Thermodynamic Assessment of the P₂O₅-Na₂O and P₂O₅-MgO Systems

Abstract

Knowledge about the thermodynamic equilibria of the P₂O₅-Na₂O and P₂O₅-MgO systems is very important for controlling the phosphorus content of steel materials in the process of steelmaking dephosphorization. The phase equilibrium and thermodynamic data of the P₂O₅-Na₂O and P₂O₅-MgO systems were critically evaluated and re-assessed by the CALPHAD (CAlculation of PHAse Diagram) approach. The liquid phase was described by the ionic two-sublattice model for the first time with the formulas (Na⁺¹)_P(O⁻², PO₃⁻¹, PO₄⁻³, PO_5/2)_Q and (Mg⁺²)_P(O⁻², PO₃⁻¹, PO₄⁻³, PO_5/2)_Q, respectively, and the selection of the species constituting the liquid phase was based on the structure of the phosphate melts. A new and improved self-consistent set of thermodynamic parameters for the P₂O₅-Na₂O and P₂O₅-MgO systems was finally obtained, and the calculated phase diagram and thermodynamic properties exhibited excellent agreement with the experimental data. The difference in the phase composition of invariant reactions from the experimentally determined values reported in the literature is less than 0.9 mol.%. The present thermodynamic modeling contributes to constructing a multicomponent oxide thermodynamic database in the process of steelmaking dephosphorization.

1. Introduction

As society progresses, industries advance to higher developmental stages, leading to more demanding usage of steel materials and increased quality requirements for steel materials across various sectors. Phosphorus, as one of the detrimental elements in steel, serves as a critical indicator of steel quality. Therefore, the control of phosphorus content in steel remains a crucial target for enterprise development. In the steelmaking process, the inclusion of alkaline earth metal oxide fluxes such as MgO can effectively diminish phosphorus in liquid steel, while alkali metal oxide fluxes like Na₂O also exhibit a strong dephosphorization effect [1]. The phase diagrams and thermodynamic properties of the P₂O₅-Na₂O and P₂O₅-MgO systems are essential to effectively control the dephosphorization effect of slags and to understand the phosphorus distribution ratio between liquid iron and oxide slags such as Na₂O and MgO. Furthermore, good thermodynamic descriptions provide phase diagrams and thermodynamic data that can also effectively provide a theoretical basis for material design [2,3,4,5,6].

Xie et al. [7] utilized the modified quasi-chemical model to describe the liquid phase and firstly optimized the thermodynamic parameters of the P₂O₅-Na₂O system based on the reliable experimental phase diagram and thermodynamic properties, and their calculations were in good agreement with the experimental data, while the description of the enthalpy of formation of Na₅P₃O₁₀ was inaccurate. In 2015, Ding et al. [8] evaluated a P₂O₅-MgO system by using the modified quasi-chemical model to describe the liquid phase, and PO₄³⁻ was considered the basic unit of P₂O₅ in the liquid phase, but the calculations showed significant discrepancies with the experimental data. Furthermore, a set of thermodynamic data describing the liquid phase with the modified quasi-chemical model does not simultaneously describe both the oxide and metal liquid phases, which limits the study of the phosphorus distribution ratio between liquid iron and oxide slags. Therefore, it is meaningful to construct a set of multicomponent thermodynamic databases that can describe both oxide and metal liquids using appropriate thermodynamic models to guide the addition of oxide fluxes in the steelmaking dephosphorization process. The ionic two-sublattice model allowing one set of the thermodynamic parameters to simultaneously describe both the oxide and metallic liquid [9] was used to describe the liquid phase for the first time in the current work. Additionally, the ionic two-sublattice model can not only rationally describe the phosphate melt structure but also adequately reproduce the thermodynamic properties of complex liquids such as slag [10,11]. This is highly beneficial to the construction of a slag system multivariate database to guide steelmaking dephosphorization.

This work aimed to conduct a phase diagram thermodynamic optimization of the P₂O₅-Na₂O and P₂O₅-MgO systems using the CALPHAD (CAlculation of PHAse Diagram) approach through establishing suitable thermodynamic models. The crystal structure, limited measured phase diagram and thermodynamic properties were optimized to construct a Gibbs energy expression for each phase in the systems to obtain a set of thermodynamic parameters reasonably describing the phase diagrams, covering the whole composition range using Thermo-Calc software.

2. Review of Literature Data

The experimental phase diagram information and thermodynamic property data of the P₂O₅-Na₂O and P₂O₅-MgO systems are systematically evaluated. The crystal structures of the solid phases in the systems are listed in

Table 1. Crystal structures of all solid phases in the P₂O₅-Na₂O and P₂O₅-MgO systems.

System	Compound	Crystal System	Space Group	Reference
P2O5-Na2O	γ-NaPO3	Orthorhombic	P21P21P21	[12]
		Orthorhombic	Pnma	[13]
	β-NaPO3	Triclinic	P21/n	[14]
	α-NaPO3	Monoclinic	P21/c	[15]
	β-Na5P3O10	Monoclinic	C2/c	[16]
	α-Na5P3O10	Monoclinic	C2/c	[17]
	α-Na4P2O7	Orthorhombic	P21P21P21	[18]
	β-Na3PO4	Tetragonal	P$\stackrel{-}{4}$21c	[19]
	α-Na3PO4	Cubic	Fm$\stackrel{-}{3}$m	[20]
		Orthorhombic	Pnma	[21]
P2O5-MgO	Mg3P2O8	Monoclinic	P21/b	[22]
		Monoclinic	P21/n	[23]
		Monoclinic	P21/n	[24]
		Monoclinic	P21/n	[25]
		Triclinic	P$\stackrel{-}{1}$	[26]
	β-Mg2P2O7	Monoclinic	P21/c	[27]
	α-Mg2P2O7	Monoclinic	C2/m	[28]
	MgP2O6	Monoclinic	C2/c	[29]
		Monoclinic	C2/c	[30]
	MgP4O11	Monoclinic	P21/c	[31]
		Monoclinic	P21/c	[32]
		Orthorhombic	Pmc21	[33]

α/β/γ: the polymorph from high temperature to low temperature.

.

2.1. P₂O₅-Na₂O System

A phase diagram of the P₂O₅-Na₂O system has been reported by several researchers [34,35,36,37,38,39]. Partridge et al. [34], using thermal, microscopic and X-ray diffraction (XRD) analysis, determined the liquidus of the NaPO₃-Na₄P₂O₇ system and confirmed the presence of the Na₅P₃O₁₀ compound. Two invariant reactions of L = β − NaPO₃ + α − Na₅P₃O₁₀ and L + α − Na₄P₂O₇ = α − Na₅P₃O₁₀ were reported to occur at 824 K and 893 K, respectively. And the melting points of NaPO₃, Na₅P₃O₁₀ and Na₄P₂O₇ were 898 K, 788 K and 1258 K, respectively. In their work [34], the Na₄P₂O₇ and NaPO₃ phases exhibited a lot of phase transitions from room temperature to melting point. The transition temperatures of Na₄P₂O₇ were found to be 673 K, 783 K, 793 K and 818 K by differential thermal analysis (DTA). Two phase transitions of NaPO₃ at 677 K and 783 K were detected. Subsequently, Morey and Ingerson [35] also studied the phase equilibria of the NaPO₃-Na₄P₂O₇ system in good agreement with the work of Partridge et al. [34]. Two invariant reactions L = β − NaPO₃ + α − Na₅P₃O₁₀ and L + α − Na₄P₂O₇ = α − Na₅P₃O₁₀ were measured to have reaction temperatures of 825 K and 895 K, and the melting points of NaPO₃ and Na₄P₂O₇ were observed to be 901 K and 1262 K, respectively, but the third structure of NaPO₃ was not found. Turkdogan et al. [36], using the thermal, microscopic, and DTA methods, determined the phase diagram of the NaPO₃-Na₃PO₄ system and did not report the presence of Na₅P₃O₁₀. Three invariant reaction temperatures of L = β − NaPO₃ + α − Na₅P₃O₁₀, L + α − Na₄P₂O₇ = α − Na₅P₃O₁₀ and L = α − Na₄P₂O₇ + β − Na₃PO₄ in the NaPO₃-Na₃PO₄ system were suggested to be 763 K, 893 K and 1218 K by Markina et al. [37], respectively. In 1970, Osterheld et al. [38] determined the phase transition temperature of the Na₄P₂O₇-Na₃PO₄ system below 1573 K by thermal analysis and high-temperature microscopy. They reported that the eutectic reaction L = α − Na₄P₂O₇ + β − Na₃PO₄ occurred at 1225 K, and two compounds (Na₄P₂O₇ and Na₃PO₄) melted congruently at 1271 K and 1785 K, respectively. In 1972, Berak et al. [39] observed three invariant reactions in the liquidus study of the Na₂O-P₂O₅ system. The liquidus data obtained from these works for the P₂O₅-Na₂O system were in reasonable agreement and were used in the optimization process of the current work. The four compounds NaPO₃, Na₅P₃O₁₀, Na₄P₂O₇ and Na₃PO₄ have polymorphic phase transitions, and the thermodynamic description of the phase transition of the compounds in the P₂O₅-Na₂O system by Xie et al. [7] based on the reliable literature is more complete, which was considered in the thermodynamic assessment of the present work with refinement and improvement. It is worth noting that there is less information about the experimental phase relation of the P₂O₅-rich and Na₂O-rich regions in the P₂O₅-Na₂O system, which still needs to be further determined experimentally.

In 1909, Mixter [40] determined the enthalpy of formation of NaPO₃ from its elements using solution calorimetry (SCA). In 1967, Irving et al. [41] also utilized SCA to measure the enthalpy of formation of Na₃PO₄ from its elements at 298 K. Subsequently, in 1968, Irving et al., [42,43] Krivtsov et al., [44] and Zhuang et al. [45] determined the enthalpies of formation of Na₄P₂O₇, Na₅P₃O₁₀ and NaPO₃ from their elements using the SCA method. In 2011, Khaled et al. [46] measured the standard enthalpy of formation of Na₄P₂O₇ from its elements using the SCA method. These experimental results were incorporated into the present study with consideration for possible error margins. Andon et al. [47] determined the heat capacities of NaPO₃, Na₅P₃O₁₀, Na₄P₂O₇ and Na₃PO₄ using adiabatic calorimetry within the temperature range of 10 to 320 K. Ashcroft et al. [48] measured the heat capacities of Na₄P₂O₇ and NaPO₃ from 298 to 620 K and determined the low-temperature transition enthalpy of Na₄P₂O₇. Lazarev et al. [49] used DSC to measure the heat capacity of Na₄P₂O₇ in the temperature range from 300 to 1000 K and measured the low-temperature enthalpy of transition of the Na₄P₂O₇. Grantscharova et al. [50] used DSC to determine the heat capacity of NaPO₃ between 468 and 675 K, but their measurements were much higher than those reported by Ashcroft et al. [48]. Considering the above-reported heat capacity data, the data reported by Andon et al. [47], Ashcroft et al. [48] and Lazarev et al. [49] were considered in the present work to optimize the heat capacities of Na₄P₂O₇ and NaPO₃.

2.2. P₂O₅-MgO System

The phase diagram of the P₂O₅-MgO system in the composition ranges from 0 to 50 mol.% P₂O₅ was investigated by Berak [51] using thermal, microscopy and XRD analyses. In this concentration range, three intermediate compounds were observed: Mg₃P₂O₈, Mg₂P₂O₇ and MgP₂O₆ with melting points at 1630 K, 1655 K and 1438 K, respectively. These phases were considered as line compounds. The temperature of three eutectic reactions L = MgO + Mg₃P₂O₈, L = Mg₃P₂O₈ + α − Mg₂P₂O₇ and L = α − Mg₂P₂O₇ + MgP₂O₆ were found to be 1598 K, 1555 K and 1423 K, respectively. Additionally, Mg₃P₂O₈ with two polymorphic forms was confirmed, and its transition temperature was 1328 K. Subsequently, Bobrownicki and Slawski [52] also measured the melting temperature of Mg₃P₂O₈ to be 1628 K and the structural transition temperature to be 1323 K. However, these two studies did not give data such as the lattice parameter and the structural transition of Mg₃P₂O₈, which have not been reported in subsequent studies [53,54]. Therefore, the optimization process of the present work did not consider the phase transformation of Mg₃P₂O₈. Bookey [55], using thermal analysis, investigated the eutectic reaction L = MgO + Mg₃P₂O₈ by means of cooling curves, which yielded a reaction temperature of 1603 K. The results were consistent with the data reported by Berak [51]. The melting points of Mg₃P₂O₈ and Mg₂P₂O₇ were investigated, and the presence of the phase transition in the Mg₂P₂O₇ was determined by Czupinska et al. [53] and Oetting et al. [54] using thermal analysis. Combined with the data obtained by Roy et al. [56] and Calvo et al. [57], only the structural transformation of the Mg₂P₂O₇ in the low-temperature region was considered in the present work. MgP₄O₁₁ was reported to melt congruently at 1183 K by Meyer et al. [32] using DTA. Rakotomahanina-Rolaisoa et al. [58] investigated the melting point of MgP₂O₆ by DTA.

In 1897, Berthelot [59] determined the enthalpy of formation of Mg₃P₂O₈ from elements using SCA. In 1952, Bookey et al. [55] investigated the enthalpy of formation of Mg₃P₂O₈. In 1954, the enthalpy of formation of Mg₃P₂O₈ from elements was measured by Stevens and Turkdogan [60] using SCA. In 1986, Lopatin et al. [61] studied the standard enthalpies of formation of Mg₂P₂O₇ and MgP₂O₆ from elements using the Knudsen cell mass spectrometry (KCMS) approach. In 1989, Lopatin et al. [62] used the KCMS method to determine the enthalpy of formation of Mg₃P₂O₈ from elements. In 1999, Abdelkader et al. [63] measured the standard enthalpy of formation of Mg₃P₂O₈ from elements using the SCA approach. These experimental data on the enthalpies of formation of the compounds in the P₂O₅-MgO system described above were accepted for the present work. Oetting and Mcdonald [54] measured the heat capacities of Mg₃P₂O₈ and Mg₂P₂O₇ using an adiabatic calorimeter and determined the heat contents of Mg₃P₂O₈ and Mg₂P₂O₇ in the temperature range from 0 to 1700 K. Furthermore, the energy change in the low-temperature phase transition of Mg₂P₂O₇ was determined. Iwase et al. [64] investigated the activity of P₂O₅ in liquid P₂O₅-MgO mixtures using solid oxide galvanic cell techniques at 1673 K. Given that the reported data were obtained from indirect calculations, the data on the activity were not used in the current work.

3. Thermodynamic Modeling

The CALPHAD method is used to formulate a comprehensive thermodynamic model to describe each phase in a system, drawing upon experimental data encompassing phase diagrams, thermodynamic properties and crystal structures. This method rationally selects undetermined parameters to represent each phase of the system as a Gibbs free energy function of variables such as temperature, pressure and composition. Ultimately, the phase diagrams and thermodynamic properties are derived through the utilization of a thermodynamic database containing these Gibbs free energy expressions. In the present study, the thermodynamic assessment of the P₂O₅-Na₂O and P₂O₅-MgO systems will be conducted using Thermo-Calc software. Employing the least-squares method, Thermo-Calc software endeavors to align the calculated values with the observed data, seeking optimized variable values that minimize the sum of squared differences between calculated and experimental data. Hence, the formulation of an appropriate thermodynamic model lays the groundwork for an excellent thermodynamic database.

The following thermodynamic models were used to model the P₂O₅-Na₂O and P₂O₅-MgO systems in the present work. The constructed thermodynamic models used for two binary systems are listed in

Table 2. The obtained thermodynamic parameters of the P₂O₅-Na₂O and P₂O₅-MgO systems in the present work.

System	Phase	Formula	Thermodynamic Parameter/J·mol−1
P2O5-Na2O	Liquid	(Na+1)p(O−2, PO3−1, PO4−3, PO5/2)q	${}_{ }{}^0{G}_{{Na}^{+1}:O^{-2}}^{Liquid}=+{}_{ }{}^0{G}_{{Na}_{2}O}^{Liquid}$
			${}_{ }{}^0{G}_{{PO}_{5/2}}^{Liquid}=+{}_{ }{}^0{G}_{P_2{O}_5}^{Liquid}$
			${}_{ }{}^0{G}_{{Na}^{+1}:{PO}_3^{-1}}^{Liquid}=+0.5{}_{ }{}^0{G}_{{Na}_{2}O}^{Liquid}+0.5{}_{ }{}^0{G}_{P_2{O}_5}^{Liquid}-223581.5-46.8T$
			${}_{ }{}^0{G}_{{Na}^{+1}:{PO}_4^{-3}}^{Liquid}=+1.5{}_{ }{}^0{G}_{{Na}_{2}O}^{Liquid}+0.5{}_{ }{}^0{G}_{P_2{O}_5}^{Liquid}-597241+62T$
			${}_{ }{}^0{L}_{{Na}^{+1}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}=-127756+18T$
			${}_{ }{}^1{L}_{{Na}^{+1}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}=-63351$
			${}_{ }{}^0{L}_{{Na}^{+1}:O^{-2},{PO}_4^{-3}}^{Liquid}=+7424$
			${}_{ }{}^0{L}_{{Na}^{+1}:{PO}_3^{-1},{PO}_{5/2}}^{Liquid}=-48065$
			${}_{}{}^1{L}_{{Na}^{+1}:{PO}_3^{-1},{PO}_{5/2}}^{Liquid}=-37884$
	Na3PO4_β	(Na+1)3(P+5)1(O−2)4	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_{3}P{O}_4\_\alpha }=+{}_{ }{}^0{G}_{{Na}_{3}P{O}_4}^{Solid}$
	Na3PO4_α	(Na+1)3(P+5)1(O−2)4	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_{3}P{O}_4\_\beta }=+{}_{ }{}^0{G}_{{Na}_{3}P{O}_4}^{Solid}+472-0.27T$
	Na4P2O7_ζ	(Na+1)4(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_4{P}_2{O}_7\_\alpha }=+{}_{ }{}^0{G}_{{Na}_4{P}_2{O}_7}^{Solid}$
	Na4P2O7_ε	(Na+1)4(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_4{P}_2{O}_7\_\beta }=+{}_{ }{}^0{G}_{{Na}_4{P}_2{O}_7}^{Solid}+10040-14.65693431T$
	Na4P2O7_δ	(Na+1)4(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_4{P}_2{O}_7\_\gamma }=+{}_{ }{}^0{G}_{{Na}_4{P}_2{O}_7}^{Solid}+13806-19.38215388T$
	Na4P2O7_γ	(Na+1)4(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_4{P}_2{O}_7\_\delta }=+{}_{ }{}^0{G}_{{Na}_4{P}_2{O}_7}^{Solid}+15061-20.94504305T$
	Na4P2O7_β	(Na+1)4(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_4{P}_2{O}_7\_\epsilon }=+{}_{ }{}^0{G}_{{Na}_4{P}_2{O}_7}^{Solid}+17153-23.47773070T$
	Na4P2O7_α	(Na+1)4(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_4{P}_2{O}_7\_\zeta }=+{}_{ }{}^0{G}_{{Na}_4{P}_2{O}_7}^{Solid}+20082-26.98551513T$
	Na5P3O10_β	(Na+1)5(P+5)3(O−2)10	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_5{P}_3{O}_{10}\_\alpha }=+{}_{ }{}^0{G}_{{Na}_5{P}_3{O}_{10}}^{Solid}$
	Na5P3O10_α	(Na+1)5(P+5)3(O−2)10	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{{Na}_5{P}_3{O}_{10}\_\beta }=+{}_{ }{}^0{G}_{{Na}_5{P}_3{O}_{10}}^{Solid}+10878-13.769620T$
	NaPO3_γ	(Na+1)1(P+5)1(O−2)3	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{NaP{O}_4\_\alpha }=+{}_{ }{}^0{G}_{NaP{O}_3}^{Solid}$
	NaPO3_β	(Na+1)1(P+5)1(O−2)3	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{NaP{O}_4\_\beta }=+{}_{ }{}^0{G}_{NaP{O}_3}^{Solid}+628-0.78795483T$
	NaPO3_α	(Na+1)1(P+5)1(O−2)3	${}_{ }{}^0{G}_{{Na}^{+1}:{P^{+5}:O}^{-2}}^{NaP{O}_4\_\gamma }=+{}_{ }{}^0{G}_{NaP{O}_3}^{Solid}+4226-5.010959525T$
P2O5-MgO	Liquid	(Mg+2)p(O−2, PO3−1, PO4−3, PO5/2)q	${}_{ }{}^0{G}_{{Mg}^{+2}:O^{-2}}^{Liquid}=+2{}_{ }{}^0{G}_{MgO}^{Liquid}$
			${}_{ }{}^0{G}_{{PO}_{5/2}}^{Liquid}=+{}_{ }{}^0{G}_{P_2{O}_5}^{Liquid}$
			${}_{ }{}^0{G}_{{Mg}^{+2}:{PO}_3^{-1}}^{Liquid}=+{}_{ }{}^0{G}_{MgO}^{Liquid}+{}_{ }{}^0{G}_{P_2{O}_5}^{Liquid}-238484-10T$
			${}_{ }{}^0{G}_{{Mg}^{+2}:{PO}_4^{-3}}^{Liquid}=+3{}_{ }{}^0{G}_{MgO}^{Liquid}+{}_{ }{}^0{G}_{P_2{O}_5}^{Liquid}-709347+185T$
			${}_{ }{}^0{L}_{{Mg}^{+2}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}=-128900+45T$
			${}_{ }{}^1{L}_{{Mg}^{+2}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}=+13365$
			${}_{ }{}^2{L}_{{Mg}^{+2}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}=-16547$
			${}_{ }{}^3{L}_{{Mg}^{+2}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}=+40441$
			${}_{ }{}^0{L}_{{Mg}^{+2}:O^{-2},{PO}_4^{-3}}^{Liquid}=-57701$
			${}_{ }{}^1{L}_{{Mg}^{+2}:O^{-2},{PO}_4^{-3}}^{Liquid}=+6541$
			${}_{ }{}^2{L}_{{Mg}^{+2}:O^{-2},{PO}_4^{-3}}^{Liquid}=-8954$
			${}_{ }{}^0{L}_{{Mg}^{+2}:{PO}_3^{-1},{PO}_{5/2}}^{Liquid}=-92054+60T$
			${}_{ }{}^1{L}_{{Mg}^{+2}:{PO}_3^{-1},{PO}_{5/2}}^{Liquid}=-25546$
			${}_{ }{}^2{L}_{{Mg}^{+2}:{PO}_3^{-1},{PO}_{5/2}}^{Liquid}=-35451$
			${}_{ }{}^0{L}_{{Mg}^{+2}:O^{-2},{PO}_3^{-1}}^{Liquid}=-66748+51T$
	Mg3P2O8	(Mg+2)3(P+5)2(O−2)8	${}_{ }{}^0{G}_{{Mg}^{+2}:{P^{+5}:O}^{-2}}^{{Mg}_3{P}_2{O}_8}=+{}_{ }{}^0{G}_{{Mg}_3{P}_2{O}_8}^{Solid}$
	Mg2P2O7_β	(Mg+2)2(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Mg}^{+2}:{P^{+5}:O}^{-2}}^{{Mg}_2{P}_2{O}_7\_\alpha }=+{}_{ }{}^0{G}_{{Mg}_2{P}_2{O}_7}^{Solid}$
	Mg2P2O7_α	(Mg+2)2(P+5)2(O−2)7	${}_{ }{}^0{G}_{{Mg}^{+2}:{P^{+5}:O}^{-2}}^{{Mg}_2{P}_2{O}_7\_\beta }=+{}_{ }{}^0{G}_{{Mg}_2{P}_2{O}_7}^{Solid}+680-2.0T$
	MgP2O6	(Mg+2)1(P+5)2(O−2)6	${}_{ }{}^0{G}_{{Mg}^{+2}:{P^{+5}:O}^{-2}}^{Mg{P}_2{O}_6}=+{}_{ }{}^0{G}_{Mg{P}_2{O}_6}^{Solid}$
	MgP4O11	(Mg+2)1(P+5)4(O−2)11	${}_{ }{}^0{G}_{{Mg}^{+2}:{P^{+5}:O}^{-2}}^{Mg{P}_4{O}_{11}}=+{}_{ }{}^0{G}_{Mg{P}_4{O}_{11}}^{Solid}$
Function	Temperature range/K
${}_{ }{}^0{G}_{P_2{O}_5}^{Liquid}$	(298.15–1000)		−1639225.067 − 230.7480381T + 21.643407TlnT − 0.1681142T2 + 1.87715$\times$10−5T3 + 1758186.5T−1 + 22900.402lnT
	(1000–6000)		−1579441.75 + 1382.959261T − 225TlnT
${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_OO}$	(298.15–1000)		−1665880.067 − 199.4980381T + 21.643407TlnT − 0.1681142T2 + 1.87715$\times$10−5T3 + 1758186.5T−1 + 22900.402lnT
	(1000–6000)		−1606096.75 + 1414.209261T − 225TlnT
${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_O}$	(298.15–1000)		−1666269.067 − 198.3480381T + 21.643407TlnT − 0.1681142T2 + 1.87715$\times$10−5T3 + 1758186.5T−1 + 22900.402lnT
	(1000–6000)		−1606485.75 + 1415.359261T − 225TlnT
${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_H}$	(298.15–1000)		−1631835.067 − 221.1390381T + 21.643407TlnT − 0.1681142T2 + 1.87715$\times$10−5T3 + 1758186.5T−1 + 22900.402lnT
	(1000–6000)		−1572051.75 + 1392.568261T − 225TlnT
${}_{ }{}^0{G}_{{Na}_{2}O}^{Liquid}$	(298.15–1405)		−380898.2803 + 340.194781T − 66.216001TlnT − 0.021932551T2 + 2.34792$\times$10−6T3 + 406685.01T−1
	(1405–1500)		−387789.21 + 580.2481164T − 104.6TlnT
${}_{ }{}^0{G}_{{Na}_{2}O}^{{Na}_{2}O\_\mathsf{\alpha }}$	(298.15–1405)		−428595.8803 + 374.143281T − 66.216001TlnT − 0.021932551T2 + 2.34792$\times$10−6T3 + 406685.01T−1
	(1405–1500)		−435486.81 + 614.1966164T − 104.6TlnT
${}_{ }{}^0{G}_{{Na}_{2}O}^{{Na}_{2}O\_\mathsf{\beta }}$	(298.15–1405)		−440520.2803 + 383.736481T − 66.216001TlnT − 0.021932551T2 + 2.34792$\times$10−6T3 + 406685.01T−1
	(1405–1500)		−447411.21 + 623.7898164T − 104.6TlnT
${}_{ }{}^0{G}_{{Na}_{2}O}^{{Na}_{2}O\_\mathsf{\gamma }}$	(298.15–1405)		−442277.5603 + 385.454281T − 66.216001TlnT − 0.021932551T2 + 2.34792$\times$10−6T3 + 406685.01T−1
	(1405–1500)		−449168.49 + 625.5076164T − 104.6TlnT
${}_{ }{}^0{G}_{{Na}_{3}P{O}_4}^{Solid}$	(298.15–6000)		+${}_{ }{}^0{G}_{{Na}_{2}O}^{{Na}_{2}O\_\mathsf{\gamma }}$ + 0.5${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_H}$ − 520688.946 − 4.353T
${}_{ }{}^0{G}_{{Na}_4{P}_2{O}_7}^{Solid}$	(298.15–967)		−3282062.70195 + 815.816308T − 145.08494TlnT − 0.215875T2 + 3.77148$\times$10−5T3 + 542554.7741T−1
	(967–1273)		−3322276.2378 + 2062.32142T − 349.82659TlnT
${}_{ }{}^0{G}_{{Na}_5{P}_3{O}_{10}}^{Solid}$	(298.15–6000)		+2.5${}_{ }{}^0{G}_{{Na}_{2}O}^{{Na}_{2}O\_\mathsf{\gamma }}$ + 1.5${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_H}$ − 1144087.404 − 20.487T
${}_{ }{}^0{G}_{NaP{O}_3}^{Solid}$	(298.15–703)		−1243133.8886 + 264.3713457T − 46.08288TlnT − 0.09082T2 + 1.80447$\times$10−5T3 + 188542.1015T−1
	(703–973)		−1256643.34549 + 704.6628499T − 119.50568TlnT
${}_{ }{}^0{G}_{MgO}^{Liquid}$	(298.15–1700)		−549098.33 + 275.724634T − 47.4817TlnT − 0.00232681T2 +4.5043$\times$10−8T3 + 516900T−1
	(1700–2450)		−585159.646 + 506.06825T − 78.3772TlnT + 0.0097344T2 −8.60338$\times$10−7T3 + 8591550T−1
	(2450–3100)		+9110429.75−42013.7634T + 5298.548TlnT − 1.30122485T2 + 5.8262601$\times$10−5T3 − 3.24037416$\times$109T−1
	(3100–5100)		−632664.468 + 589.239555T − 84TlnT
${}_{ }{}^0{G}_{MgO}^{Solid}$	(298.15–1700)		−619428.502 + 298.253571T − 47.4817TlnT − 0.00232681T2 +4.5043$\times$10−8T3 + 516900T−1
	(1700–3100)		−655489.818 + 528.597187T − 78.3772TlnT + 0.0097344T2 −8.60338$\times$10−7T3 + 8591550T−1
	(3100–5000)		−171490.159–1409.43369T + 163.674142TlnT − 0.044009535T2 + 1.374896$\times$10−6T3−1.72665403$\times$108T−1
	(5000–5100)		−722412.718 + 617.657452T−84TlnT
${}_{ }{}^0{G}_{{Mg}_3{P}_2{O}_8}^{Solid}$	(298.15–1800)		−3863914.664 + 1191.277265T − 195.04422TlnT−0.098665T2 + 9.22527$\times$10−6T3 + 1562390.321T−1
${}_{ }{}^0{G}_{{Mg}_2{P}_2{O}_7}^{Solid}$	(298.15–1800)		−3217696.802 + 1003.594497T − 165.99611TlnT − 0.07885T2 + 7.96388$\times$10−6T3 + 1371185.587T−1
${}_{ }{}^0{G}_{Mg{P}_2{O}_6}^{Solid}$	(298.15–6000)		+${}_{ }{}^0{G}_{MgO}^{Solid}$ + ${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_H}$ − 240840 + 2.95T
${}_{ }{}^0{G}_{Mg{P}_4{O}_{11}}^{Solid}$	(298.15–6000)		+${}_{ }{}^0{G}_{MgO}^{Solid}$ + ${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_H}$ – 269030 − 4.29T

and will be described below in more detail.

3.1. Pure Unary Component

The Gibbs energy G_i(T) of pure unary component i can be expressed as follows:

$$G_i\left(T\right)-H_i^{SER}=a+bT+cTlnT+d{T}^2+e{T}^{-2}+f{T}^3+g{T}^7+h{T}^{-9}$$

(1)

where H_i^SER is the standard molar enthalpy of pure unary component i at 298.15 K and 101,325 Pa, J·mol⁻¹; a~h are the parameters to be optimized; T is the thermodynamic temperature, K.

3.2. Liquid Phase

In the current assessment, the ionic two-sublattice model is used to describe the liquid phase of the P₂O₅-Na₂O and P₂O₅-MgO systems. The ionic two-sublattice model assumes that cations only mix with each other, and anions only mix with each other. This model comprises two sublattices: one for cations and the other for anions, neutrals, and vacancies.

In the liquid phase of the P₂O₅-Na₂O and P₂O₅-MgO systems, the content of anions such as PO₃⁻¹, P₂O₇⁻⁴, PO₄⁻³ varies with the composition of the system oxides [8]. To simplify the thermodynamic model by reducing the thermodynamic parameters, only the two anions (PO₃⁻¹ and PO₄⁻³) are considered in the optimized modeling process. Therefore, the thermodynamic models of the liquid phase of the P₂O₅-Na₂O and P₂O₅-MgO systems are (Na⁺¹)_P(O⁻², PO₃⁻¹, PO₄⁻³, PO_5/2)_Q and (Mg⁺²)_P(O⁻², PO₃⁻¹, PO₄⁻³, PO_5/2)_Q, where P and Q denote the total valence of the anion sublattice and the total valence of the cation sublattice, respectively. To maintain the electroneutrality of the liquid phase of the systems, the stoichiometric factors P and Q are allowed to change with the composition of the system oxides. Taking the P₂O₅-Na₂O system as an example, the Gibbs energy of the liquid phase is expressed as follows:

$$\begin{array}{cc}{G}_m^{Liquid}-H_i^{SER}=& y_{{Na}^{+1}}{y}_{O^{-2}}{G}_{{Na}^{+1}:O^{-2}}^{Liquid}+y_{{Na}^{+1}}{y}_{{PO}_3^{-1}}{G}_{{Na}^{+1}:{PO}_3^{-1}}^{Liquid}+y_{{Na}^{+1}}{y}_{{PO}_4^{-3}}{G}_{{Na}^{+1}:{PO}_4^{-3}}^{Liquid}+Q\left(y_{{PO}_{5/2}}\ln y_{{PO}_{5/2}}\right)\hfill \\ & +PRT\left(y_{{Na}^{+1}}\ln y_{{Na}^{+1}}\right)+QRT\left(y_{O^{-2}}\ln y_{O^{-2}}+y_{{PO}_3^{-1}}\ln y_{{PO}_3^{-1}}+y_{{PO}_4^{-3}}\ln y_{{PO}_4^{-3}}+y_{{PO}_{5/2}}\ln y_{{PO}_{5/2}}\right)\\ & +{}_{ }{}^E{G}_m^{Liquid}\hfill \end{array}$$

(2)

where H_i^SER is the molar enthalpy of the pure unary component in the reference state of the standard element at 298.15 K and 101,325 Pa, J·mol⁻¹; y is the site fraction of each species in the liquid phase in their respective sublattices; G is the Gibbs energy for the formation of the end-member, J·mol⁻¹; R is the gas constant (R = 8.314 J·(mol·K)⁻¹); ${}_{ }{}^E{G}_m^{Liquid}$ is the excess Gibbs energy, J·mol⁻¹, which is denoted as follows:

$$\begin{array}{cc}{}_{ }{}^E{G}_m^{Liquid}=& y_{{Na}^{+1}}{y}_{O^{-2}}{y}_{{PO}_3^{-1}}{}_{ }{}^0{L}_{{Na}^{+1}:O^{-2},{PO}_3^{-1}}^{Liquid}+y_{{Na}^{+1}}{y}_{O^{-2}}{y}_{{PO}_4^{-3}}({}_{ }{}^0{L}_{{Na}^{+1}:O^{-2},{PO}_4^{-3}}^{Liquid}+{}_{ }{}^1{L}_{{Na}^{+1}:O^{-2},{PO}_4^{-3}}^{Liquid}(y_{O^{-2}}-y_{{PO}_4^{-3}})\hfill \\ & +{}_{ }{}^2{L}_{{Na}^{+1}:O^{-2},{PO}_4^{-3}}^{Liquid}{\left(y_{O^{-2}}-y_{{PO}_4^{-3}}\right)}^2)+y_{{Na}^{+1}}{y}_{O^{-2}}{y}_{{PO}_{5/2}}{}_{ }{}^0{L}_{{Na}^{+1}:O^{-2},{PO}_{5/2}}^{Liquid}\hfill \\ & +y_{{Na}^{+1}}{y}_{{PO}_3^{-1}}{y}_{{PO}_4^{-3}}({}_{ }{}^0{L}_{{Na}^{+1}:{PO}_3^{-1} ,{PO}_4^{-3}}^{Liquid}+{}_{ }{}^1{L}_{{Na}^{+1}:{PO}_3^{-1} ,{PO}_4^{-3}}^{Liquid}(y_{{PO}_3^{-1}}-y_{{PO}_4^{-3}})\hfill \\ & +{}_{ }{}^2{L}_{{Na}^{+1}:{PO}_3^{-1} ,{PO}_4^{-3}}^{Liquid}{\left(y_{{PO}_3^{-1}}-y_{{PO}_4^{-3}}\right)}^2+{}_{ }{}^3{L}_{{Na}^{+1}:{PO}_3^{-1} ,{PO}_4^{-3}}^{Liquid}{\left(y_{{PO}_3^{-1}}-y_{{PO}_4^{-3}}\right)}^3)\hfill \\ & +y_{{Na}^{+1}}{y}_{{PO}_3^{-1}}{y}_{{PO}_{5/2}}({}_{ }{}^0{L}_{{Na}^{+1}:{PO}_3^{-1},{PO}_{5/2}}^{Liquid}+{}_{ }{}^1{L}_{{Na}^{+1}:{PO}_3^{-1} ,{PO}_{5/2}}^{Liquid}(y_{{PO}_3^{-1}}-y_{{PO}_{5/2}})\hfill \\ & +{}_{ }{}^2{L}_{{Na}^{+1}:{PO}_3^{-1} ,{PO}_{5/2}}^{Liquid}{\left(y_{{PO}_3^{-1}}-y_{{PO}_{5/2}}\right)}^2)+y_{{Na}^{+1}}{y}_{{PO}_4^{-3}}{y}_{{PO}_{5/2}}{}_{ }{}^0{L}_{{Na}^{+1}:{PO}_4^{-3} ,{PO}_{5/2}}^{Liquid}\hfill \end{array}$$

(3)

where ⁱL^Liquid(i = 0, 1, 2, 3) represents the interaction of the species in each sublattice and is the interaction parameter to be optimized.

3.3. Intermediate Compounds

In this work, all solid phases Na₃PO₄, Na₄P₂O₇, Mg₃P₂O_8, Mg₂P₂O₇, etc., are described as stoichiometric compounds. For the solid phase with heat capacity data, the Gibbs free energy G_m is expressed as follows:

$$G_m-H^{SER}=a+bT+cTlnT+d{T}^2+e{T}^{-1}$$

(4)

where H^SER is the molar enthalpy of the pure elements (Na, Mg, P and O) in the reference state of the standard element at 298.15 K and 101,325 Pa, J·mol⁻¹; a~e are the parameters which will be optimized; T is the thermodynamic temperature, K.

For the solid phase lacking heat capacity data, taking the P₂O₅-Na₂O system as an example, the Gibbs energy ⁰G_m of the solid is expressed as follows:

$${}_{ }{}^0{G}_m-H^{SER}=x{}_{ }{}^0{G}_{{Na}_{2}O}^{{Na}_{2}O\_\gamma }+y{}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_H}+A+BT$$

(5)

where x and y are the ratios of Na₂O and P₂O₅ in the solid phase; ${}_{ }{}^0{G}_{{Na}_{2}O}^{{Na}_{2}O\_\gamma }$ and ${}_{ }{}^0{G}_{P_2{O}_5}^{P_2{O}_5\_H}$ are the Gibbs energy of the solid phase of Na₂O and P₂O₅, respectively, J·mol⁻¹; A and B are the parameters which will be evaluated.

4. Results and Discussion

The thermodynamic optimization of the P₂O₅-Na₂O and P₂O₅-MgO binary systems was carried out based on the critical evaluation of phase equilibrium and thermodynamic property data using Thermo-Calc software in the current study. During the optimization process, certain emphasis was given to each dataset of phase equilibrium and thermodynamic property data, taking into account their reliability. By adjusting the thermodynamic parameters for each phase within the systems, the calculated results could reasonably describe the experimental data within the acceptable error range.

Initially, the solid-phase parameters were optimized using experimental data, encompassing the heat capacity and formation enthalpy of the intermediate phases. Subsequently, liquid-phase parameters were incorporated to replicate the liquidus and invariant reactions of the systems. Finally, all parameters were simultaneously optimized by considering all reliable experimental data to obtain a set of thermodynamic parameters capable of effectively describing the P₂O₅-Na₂O and P₂O₅-MgO binary systems, as presented in Table 2.

4.1. P₂O₅-Na₂O System

The Gibbs energy functions for the components P₂O₅ and Na₂O were sourced from the works of Jung et al. [65] and Wu et al. [66], respectively. Initially, the heat capacities of polymorphic forms of Na₄P₂O₇ and NaPO₃ were determined by fitting experimental data from Andon et al. [47], Ashcroft et al. [48] and Lazarev et al. [49], which were treated as identical within this study. Subsequently, the formation enthalpies of the four intermediate phases were optimized using experimental data on formation enthalpies from elements [40,41,42,43,44,45,46]. Then, the liquid parameters such as ${}_{ }{}^0{L}_{{Na}^{+1}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}$, ${}_{ }{}^1{L}_{{Na}^{+1}:{PO}_3^{-1},{PO}_4^{-3}}^{Liquid}$, etc., were adjusted to replicate the liquidus and invariant reactions of the P₂O₅-Na₂O binary system. Finally, all parameters were optimized simultaneously by considering all available experimental data.

Figure 1 presents the optimized phase diagram of the P₂O₅-Na₂O binary system in comparison with experimental data [34,35,36,37,38,39]. The eutectic reactions L = β − NaPO₃ + α − Na₅P₃O₁₀ and L = α − Na₄P₂O₇ + β − Na₃PO₄ are calculated to occur at temperatures of 820 K and 1212 K, respectively, while the peritectic reaction L + α − Na₄P₂O₇ = α − Na₅P₃O₁₀ takes place at 895 K. The difference in the calculated X(Na₂O) from the experimentally determined values reported in the literature is less than 0.9 mol.%, marking a substantial improvement over the calculation of Xie et al. [7] and aligning more closely with the experimental data. Due to the limited availability of liquidus data for the P₂O₅-Na₂O system, the predicted temperatures for the L = γ − NaPO₃ + O’ − P₂O₅ and L = β − Na₃PO₄ + β − Na₂O reactions calculated in this study are 560 K and 1220 K, respectively, which warrant validation through further experiments. The calculated liquidus points generally correspond well with the experimental data in the literature.

Table 3. Calculated invariant reactions involving the liquid phase in the P₂O₅-Na₂O binary system.

Reaction	Type	Liquid Composition/Mole Fraction Na2O	Temperature/K	Reference
L = γ − NaPO3 + O’ − P2O5	Eutectic	0.276	560	This work
L = β − NaPO3 + α − Na5P3O10	Eutectic	0.559	824	[34]
		0.556	825	[35]
		0.57	819	[36]
		0.543	763	[37]
		0.56	819	[39]
		0.56	833	[7]
		0.563	820	This work
L + α − Na4P2O7 = α − Na5P3O10	Peritectic	0.587	893	[34]
		0.588	895	[35]
		0.585	893	[37]
		0.589	893	[39]
		0.575	898	[7]
		0.576	895	This work
L = α − Na4P2O7 + β − Na3PO4	Eutectic	0.684	1218	[37]
		0.694	1225	[38]
		0.6975	1217	[39]
		0.691	1209	[7]
		0.6999	1212	This work
L = β − Na3PO4 + β − Na2O	Eutectic	0.814	1220	This work

provides a comparison of the calculated invariant reactions involving the liquid phase in the P₂O₅-Na₂O binary system with experimental data. It is evident that the calculated results of this study can effectively describe most of the available experimental information.

Utilizing the optimized thermodynamic parameters, the thermodynamic properties of the P₂O₅-Na₂O system are computed. Figure 2a and b show the calculated heat capacities of Na₄P₂O₇ and NaPO₃, respectively, obtained in this study, juxtaposed with experimental data measured by Andon et al. [47], Ashcroft et al. [48] and Lazarev et al. [49]. The calculated results exhibit satisfactory agreement with the measured values. For Na₅P₃O₁₀ and Na₃PO₄, the Neumann–Kopp equation was employed to describe their heat capacities due to the limited experimental data available. The standard enthalpies of formation of the intermediate compounds from elements (BCC_A2 for sodium and white phosphorus) at 298 K are also calculated in this work, as depicted in Figure 3. The graph illustrates that our calculated results are generally consistent with the experimental values from Refs. [40,41,42,43,44,45,46]. Considering experimental uncertainties, the calculations are deemed acceptable.

4.2. P₂O₅-MgO System

The Gibbs energy functions for the components P₂O₅ and MgO utilized in this study were sourced from Jung et al. [65] and Mao et al. [67], respectively. The heat capacities of Na₄P₂O₇ and NaPO₃ were modeled using experimental data from Oetting et al. [54]. In the present research, it is assumed that the heat capacities of both allotropic forms of Mg₂P₂O₇ were equal. The optimized phase diagram of the P₂O₅-MgO system, presented in Figure 4, is compared with experimental data [32,51,53,54,55,56,57,58]. Additionally, the temperature and phase composition details of invariant reactions are juxtaposed with the experimental data reported in the literature, as shown in

Table 4. Calculated invariant reactions involving the liquid phase in the P₂O₅-MgO binary system.

Reaction	Type	Liquid Composition/Mole Fraction P2O5	Temperature/K	Reference
L = MgO + Mg3P2O8	Eutectic	0.23	1598	[51]
		-	1603	[55]
		0.23	1596	[8]
		0.23	1602	This work
L = Mg3P2O8 + α − Mg2P2O7	Eutectic	0.276	1555	[51]
		0.277	1563	[8]
		0.276	1558	This work
L = α − Mg2P2O7 + MgP2O6	Eutectic	0.468	1423	[51]
		0.469	1410	[8]
		0.469	1421	This work
L = MgP2O6 + MgP4O11	Eutectic	0.62	1149	This work
L = MgP4O11 + O’ − P2O5	Eutectic	0.91	773	This work

. It is evident that the calculated phase boundaries align well with the experimental information found in the literature. The present study provides a better and more reasonable description of the experimental data for the P₂O₅-MgO system compared to the results of Ding et al. [8].

The phase relationship in the composition range above 50 mol.% P₂O₅ remains to be definitively determined experimentally, owing to the limited available experimental data. In the optimization process, two eutectic reactions were predicted in this portion of the phase diagram. The calculated reaction temperatures are 1149 K for the reaction L = MgP₂O₆ + MgP₄O₁₁ and 773 K for the reaction L = MgP₄O₁₁ + O’ − P₂O₅. Correspondingly, the calculated X(P₂O₅) values are 62 mol.% and 91 mol.%, respectively.

The heat capacities of Mg₃P₂O₈ and Mg₂P₂O₇ obtained by optimization in this work are illustrated in Figure 5a,b. Reasonable agreement is obtained between our calculated results and the heat capacities of Mg₃P₂O₈ and Mg₂P₂O₇ in the temperature range from 298.15 K to 1800 K determined by Oetting et al. [54]. To describe the heat capacities of MgP₂O₆ and MgP₄O₁₁, the Neumann–Kopp equation was employed during the optimization process to sum the heat capacities of Na2O_γ and P2O5_H. Figure 6a,b present the calculated heat contents of Mg₃P₂O₈ and Mg₂P₂O₇ based on the obtained thermodynamic parameters, compared with the experimental data [54]. The results indicate a close alignment with the experimental values, with acceptable deviations considering experimental errors. The calculated melting enthalpy of Mg₃P₂O₈($∆H_{melt}$ = 97.449 kJ·mol⁻¹) is slightly lower than the experimental value reported by Oetting et al. [54], while the calculated melting enthalpy of Mg₂P₂O₇($∆H_{melt}$ = 160.03 kJ·mol⁻¹) is slightly higher than the experimental value. This brings the calculated values much closer to the experimental results compared to the study by Ding et al. [8]. Additionally, the calculated enthalpy of transition from Mg₂P₂O₇_β to Mg₂P₂O₇_α at 340 K is determined to be 0.68 kJ·mol⁻¹. Figure 7 shows the calculated standard enthalpies of formation for the intermediate compounds from elements at 298 K compared with the experimental data [55,59,60,61,62,63] and calculated results from the literature [8]; the reference states are the Mg of HCP_A3 and white phosphorus, which reproduce the standard enthalpy of formation for the compounds from elements very well. As can be seen, a precise description of the experimental thermodynamic properties of the system can be provided by utilizing the calculated thermodynamic parameters within the acceptable margin of error.

5. Conclusions

The CALPHAD method was utilized to critically evaluate and assess the P₂O₅-Na₂O and P₂O₅-MgO binary systems. The main conclusions are summarized below:

• A set of self-consistent thermodynamic parameters is derived for the P₂O₅-Na₂O and P₂O₅-MgO binary systems based on a critical evaluation of the available phase diagram and thermodynamic property data. The calculated phase diagrams and thermodynamic properties employing the obtained thermodynamic parameters well reproduce the data reported in the literature.
• In comparison with the previous assessments using the modified quasi-chemical model for the liquid phase, the present study using the ionic two-sublattice model to express the liquid phase for the first time can describe the experimental data of the P₂O₅-Na₂O and P₂O₅-MgO binary systems in a better and more reasonable way, particularly the invariant reactions involving the liquid phase. The difference in the phase composition and temperature of invariant reactions from the experimentally determined values reported in the literature is less than 0.9 mol.% and 5K, respectively.
• Four eutectic reactions (L = γ − NaPO₃ + O’-P₂O₅, L = β − Na₃PO₄ + β − Na₂O, L = MgP₂O₆+ MgP₄O₁₁ and L = MgP₄O₁₁ + O’ − P₂O₅) are predicted in the P₂O₅-Na₂O and P₂O₅-MgO binary systems. The predicted temperatures of these eutectic reactions are 560 K, 1220 K, 1149 K and 773 K, with the corresponding phase compositions X(P₂O₅) being 82.4 mol%, 18.6 mol%, 62 mol% and 91 mol%, respectively. These predictions await further experimental validation.