₃) and ΔH⁰_f(LnSalt) in all cases, despite the low r² values found for LnPO₄ and LnOCl.

Table 2 Statistical analysis of the linear regression of lanthanide compounds using experimental values

LnSalt	r ^2	F calc.	F critical
LnCl3	0.7806	35.58	1.38 × 10^−4
LnOCl	0.3768	6.04	3.37 × 10^−2
LnPO4	0.0845	0.92	3.59 × 10^−1
Ln2O2S	0.9098	101.81	1.53 × 10^−6
Ln2O2CO3	0.9864	72.40	7.44 × 10^−2

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Group contribution method

The values of ΔH⁰_f(LnSalt) calculated using the group contribution method developed by Mostafa et al. are presented in Table S2 (ESI†), where all the values are within 9.0% of available experimental data. Promethium compounds were not calculated because the unknown contribution value for Pm³⁺. According to the literature review carried out, the type I crystal structure of the oxycarbonates is the most stable at low temperature. However, thermogravimetric studies,^8,69 have confirmed the co-existence of type I and type II oxy-carbonates for the light rare earth elements from 400 to 600 °C as a consequence of their polymorphic nature, which suggests that little difference may exist between the enthalpy of formation of the type I and type II oxy-carbonates.

No experimental values of the enthalpy of formation were found for the sodium lanthanide oxides. Only Barker et al.⁷⁰ reported the experimental value of the enthalpy of formation of NaCeO₂ as −1127.0 kJ mol⁻¹ at 352 °C, which is within reasonable limits of our estimation using Mostafa's et al. group contribution method.

Volume based thermodynamics

The VBT method relies on accurate determination of molecular volumes to calculate the lattice energy of solid phases. It is important to bear in mind that propagation of uncertainties of ±10% in the molecular volume can lead to ±20% uncertainties in the calculated enthalpy of formation. In addition, the VBT method assumes ionic character of the compounds studied. The values of the ΔH⁰_f(LnSalt) calculated using the volume based thermodynamics are presented in Table S3 (ESI†). As can be observed from Table S3 (ESI†), the average deviation from experimental available data is higher than 14.0%. One of the main reasons for the inaccuracies observed is that the most recent crystal data for the chlorides and oxychlorides are from eighties. In addition, when covalent forces play an important role, eqn (1) and (2) become increasingly unreliable.³ The oxysalts (LnOCl, Ln₂O₂CO₃ and Ln₂O₂S) and the A-Ln₂O₃, can be structurally defined as tri or bi-dimensional packing of OLn₄ tetrahedra sharing edges that appear as an infinite polymeric complex cation (LnO)_nⁿ⁺ which alternate with sheets of the anions. Considering the OLn₄ tetrahedra as the basic structural unit, directional bonding of the oxygen atom might be expected, meaning that a large, but not dominant, covalent contribution may be present in these compounds.⁷¹ Similar structural features may apply to the NaLnO₂. However, there is little information available in the literature about them.

EUGEN calculations

To attempt to improve estimations of enthalpies of formation calculated using the BHFC, electrostatic lattice energies determined using the EUGEN code⁵⁸ were used as an approximation for lattice potential energies (U_POT) in eqn (3). These lead to significantly improved values, except for the phosphates.

Method comparison

In Fig. 2 we present, the absolute error calculated according to eqn (4), for the rare earth chlorides, oxychlorides, oxysulfides and phosphates. When no experimental values were available or the enthalpy of formation could not be predicted by a particular method for some compounds the error was not calculated.Comparison of the methods used with the experimental values for LnCl₃, LnOCl, LnPO₄ and Ln₂O₂S, indicate that, in general, the group contribution method and the linear relationship provide better values for the enthalpy of formation. Note that, for these four compounds, the linear relationship was calculated by fitting to all the experimental data and so the percentage error simply represents the misfit between the relationship and experimental results, rather than a prediction. In addition, it should be considered that the linear relationship does not give good estimations for Eu and Yb due to their exclusion from the linear analysis. For Ln₂O₂S, the linear relationship predicts values in better agreement with experimental results, due to the contribution of the S²⁻ ion being estimated from the original work of Mostafa et al.¹ That said, reasonable estimations were achieved with all estimated values within 12.0% of available experimental data.

Fig. 2 Comparison between estimated and experimental values of the enthalpy of formation for LnCl₃, LnOCl, LnPO₄ and Ln₂O₂S.

The VBT method appears to be unsuitable for estimating the enthalpy of formation of LnCl₃ with absolute errors between about 40.0–90.0%. The situation is better for LnOCl and Ln₂O₂S with absolute errors of about 23.0–28.0% and 12.0–19.0% respectively. For LnPO₄, the VBT method shows high variability, with absolute errors of between 0.9 to 26%. The correction of the VBT method carried out using EUGEN to calculate U_POT rather than eqn (1) and (2) improves the estimations of all the compounds except for LnPO₄. These differences are attributed to the values of the parameters A and α, which seem to be unsuitable for lanthanide compounds.

Due to scarce experimental values for Ln₂O₂CO₃ and NaLnO₂, the group contribution method was taken as a reference for comparison with the other methods, which is presented in Fig. 3. It can be observed that the linear relationship presented in Table 2 for Ln₂O₂CO₃ agrees well with the values predicted using the group contribution method, with absolute errors ranging from 0.6–9.0%. This agreement is excellent, considering that the linear relationship was calculated using the enthalpies of formation of just three of the oxycarbonates, and illustrates the predictive power of the method. The VBT method fails in the prediction of the enthalpy of formation of Ln₂O₂CO₃, because of the important covalent contribution present. It was not possible to predict values for Ln₂O₂CO₃ using the EUGEN code, due to the crystal structure having partial occupancies.

Fig. 3 Comparison between estimated values of the enthalpy of formation with the group contribution, VBT and Madelung methods for Ln₂O₂CO₃ and NaLnO₂.

For NaLnO₂ discrepancies between 19.0–32.0% are reported when the VBT method is used. Better estimations are achieved when for calculations based on lattice potential energies calculated using the EUGEN code, as shown in Fig. 3.

Conclusions

In this work we have compared several methods for calculating the enthalpies of formation of the rare earth compounds: LnCl₃, LnOCl, LnPO₄, Ln₂O₂S, Ln₂O₂CO₃ and NaLnO₂. We find that the method of Mostafa et al.¹ and our method based on a linear relationship are the most accurate. However, the method of Mostafa et al.¹ requires that the contribution of each group is known, while our method based on a linear relationship necessitates that experimental data is available for some of the rare earth compounds. In contrast, the VBT method only requires the molecular volume of a compound, but the accuracy of predicted values is poor when covalent bonding contributions are significant. The calculation of the lattice potential energy using EUGEN instead of eqn (1) and (2) improves the estimation using the BHFC but requires the full crystal structure. This improvement suggests that the value of the parameters α = 139.0 kJ mol⁻¹ and A = 123.0 kJ mol⁻¹ are unsuitable for the compounds studied. Taken together, these methods provide a complementary suite of techniques for predicting the enthalpies of formation of rare earth compounds, depending on the experimental data available and accuracy desired. Our recommendations are as follows. If enthalpies of formation are available for some rare earth compounds in the series, use the linear relationship. If no enthalpies of formation exist, and all group contributions are available, use the method of Mostafa et al.¹ Failing this, use the BHFC. If using the BHFC, and the crystal structure is available, calculating the lattice potential energy using the EUGEN code. If only the volume is available, use the VBT method to calculate the lattice potential energy.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge financial support received from the Engineering and Physical Sciences Research Council (EPSRC) (Grants GR/T19889/01 and GR/L95977/01), and Natural Environment Research Council (NERC) (Grants NE/M01147X/1, and NE/L002280/1). Dr Sanchez-Segado acknowledges the support from the European Union's Marie Curie Fellowship grant number 331385 and from The Ministry of Science, Innovation and University of Spain (“Beatriz Galindo” Fellowship BEAGAL18/00079). The authors thank Samuel Tan and Ekaterina I. Izgorodina, for providing the modified version of the EUGEN code.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1cp03280a

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