Experimental Determination of the Standard Gibbs Energy of Formation of Fe_3–xV_xO₄ at 1473 K

Abstract

In the present study, an approach of determining the standard Gibbs energy of formation of Fe_3–xV_xO₄ was proposed firstly, then the standard Gibbs energies of formation of a variety of Fe_3–xV_xO₄ were determined experimentally, and finally, a calculating model of the standard Gibbs energy of formation of Fe_3–xV_xO₄ was established. The detailed results are as follows: (1) the standard Gibbs energy of formation of Fe_3–xV_xO₄ can be determined successfully by two steps; the first is to measure the chemical potential of Fe in Fe_3–xV_xO₄ under fixed oxygen partial pressure, the second is to derive the chemical potential of V in Fe_3–xV_xO₄ by Gibbs–Duhem relation; (2) the standard Gibbs energies of formation of Fe_3–xV_xO₄ are mainly decided by the Fe/V molar ratio, and almost not influenced by the oxygen partial pressure in the range from 2.39 × 10⁻¹² to 3.83 × 10⁻¹¹ atm; (3) in this oxygen partial pressure range, the standard Gibbs energies of formation of Fe_3–xV_xO₄ can be calculated satisfactorily by the following model: ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }\left(J/mol\right)=\left(1-x/2\right){\Delta }_f{G}_{F{e}_3{O}_4}^{\theta }+\left(x/2\right){\Delta }_f{G}_{Fe{V}_2{O}_4}^{\theta }+\left(1-x/2\right)RTln\left(1-x/2\right)+\left(x/2\right)RTln\left(x/2\right)$$- 168627.48\left(1-x/2\right)\left(x/2\right)$.

1. Introduction

The Fe–V–O system has attracted huge interest for decades due to the variety of the iron vanadate oxides formed in this system and the diverse applications of the oxides [1,2]. For instance, more than twenty oxides of iron vanadate such as Fe_0.1V₂O_5.15, Fe_0.33V₂O₅, Fe_0.83V_1.17O₄, Fe₂V₂O₅, Fe₂V₄O_12.29, Fe₂V₄O₁₃, FeV₂O₆, FeV₃O₈, FeVO₃, Fe₂VO₄, FeV₂O₄, FeVO₄, and Fe₄V₆O₂₁. have been reported [3,4,5,6], and their potential application in rechargeable batteries, electrical and optical switching devices, and heterogeneous catalysis fields have also been investigated [7,8].

Among the reported iron vanadate oxides, the oxides Fe₂VO₄ and FeV₂O₄ have been the focus of special attention [9,10] because they possess a spinel structure and key structural-related magnetic or electrical properties [11,12]. Fe₂VO₄ has the inverse spinel structure of Fe³⁺[Fe²⁺V³⁺]O₄, like magnetite Fe³⁺[Fe²⁺Fe³⁺]O₄. However, FeV₂O₄ changes to the normal spinel structure of Fe²⁺[V³⁺]₂O₄ with divalent iron (Fe²⁺) occupying tetrahedral sites and trivalent vanadium (V³⁺) occupying octahedral sites due to the strong affinity of V³⁺ ions for octahedral sites.

In addition to the stoichiometric compounds Fe₂VO₄ and FeV₂O₄, a complete solid solution Fe_3–xV_xO₄ can be formed by Fe³⁺[Fe²⁺Fe³⁺]O₄ and Fe²⁺[V³⁺]₂O₄. The cation distribution in the spinel solid solution Fe_3–xV_xO₄ has been determined by Mössbauer spectroscopy [13]. For 0 < x < 1.0, the solid solution remains an inverse spinel structure as Fe³⁺[Fe²⁺Fe³⁺_1-xV³⁺_x]O₄. For 1.64 < x < 2.0, the solid solution remains a normal spinel structure as Fe²⁺[Fe³⁺_1–x/2V³⁺_x/2]₂O₄. The transition from normal to inverse is observed for 1.0 < x < 1.64.

Due to the similar structure to magnetite Fe₃O₄, great consideration has been given to the magnetic properties of Fe_3–xV_xO₄. Wakihara et al. [14] synthesized the spinel solid solution with stoichiometric composition at 1500 K under controlled CO₂–H₂ atmospheres, measured the Curie temperature and the magnetization behaviors of a series of solid-solution compounds, and proposed a magnetic structure model. Jin et al. [15] fabricated epitaxial Fe_3–xV_xO₄ (0 ≤ x ≤ 0.6) films by using reactive co-sputtering from pure Fe and V in a gas mixture of Ar and O₂, and the magnetic and magnetotransport properties of the film spinels were studied at room temperature. Pool et al. [16] synthesized nanoparticles of Fe_3–xV_xO₄ with up to 33% vanadium substitution (x = 0 to 1) by mixing appropriate ratios of solutions of 0.5 mmol V(acac)₃ and Fe(acac)₃, 1,2-hexadecanediol with benzyl ether, oleic acid(1.5 mmol), and oleylamine (1.5 mmol) under evacuated conditions. The site preference of the vanadium and the magnetic behavior of the nanoparticles were investigated through L23edge X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (MCD) spectra.

Research on the optical absorption properties as well as the catalytic and stable properties of Fe_3–xV_xO₄ have also been reported. Kim et al. [17] prepared the Fe_3–xV_xO₄ thin film on Si (100) substrates by using a sol–gel method. The valence and occupying sites of V ions in the Fe_3–xV_xO₄ (x < 1.0) compounds were revealed by using X-ray diffraction, X-ray photoelectron spectroscopy, and Mössbauer spectroscopy, while the optical absorption properties of the films were measured by spectroscopic ellipsometry (SE). Häggblad et al. [18] investigated the catalytic effect of Fe_3–xV_xO₄ (x < 1.37) on the oxidation of methanol to produce formaldehyde, and the valence change of V ion was revealed by in and ex situ analyses of the samples with X-ray absorption near edge structure (XANES) spectroscopy.

Apart from the research on the physical properties of Fe_3–xV_xO₄, a number of works relating to the phase equilibria of Fe_3–xV_xO₄ have been carried out. Coetsee et al. [19] determined the existing phases with the change of Fe/V molar ratio in the V₂O₃–FeO system from 1673 K to 1873 K by using CO/CO₂ = 3 gas mixture to control the oxygen partial pressure. Wakihara et al. [20] determined the phase equilibria in FeO–Fe₂O₃–V₂O₃ system at 1500 K by varying the oxygen partial pressure and proposed the phase boundaries of Fe_3–xV_xO₄. In our previous publication [21], the phase equilibria of the FeO–V₂O₃ system at 1473 K under various oxygen partial pressures were determined, and a phase diagram of $\lg \left(p_{{\mathrm{O}}_2}/atm\right)$ vs. the molar ratio of V/(Fe + V) was presented. Based on the available phase equilibria and thermodynamic data, the Fe–V–O system was assessed and optimized through the CALPHAD method. Xie et al. [2] performed the assessment by adopting the modified quasichemical model to describe the liquid phase in the system and the sublattice model based on the compound energy formalism to describe the spinel solid solution Fe_3–xV_xO₄. Du et al. [22] also performed an evaluation and modeling of the system by using the modified quasichemical model to describe the liquid oxide solution and the two sublattice spinel solution model within the framework of the compound energy formalism to the spinel solid solution Fe_3–xV_xO₄. In addition, the modeling works of some more complex systems, such as the Fe–Ti–V–O system [23] and CaO–FeO–Fe₂O₃–MgO–SiO₂-containing V₂O₃ and V₂O₅ system [24], were reported.

However, regarding the elementary thermodynamic data, i.e., the standard Gibbs energy of Fe_3–xV_xO₄, only the data of FeV₂O₄ was reported. The first Gibbs energy measurement was performed by Chipman and Dastur [25] by the equilibration of FeV₂O₄ with liquid Fe–V solution under a controlled H₂–H₂O gas atmosphere at 1873 K. Wakihara et al. [20] determined the standard Gibbs energy of the reaction of Fe + V₂O₃ + 0.5O₂ = FeV₂O₄ and 0.05Fe + V₂O₃ + Fe_0.95O = FeV₂O₄ at 1500 K, according to the equilibrated oxygen partial pressure. And even in the database of commercial thermodynamic software FactSage [26], only the data of FeV₂O₄ can be retrieved. Therefore, the present study aimed at providing the fundamental thermodynamic data of Fe_3–xV_xO₄ and understanding the effect of V substituting on the thermodynamic property of the spinel phase; the standard Gibbs energy of formation of Fe_3–xV_xO₄ was determined by combining theoretical derivation with experimental measurements.

2. Principle and Experimental

2.1. Principle of Determining the Gibbs Energy of Fe_3–xV_xO₄

The molar Gibbs energy of Fe_3–xV_xO₄ can be calculated according to Equation (1).

$$G_{F{e}_{3-x}{V}_x{O}_4}=\left(3-x\right){\mu }_{Fe}+x{\mu }_V+2{\mu }_{O_2}$$

(1)

where ${\mu }_{Fe}$, ${\mu }_V$, and ${\mu }_{O_2}$ are the chemical potentials of Fe, V, and O₂, respectively. Furthermore, according to the Gibbs–Duhem relation [27], the chemical potentials of Fe, V, and O₂ can be connected by Equation (2).

$$\left(3-x\right)·d{\mu }_{Fe}+x·d{\mu }_V+2·d{\mu }_{O_2}=0$$

(2)

If the oxygen partial pressure is fixed unchangeably, the $d{\mu }_{O_2}$ will be zero, and then Equation (2) can be simplified as Equation (3).

$$\left(3-x\right)·d{\mu }_{Fe}+x·d{\mu }_V=0$$

(3)

Then, Equation (4) can be obtained by dividing both sides of Equation (3) by x.

$$d{\mu }_V=-\frac{3-x}{x}d{\mu }_{Fe}$$

(4)

where $\frac{3-x}{x}$ represent the molar ratio of Fe to V in Fe_3–xV_xO₄. If it is replaced by y, Equation (4) becomes Equation (5).

$$d{\mu }_V=-y·d{\mu }_{Fe}$$

(5)

By integrating Equation (5) from $y_s$ to $y$, the following relations can be obtained.

$${{\displaystyle \int }}_{{\mu }_{V,y_s}}^{{\mu }_V}d{\mu }_V={{\displaystyle \int }}_{y_s}^y-y·d{\mu }_{Fe}$$

(6)

where $y_s$ refers to the molar ratio of Fe to V in Fe_3–xV_xO₄ when it is saturated or equilibrated with V₂O₃, i.e., the minimum molar ratio of Fe to V in Fe_3–xV_xO₄ at the fixed oxygen partial pressure. Correspondingly, ${\mu }_{V,y_s}$ refers to the chemical potential of V in the Fe_3–xV_xO₄ that is saturated or equilibrated with V₂O₃ at the fixed oxygen partial pressure.

Next, using the method of integration by parts, Equation (6) can be changed to Equation (7).

$${\mu }_V-{\mu }_{V, y_s}={\left[-y·{\mu }_{Fe}\right]}_{y_s,{\mu }_{Fe,y_s}}^{y,{\mu }_{Fe}}+{{\displaystyle \int }}_{y_s}^y{\mu }_{Fe}·dy$$

(7)

Rearranging Equation (7), a relation of Equation (8) can be obtained.

$${\mu }_V={\mu }_{V, y_s}+\left(y_s·{\mu }_{Fe,y_s}-y·{\mu }_{Fe}\right)+{{\displaystyle \int }}_{y_s}^y{\mu }_{Fe}·dy$$

(8)

Equation (8) tells us if we can know the functional relationship between ${\mu }_{Fe}$ and $y$, as well as the value of $y_s$ and ${\mu }_{V, y_s}$, then the chemical potential of V in Fe_3–xV_xO₄ can be calculated through Equation (8).

Regarding the parameters $y_s$ and ${\mu }_{V, y_s}$, they can be determined by using the following approaches. For $y_s$, it can be read directly from the phase diagram of the Fe_tO–V₂O₃ system at 1473 K under various oxygen partial pressures, which we have reported in a previous publication [21]. For ${\mu }_{V, y_s}$, it can be determined by the formation reaction of V₂O₃ shown as Equation (9), because the chemical potentials of V in all the equilibrated phases are equal.

$$V+3/4O_2=1/2V_2{O}_3$$

(9)

According to Equation (9), the ${\mu }_{V, y_s}$ can be determined by Equation (10).

$${\mu }_{V, y_s}=1/2{\mu }_{V_2{O}_3}^{\theta }-3/4\left[{\mu }_{O_2}^{\theta }+RTln\left(p_{O_2}/p^{\theta }\right)\right]$$

(10)

where ${\mu }_{V_2{O}_3}^{\theta }$ is the chemical potential of pure V₂O₃ compound, ${\mu }_{O_2}^{\theta }$ is the chemical potential of O₂ at 1atm, and $p_{O_2}$ is the oxygen partial pressure adopted in experiments.

From the above analysis, it can be concluded that when the dependence of ${\mu }_{Fe}$ on $y$ can be obtained, the ${\mu }_V$ will be calculated through Equation (8), and the $G_{F{e}_{3-x}{V}_x{O}_4}$ (molar Gibbs energy of Fe_3–xV_xO₄) can be calculated by inserting ${\mu }_V$ and ${\mu }_{Fe}$ into Equation (1). Therefore, to measure the ${\mu }_{Fe}$ in Fe_3–xV_xO₄ is a prerequisite for the determination of the molar Gibbs energy of Fe_3–xV_xO₄.

2.2. Experimental Procedure for Measuring ${\mu }_{Fe}$ in the Solid-Solution Phase Fe_3–xV_xO₄

The experiments to measure the values of the chemical potential of Fe in the solid solution phase Fe_3–xV_xO₄ consisted of three steps. The first was to prepare the pure Fe_3–xV_xO₄ compounds with different Fe/V molar ratios at fixed oxygen partial pressure; the second was to equilibrate the prepared Fe_3–xV_xO₄ with liquid copper; the third was to measure the content of Fe in the copper by Inductive Coupled Plasma–Optical Emission Spectrometer (ICP–OES, OPTIMA 7000DV, Waltham, MA, USA).

Pure Fe_3–xV_xO₄ compounds with different Fe/V molar ratios were prepared through the conventional solid-state reaction method under CO–CO₂ gas mixture atmosphere at 1473 K, where CO–CO₂ gas mixture was adopted to control the oxygen partial pressure in the reaction chamber. The experimental apparatus and the operating details can be found in our previous publications [21,28,29]. The reactants, composed of analytical reagents of Fe, Fe₂O₃, and V₂O₃ with purity of 99.99%, were ground homogeneously in an agate mortar and pressed into pellets. Then, the pellets were placed into an alumina crucible and reacted in a pre-designed atmosphere for 72 h. Detailed information about the composition of prepared samples and the preparing atmospheres is shown in Table 1.

Table 1. Composition and preparing conditions of Fe_3–xV_xO₄ compounds.

No. of Samples	Fe/V Molar Ratio	Volume Ratio of CO to CO2	Oxygen Partial Pressure (atm)
1-1	1.6	2	2.39 × 10−12
1-2	1.4
1-3	1.2
1-4	1.0
1-5	0.8
1-6	0.6
2-1	1.8	1	9.57 × 10−12
2-2	1.6
2-3	1.4
2-4	1.2
2-5	1.0
2-6	0.8
3-1	2.8	0.5	3.83 × 10−11
3-2	2.6
3-3	2.2
3-4	2.0
3-5	1.6
3-6	1.0

After the pure Fe_3–xV_xO₄ was prepared, it was equilibrated with liquid copper at 1473 K for 72 h under the same atmosphere as the preparing condition. Here, the liquid copper was selected as the metal to equilibrate with the Fe_3–xV_xO₄ sample. This is because the liquid copper is stable and cannot be oxidized during the present investigated oxygen partial pressure range (2.39 × 10⁻¹²~3.83 × 10⁻¹¹ atm), and also, the activity coefficient of Fe in liquid copper is far greater than unity [30], which can ensure that the concentration of Fe in liquid copper is very small. For a typical run of the equilibrating experiment, about 5 g copper powder with a purity of 99.99% was placed into an alumina crucible, a tablet of about 2.5 g of Fe_3–xV_xO₄ was placed above the copper powder, and then the crucible containing the samples was put into a furnace and heated to 1473 K. When the temperature reached 1473 K, the samples were held for 72 h to ensure the equilibrium between Fe_3–xV_xO₄ and liquid copper. Then, the equilibrated samples were quenched by lifting the sample from the high-temperature zone to the water-cooled quenching chamber. The quenched sample was taken out from the quenching chamber, and the solidified copper was separated from the Fe_3–xV_xO₄ by a mechanical method. Then, the surface of the separated copper was cleaned mechanically for ICP–OES analysis to measure the content of Fe. For the ICP–OES analysis of Fe content, each sample was measured three times to ensure the error was less than 10 ppm; then, the measured three values were averaged as the final results.

When the content of Fe in the copper sample was measured, the chemical potential of Fe in the Fe_3–xV_xO₄ sample can be deduced by using the following method. As the Fe_3–xV_xO₄ sample was equilibrated with the liquid copper chemically, the chemical potential of Fe in the Fe_3–xV_xO₄ sample was equal to that in the copper sample. So, by using the measured content of Fe in copper, the chemical potential of Fe can be determined by Equation (11).

$${\mu }_{Fe}={\mu }_{Fe}^{\theta }+RTln\left({\gamma }_{Fe\left(Cu\right)}·x_{Fe\left(Cu\right)}\right)$$

(11)

where ${\mu }_{Fe}^{\theta }$ is the chemical potential of pure iron, ${\gamma }_{Fe\left(Cu\right)}$ is the activity coefficient of Fe in liquid copper, and $x_{Fe\left(Cu\right)}$ is the molar fraction of Fe in the liquid copper. Furthermore, if the content of Fe in the liquid copper is low and can be treated as a dilute solution, the activity coefficient ${\gamma }_{Fe\left(Cu\right)}$ can be replaced by the Henry constant of Fe in liquid copper ${\gamma }_{Fe\left(Cu\right)}^0$, which was reported to be 24.1 relative to the pure solid Fe [30]. Therefore, in the present study, the chemical potential of Fe would be determined by Equation (12).

$${\mu }_{Fe}={\mu }_{Fe}^{\theta }+1473·R·ln\left(24.1·x_{Fe\left(Cu\right)}\right)$$

(12)

After ${\mu }_{Fe}$ in the Fe_3–xV_xO₄ sample with different Fe/V molar ratio was determined, the dependence of ${\mu }_{Fe}$ on $y$ can be deduced. When the relationship between ${\mu }_{Fe}$ and $y$ was obtained, the molar Gibbs energy of Fe_3–xV_xO₄ can then be obtained by the approach presented in the previous section.

3. Results and Discussion

3.1. Chemical Potential of Fe in the Fe_3–xV_xO₄ under Different Oxygen Partial Pressures

In our previous publication [21], the phase diagram of the “FeO”–V₂O₃ system at 1473 K in the oxygen partial pressure range of 5.98 × 10⁻¹⁵ atm to 2.39 × 10⁻⁸ atm was reported. Under the present investigated oxygen partial pressures, the Fe/V molar ratio for the single-phase area of Fe_3–xV_xO₄ is summarized in Table 2. Where $y_L$ and $y_s$ correspond to the maximal and minimum Fe/V molar ratio of Fe_3–xV_xO₄ single-phase area. When the Fe/V molar ratio in the system is larger than $y_L$, Fe_3–xV_xO₄ will coexist with the FeO phase. And when the Fe/V in the system is less than $y_s$, Fe_3–xV_xO₄ will coexist with the V₂O₃ phase.

Table 2. Compositional limitations of Fe_3–xV_xO₄ single-phase area.

Oxygen Partial Pressure	Compositional Limitations
${\mathit{p}}_{{\mathit{O}}_2}/\mathit{a}\mathit{t}\mathit{m}$	${\mathit{y}}_{\mathit{s}}$	${\mathit{y}}_{\mathit{L}}$
2.39 × 10−12	0.524	1.604
9.57 × 10−12	0.565	2.300
3.83 × 10−11	0.567	3.098

From Table 2, it can be clearly seen that the Fe/V ratio of the presently investigated samples is located in the single-phase area of Fe_3–xV_xO₄. So this can ensure the prepared compounds are pure phase with the pre-designed Fe/V molar ratio.

In Table 3, the ICP-OES measurement results for the contents of Fe in the quenched copper phase that had been equilibrated with Fe_3–xV_xO₄ are present. For the sake of clarity, the Fe/V molar ratio of the equilibrating Fe_3–xV_xO₄ compound and the equilibrating oxygen partial pressure are also listed together in Table 3. By using the measured contents of Fe in the quenched copper phase, the chemical potential of Fe in Fe_3–xV_xO₄ was calculated according to Equation (12), and the calculated values are shown in the form of ${\mu }_{Fe}-{\mu }_{Fe}^{\theta }$ in the last column of Table 3.

Table 3. The equilibrating experimental conditions and the related results.

Equilibrating Experimental Conditions			Results
No.	$\mathit{y}$	${\mathit{p}}_{{\mathit{O}}_2}/\mathit{a}\mathit{t}\mathit{m}$	Mass % of Fe in Cu	${\mathit{x}}_{\mathit{F}\mathit{e}\left(\mathit{C}\mathit{u}\right)}$	${\mathit{\mu }}_{\mathit{F}\mathit{e}}-{\mathit{\mu }}_{\mathit{F}\mathit{e}}^{\mathit{\theta }}$ (J/mol)
1-1	1.6	2.39 × 10−12	0.52	0.0059	−23,861.3
1-2	1.4		0.45	0.0051	−25,630.8
1-3	1.2		0.38	0.0043	−27,700.2
1-4	1.0		0.35	0.0040	−28,706.8
1-5	0.8		0.24	0.0027	−33,325.5
1-6	0.6		0.18	0.0020	−36,847.6
2-1	1.8	9.57 × 10−12	0.41	0.0047	−26,770.1
2-2	1.6		0.36	0.0041	−28,362.0
2-3	1.4		0.31	0.0035	−30,192.4
2-4	1.2		0.24	0.0027	−33,325.5
2-5	1.0		0.15	0.0017	−39,079.9
2-6	0.8		0.08	0.0009	−46,777.0
3-1	2.8	3.83 × 10−11	0.31	0.0035	−30,192.4
3-2	2.6		0.28	0.0032	−31,438.4
3-3	2.2		0.20	0.0023	−35,557.6
3-4	2.0		0.19	0.0022	−36,185.6
3-5	1.6		0.17	0.0019	−37,547.4
3-6	1.0		0.08	0.0009	−46,777.0

$y$ is the molar ratio of Fe to V in Fe_3–xV_xO₄.

From the calculated results of ${\mu }_{Fe}-{\mu }_{Fe}^{\theta }$ shown in the last column of Table 3, it can be found that both the Fe/V molar ratio of Fe_3–xV_xO₄ and the equilibrating oxygen partial pressure have a significant effect on the chemical potential of Fe in the Fe_3–xV_xO₄ compound. At each fixed oxygen partial pressure, the chemical potential of Fe increased gradually with the increase of the Fe/V molar ratio. By comparing the results with the same Fe/V ratio, it can be found that the chemical potential of Fe decreased gradually with the increase of the equilibrating oxygen partial pressures. For instance, the ${\mu }_{Fe}-{\mu }_{Fe}^{\theta }$ for Fe_3–xV_xO₄(Fe/V = 1.6) decreased from −23,861.3 J/mol to −28,362.0 J/mol and then to −37,547.4 J/mol, while the oxygen partial pressure increased from 2.39 × 10⁻¹² atm to 9.57 × 10⁻¹² atm and then to 3.83 × 10⁻¹¹ atm.

In view of the above-indicated situation, it is an expedient approach to deduce the functional relations between ${\mu }_{Fe}$ and y separately at each experimental oxygen partial pressure. It was assumed that the relations between ${\mu }_{Fe}$ and y could be expressed by the relation shown in Equation (13).

$${\mu }_{Fe}={\mu }_{Fe}^{\theta }+RTln\left(a+by\right)$$

(13)

To determine the model parameters $a$ and $b$, $\exp \left[\left({\mu }_{Fe}-{\mu }_{Fe}^{\theta }\right)/\mathrm{RT}\right]$ was plotted against y, and the results are shown in Figure 1.

Figure 1. Plots of $\exp \left[\left({\mu }_{Fe}-{\mu }_{Fe}^{\theta }\right)/\mathrm{RT}\right]$ against Fe/V ratio of Fe_3–xV_xO₄ and the linear fitting results.

From Figure 1, it can be seen that satisfactory linear relationships exist between $\exp \left[\left({\mu }_{Fe}-{\mu }_{Fe}^{\theta }\right)/\mathrm{RT}\right]$ and y. This implies that the assumed functional relation by Equation (13) is acceptable in the current investigated conditions, and the obtained functional relations between ${\mu }_{Fe}$ and y at different oxygen partial pressures are summarized and presented in Table 4.

Table 4. The functional relations between ${\mu }_{Fe}$ and y at different oxygen partial pressure.

$\mathbf{Equilibrating} {\mathit{p}}_{{\mathit{O}}_2}/\mathit{a}\mathit{t}\mathit{m}$	$\mathbf{Functional} \mathbf{Relations} \mathbf{between} {\mathit{\mu }}_{\mathit{F}\mathit{e}} \mathbf{and} \mathbf{y}$
2.39 × 10−12	${\mu }_{Fe}={\mu }_{Fe}^{\theta }+RTln\left(-0.0048+0.092y\right)$
9.57 × 10−12	𝜇𝐹𝑒 = 𝜇𝐹𝑒𝜃 + 𝑅𝑇𝑙𝑛 − 0.0488 + 0.092𝑦
3.83 × 10−11	${\mu }_{Fe}={\mu }_{Fe}^{\theta }+RTln\left(-0.0116+0.033y\right)$

3.2. Chemical Potential of V in the Fe_3–xV_xO₄ under Different Oxygen Partial Pressure

After obtaining the functional relationships between the ${\mu }_{Fe}$ and y, the chemical potential of V in Fe_3–xV_xO₄ under different oxygen partial pressures were derived. The detailed procedure is as follows. Firstly, the functional relationship between ${\mu }_{Fe}$ and y, as shown in Equation (13), was inserted into Equation (8), yielding Equation (14).

$$\begin{gathered} {\mu }_V={\mu }_{V, y_s}+\left[y_s·\left({\mu }_{Fe}^{\theta }+RTln\left(a+b{y}_s\right)\right)-y·\left({\mu }_{Fe}^{\theta }+RTln\left(a+by\right)\right)\right] \\ +{{\displaystyle \int }}_{y_s}^y\left({\mu }_{Fe}^{\theta }+RTln\left(a+by\right)\right)·dy \end{gathered}$$

(14)

Rearranging Equation (14), Equation (15) was obtained.

$${\mu }_V={\mu }_{V, y_s}+\left[y_{s}RTln\left(a+b{y}_s\right)-yRTln\left(a+by\right)\right]+RT{{\displaystyle \int }}_{y_s}^{y}ln\left(a+by\right)·dy$$

(15)

By integrating Equation (15) by parts, Equation (16) was obtained.

$$\begin{gathered} {\mu }_V={\mu }_{V, y_s}+\left[y_{s}RTln\left(a+b{y}_s\right)-yRTln\left(a+by\right)\right]+{\left[y·RTln\left(a+by\right)\right]}_{y_s}^y \\ -RT{{\displaystyle \int }}_{y_s}^y\left[1-\frac{a}{\left(a+by\right)}\right]·dy \end{gathered}$$

(16)

Equation (16) was rearranged and integrated again, and Equation (17) was obtained.

$${\mu }_V={\mu }_{V, y_s}-RT\left(y-y_s\right)+\left(\frac{a}{b}\right){\left[RTln\left(a+by\right)\right]}_{y_s}^y$$

(17)

Then, inserting Equation (10) into Equation (17) yielded Equation (18)

$${\mu }_V-{\mu }_V^{\theta }=\frac{1}{2}{\Delta }_f{G}_{V_2{O}_3}^{\theta }-3/4RTln\left(p_{O_2}/p^{\theta }\right)-RT\left(y-y_s\right)+\left(\frac{a}{b}\right)RTln\left(\frac{a+by}{a+b{y}_s}\right)$$

(18)

where ${\Delta }_f{G}_{V_2{O}_3}^{\theta }$ is the standard Gibbs energy of formation of V₂O₃, and its reported value at 1473 K is −864,690.7 J/mol. The values of $y_s$ at 2.39 × 10⁻¹² atm, 9.57 × 10⁻¹² atm, and 3.83 × 10⁻¹¹ are 0.524, 0.565, and 0.567, respectively.

Now, through Equation (18), the values of ${\mu }_V-{\mu }_V^{\theta }$ for Fe_3–xV_xO₄ with various Fe/V ratios under different oxygen partial pressures can be calculated. The calculated results are summarized and presented in Table 5.

Table 5. The derived ${\mu }_V-{\mu }_V^{\theta }$ as well as the measured ${\mu }_{Fe}-{\mu }_{Fe}^{\theta }$ in Fe_3–xV_xO₄ compounds.

Equilibrating Experimental Conditions			Results
No.	$\mathit{y}$	${\mathit{p}}_{{\mathit{O}}_2}/\mathit{a}\mathit{t}\mathit{m}$	${\mathit{\mu }}_{\mathit{F}\mathit{e}}-{\mathit{\mu }}_{\mathit{F}\mathit{e}}^{\mathit{\theta }}$ (J/mol)	${\mathit{\mu }}_{\mathit{V}}-{\mathit{\mu }}_{\mathit{V}}^{\mathit{\theta }}$ (J/mol)
1-1	1.6	2.39 × 10−12	−23,861.3	−200,496.5
1-2	1.4		−25,630.8	−197,958.8
1-3	1.2		−27,700.2	−195,406.8
1-4	1.0		−28,706.8	−192,835.2
1-5	0.8		−33,325.5	−190,234.6
1-6	0.6		−36,847.6	−187,586.3
2-1	1.8	9.57 × 10−12	−26,770.1	−237,836.0
2-2	1.6		−28,362.0	−234,273.1
2-3	1.4		−30,192.4	−230,479.1
2-4	1.2		−33,325.5	−226,331.9
2-5	1.0		−39,079.9	−221,577.7
2-6	0.8		−46,777.0	−215,523.2
3-1	2.8	3.83 × 10−11	−30,192.4	−249,849.2
3-2	2.6		−31,438.4	−247,033.0
3-3	2.2		−35,557.6	−241,291.1
3-4	2.0		−36,185.6	−238,348.9
3-5	1.6		−37,547.4	−232,253.9
3-6	1.0		−46,777.0	−222,086.1

$y$ is the molar ratio of Fe to V in Fe_3-xV_xO₄.

3.3. Standard Gibbs Energy of Formation of Fe_3–xV_xO₄ at 1473 K

According to the definition of standard Gibbs energy of formation, the standard Gibbs energy of Fe_3–xV_xO₄ can be determined by the following relation.

$${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }={\mu }_{F{e}_{3-x}{V}_x{O}_4}^{\theta }-\left(3-x\right){\mu }_{Fe}^{\theta }-x{\mu }_V^{\theta }-2{\mu }_{O2}^{\theta }$$

(19)

where ${\mu }_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$, the chemical potential of pure Fe_3–xV_xO₄, is equal to the molar Gibbs energy of pure Fe_3–xV_xO₄, so Equation (1) can be inserted into Equation (19) to replace the ${\mu }_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$. Then, a relation shown as Equation (20) can be obtained.

$${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }=\left(3-x\right)\left({\mu }_{Fe}-{\mu }_{Fe}^{\theta }\right)+x\left({\mu }_V-{\mu }_V^{\theta }\right)+2\left({\mu }_{O_2}-{\mu }_{O2}^{\theta }\right)$$

(20)

Furthermore, according to the relationship between ${\mu }_{O_2}$ and oxygen partial pressure, Equation (20) can be changed to Equation (21).

$${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }=\left(3-x\right)\left({\mu }_{Fe}-{\mu }_{Fe}^{\theta }\right)+x\left({\mu }_V-{\mu }_V^{\theta }\right)+2RTln\left(p_{O_2}/p^{\theta }\right)$$

(21)

Now, by inserting the values of ${\mu }_{Fe}-{\mu }_{Fe}^{\theta }$ and ${\mu }_V-{\mu }_V^{\theta }$ presented in Table 4 into Equation (21), the standard Gibbs energy of formation of Fe_3–xV_xO₄ with different Fe/V ratios under experimental oxygen partial pressures was calculated, and the calculated results are listed in Table 6.

Table 6. The standard Gibbs energy of formation of Fe_3–xV_xO₄.

No.	$\mathit{y}$	${\mathit{p}}_{{\mathit{O}}_2}/\mathit{a}\mathit{t}\mathit{m}$	${\Delta }_{\mathit{f}}{\mathit{G}}_{\mathit{F}{\mathit{e}}_{3-\mathit{x}}{\mathit{V}}_{\mathit{x}}{\mathit{O}}_4}^{\mathit{\theta }} \mathbf{J}/\mathbf{mol}$
1-1	1.6	2.39 × 10−12	−930,820.9
1-2	1.4		−947,729.6
1-3	1.2		−967,218.6
1-4	1.0		−987,740.2
1-5	0.8		−1,016,918.9
1-6	0.6		−1,048,605.1
2-1	1.8	9.57 × 10−12	−927,899.3
2-2	1.6		−944,122.7
2-3	1.4		−962,382.6
2-4	1.2		−984,614.1
2-5	1.0		−1,012,433.4
2-6	0.8		−1,043,021.7
3-1	2.8	3.83 × 10−11	−851,470.1
3-2	2.6		−861,457.0
3-3	2.2		−887,027.6
3-4	2.0		−898,199.8
3-5	1.6		−924,783.2
3-6	1.0		−990,774.3

$y$ is the molar ratio of Fe to V in Fe_3-xV_xO₄.

To further understand the influence of Fe/V ratio and oxygen partial pressure on the standard Gibbs energy of formation of Fe_3–xV_xO₄, the calculated values presented in Table 5 were demonstrated in Figure 2 by plotting the ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ against the Fe/V molar ratio. From Figure 2, it can be seen apparently that the data of ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ determined under different oxygen partial pressures shown not only very similar changing trends but also very small differences. This implies that in the present investigated composition and atmosphere ranges, the standard Gibbs energy of formation of Fe_3–xV_xO₄ is mainly influenced by the Fe/V molar ratio, and the effect of oxygen partial pressure is almost negligible.

Figure 2. The changing trend of ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ with Fe/V ratio under different oxygen partial pressures.

In Figure 2, apart from the data determined in the present study, the standard Gibbs energy of formation of FeV₂O₄ reported as −1,061,678.1 J/mol at 1473 K [26] was also denoted by using a star marker. It can be found that the ${\Delta }_f{G}_{Fe{V}_2{O}_4}^{\theta }$ shows excellent agreement and compatibility with the present determined data.

From Figure 2, although the trend of the change of the standard Gibbs energy of formation of Fe_3–xV_xO₄ with the increasing of Fe/V molar ratio can be seen explicitly, the functional relationship between them is implicit. To derive it, the standard Gibbs energies of the formation of FeV₂O₄ and Fe₃O₄ were considered because Fe_3–xV_xO₄ can be considered as a complete solid solution formed with FeV₂O₄ and Fe₃O₄ FeV₂O₄ and Fe₃O₄ as the end members.

In Figure 3, the standard Gibbs energies of the formation of FeV₂O₄ and Fe₃O₄ were plotted together with that of Fe_3–xV_xO₄ against $x/2$. Where $x/2$ can be regarded as the molar fraction of the end member FeV₂O₄. The adopted data of the standard Gibbs energies of the formation of Fe₃O₄ were –648765.3 J/mol [26].

Figure 3. Comparing the present determined ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ with the model calculating results.

From Figure 3, it can be seen that the present determined standard Gibbs energy of formation of Fe_3–xV_xO₄ has high relevance with that of their end members. Therefore, the regular solution model was adopted tentatively to establish a calculating model of the standard Gibbs energy of formation of Fe_3–xV_xO₄.

According to the regular solution model, the Gibbs energy of a solution composed of A and B can be expressed by Equation (22).

$$G_{A-B}=x_A{G}_A^{\theta }+x_B{G}_B^{\theta }+x_{A}RTln{x}_A+x_{B}RTln{x}_B+\mathsf{\Omega }{x}_A{x}_B$$

(22)

where $G_A^{\theta }$ and $G_B^{\theta }$ are Gibbs energy of pure components A and B, $x_A$ and $x_B$ are molar fractions of A and B, and $\mathsf{\Omega }$ is the model parameter obtained from experimental data. In the present study, by using Fe₃O₄ to replace A, FeV₂O₄ to replace B, and fitting the model to the determined values of ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$, the following expression of calculating the standard Gibbs energy of formation of Fe_3–xV_xO₄ was obtained.

$$\begin{array}{ll}{\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }\left(J/mol\right)& \\ & =\left(1-x/2\right){\Delta }_f{G}_{F{e}_3{O}_4}^{\theta }+\left(x/2\right){\Delta }_f{G}_{Fe{V}_2{O}_4}^{\theta }\\ & +\left(1-x/2\right)RTln\left(1-x/2\right)+\left(x/2\right)RTln\left(x/2\right)\\ & -168627.48\left(1-x/2\right)\left(x/2\right)\end{array}$$

(23)

To evaluate the estimating effect of the derived model, Equation (23) was used to calculate the ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ with the $x$ ranging from 0 to 2, and the calculated results are presented in Figure 3 using a solid line. By comparing the calculated values with that determined experimentally, it can be seen apparently that the derived model can estimate the standard Gibbs energy of formation of Fe_3–xV_xO₄ satisfactorily.

4. Conclusions

In view of the lack of primary thermodynamic data for the solid-solution compound Fe_3–xV_xO₄, an approach of determining the standard Gibbs energy of formation of Fe_3–xV_xO₄ was proposed, and its applicability was confirmed. The detailed procedure of the approach and the obtained results are as follows:

• The approach of determining ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ consists of two steps. The first is to measure the chemical potential of Fe (${\mu }_{Fe}$) in Fe_3–xV_xO₄ under fixed oxygen partial pressure by using liquid copper as the equilibrating phase, thereby obtaining the composition-dependent relation of ${\mu }_{Fe}$. The second is to calculate the ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ by utilizing the Gibbs–Duhem relation.
• The standard Gibbs energies of the formation of a variety of Fe_3–xV_xO₄ compounds were determined through this approach, and its reliability was verified by comparing the determined ${\Delta }_f{G}_{F{e}_{3-x}{V}_x{O}_4}^{\theta }$ with the reported value of FeV₂O₄.
• In the oxygen partial pressure range from 2.39 × 10⁻¹² to 3.83 × 10⁻¹¹ atm, the standard Gibbs energies of formation of Fe_3–xV_xO₄ is mainly decided by the Fe/V molar ratio and almost not related to the oxygen partial pressure.
• The standard Gibbs energies of the formation of Fe_3–xV_xO₄ can be modeled satisfactorily using the regular solution model with Fe₃O₄ and FeV₂O₄ as the solution end members.