Experimental Determination of the Standard Gibbs Energy of Formation of Fe 3–x V x O 4 at 1473 K

: In the present study, an approach of determining the standard Gibbs energy of formation of Fe 3–x V x O 4 was proposed ﬁrstly, then the standard Gibbs energies of formation of a variety of Fe 3–x V x O 4 were determined experimentally, and ﬁnally, a calculating model of the standard Gibbs energy of formation of Fe 3–x V x O 4 was established. The detailed results are as follows: (1) the standard Gibbs energy of formation of Fe 3–x V x O 4 can be determined successfully by two steps; the ﬁrst is to measure the chemical potential of Fe in Fe 3–x V x O 4 under ﬁxed oxygen partial pressure, the second is to derive the chemical potential of V in Fe 3–x V x O 4 by Gibbs–Duhem relation; (2) the standard Gibbs energies of formation of Fe 3–x V x O 4 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–x V x O 4 can be calculated satisfactorily by the following model: ∆ f G θ Fe 3 − x V x O 4 ( J / mol ) = ( 1 − x /2 ) ∆ f G θ Fe 3 O 4 + ( x /2 ) ∆ f G θ FeV 2 O 4 + ( 1 − x /2 ) RTln ( 1 − x /2 ) + ( x /2 ) RTln ( x /2 ) − 168627.48 ( 1 − x /2 )( x /2 ) .

Among the reported iron vanadate oxides, the oxides Fe 2 VO 4 and FeV 2 O 4 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].Fe 2 VO 4 has the inverse spinel structure of Fe 3+ [Fe 2+ V 3+ ]O 4 , like magnetite Fe 3+ [Fe 2+ Fe 3+ ]O 4 .However, FeV 2 O 4 changes to the normal spinel structure of Fe 2+ [V 3+ ] 2 O 4 with divalent iron (Fe 2+ ) occupying tetrahedral sites and trivalent vanadium (V 3+ ) occupying octahedral sites due to the strong affinity of V 3+ ions for octahedral sites.
In addition to the stoichiometric compounds Fe 2 VO 4 and FeV 2 O 4 , a complete solid solution Fe 3-x V x O 4 can be formed by Fe 3+ [Fe 2+ Fe 3+ ]O 4 and Fe 2+ [V 3+ ] 2 O 4 .The cation distribution in the spinel solid solution Fe 3-x V x O 4 has been determined by Mössbauer spectroscopy [13].For 0 < x < 1.0, the solid solution remains an inverse spinel structure as Fe 3+ [Fe 2+ Fe 3+  1-x V 3+ x ]O 4 .For 1.64 < x < 2.0, the solid solution remains a normal spinel structure as Fe 2+ [Fe 3+  1-x/2 V 3+ x/2 ] 2 O 4 .The transition from normal to inverse is observed for 1.0 < x < 1.64.Due to the similar structure to magnetite Fe 3 O 4 , great consideration has been given to the magnetic properties of Fe 3-x V x O 4 .Wakihara et al. [14] synthesized the spinel solid solution with stoichiometric composition at 1500 K under controlled CO 2 -H 2 atmospheres, measured the Curie temperature and the magnetization behaviors of a series of solidsolution compounds, and proposed a magnetic structure model.Jin et al. [15] fabricated epitaxial Fe 3-x V x O 4 (0 ≤ x ≤ 0.6) films by using reactive co-sputtering from pure Fe and V in a gas mixture of Ar and O 2 , and the magnetic and magnetotransport properties of the film spinels were studied at room temperature.Pool et al. [16] synthesized nanoparticles of Fe 3-x V x O 4 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-x V x O 4 have also been reported.Kim et al. [17] prepared the Fe 3-x V x O 4 thin film on Si (100) substrates by using a sol-gel method.The valence and occupying sites of V ions in the Fe 3-x V x O 4 (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-x V x O 4 (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-x V x O 4 , a number of works relating to the phase equilibria of Fe 3-x V x O 4 have been carried out.Coetsee et al. [19] determined the existing phases with the change of Fe/V molar ratio in the V 2 O 3 -FeO system from 1673 K to 1873 K by using CO/CO 2 = 3 gas mixture to control the oxygen partial pressure.Wakihara et al. [20] determined the phase equilibria in FeO-Fe 2 O 3 -V 2 O 3 system at 1500 K by varying the oxygen partial pressure and proposed the phase boundaries of Fe 3-x V x O 4 .In our previous publication [21], the phase equilibria of the FeO-V 2 O 3 system at 1473 K under various oxygen partial pressures were determined, and a phase diagram of lg p O 2 /atm 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-x V x O 4 .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-x V x O 4 .In addition, the modeling works of some more complex systems, such as the Fe-Ti-V-O system [23] and CaO-FeO-Fe 2 O 3 -MgO-SiO 2containing V 2 O 3 and V 2 O 5 system [24], were reported.
However, regarding the elementary thermodynamic data, i.e., the standard Gibbs energy of Fe 3-x V x O 4 , only the data of FeV 2 O 4 was reported.The first Gibbs energy measurement was performed by Chipman and Dastur [25] by the equilibration of FeV 2 O 4 with liquid Fe-V solution under a controlled H 2 -H 2 O gas atmosphere at 1873 K. Wakihara et al. [20] determined the standard Gibbs energy of the reaction of Fe + V 2 O 3 + 0.5O 2 = FeV 2 O 4 and 0.05Fe + V 2 O 3 + Fe 0.95 O = FeV 2 O 4 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 FeV 2 O 4 can be retrieved.Therefore, the present study aimed at providing the fundamental thermodynamic data of Fe 3-x V x O 4 and understanding the effect of V substituting on the thermodynamic property of the spinel phase; the standard Gibbs energy of formation of Fe 3-x V x O 4 was determined by combining theoretical derivation with experimental measurements.The molar Gibbs energy of Fe 3-x V x O 4 can be calculated according to Equation (1).
where µ Fe , µ V , and µ O 2 are the chemical potentials of Fe, V, and O 2 , respectively.Furthermore, according to the Gibbs-Duhem relation [27], the chemical potentials of Fe, V, and O 2 can be connected by Equation ( 2). ( If the oxygen partial pressure is fixed unchangeably, the dµ O 2 will be zero, and then Equation ( 2) can be simplified as Equation (3).
Then, Equation ( 4) can be obtained by dividing both sides of Equation (3) by x.
where 3−x  x represent the molar ratio of Fe to V in Fe 3-x V x O 4 .If it is replaced by y, Equation (4) becomes Equation (5).
By integrating Equation (5) from y s to y, the following relations can be obtained.
where y s refers to the molar ratio of Fe to V in Fe 3-x V x O 4 when it is saturated or equilibrated with V 2 O 3 , i.e., the minimum molar ratio of Fe to V in Fe 3-x V x O 4 at the fixed oxygen partial pressure.Correspondingly, µ V,y s refers to the chemical potential of V in the Fe 3-x V x O 4 that is saturated or equilibrated with V 2 O 3 at the fixed oxygen partial pressure.Next, using the method of integration by parts, Equation ( 6) can be changed to Equation (7).
Equation ( 8) tells us if we can know the functional relationship between µ Fe and y, as well as the value of y s and µ V, y s , then the chemical potential of V in Fe 3-x V x O 4 can be calculated through Equation (8).
Regarding the parameters y s and µ 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 t O-V 2 O 3 system at 1473 K under various oxygen partial pressures, which we have reported in a previous publication [21].For µ V, y s , it can be determined by the formation reaction of V 2 O 3 shown as Equation ( 9), because the chemical potentials of V in all the equilibrated phases are equal.
According to Equation ( 9), the µ V, y s can be determined by Equation (10). where O 2 is the chemical potential of O 2 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 µ Fe on y can be obtained, the µ V will be calculated through Equation (8), and the G Fe 3−x V x O 4 (molar Gibbs energy of Fe 3-x V x O 4 ) can be calculated by inserting µ V and µ Fe into Equation (1).Therefore, to measure the µ Fe in Fe 3-x V x O 4 is a prerequisite for the determination of the molar Gibbs energy of Fe 3-x V x O 4 .

Experimental Procedure for Measuring µ
The experiments to measure the values of the chemical potential of Fe in the solid solution phase Fe 3-x V x O 4 consisted of three steps.The first was to prepare the pure Fe 3-x V x O 4 compounds with different Fe/V molar ratios at fixed oxygen partial pressure; the second was to equilibrate the prepared Fe 3-x V x O 4 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-x V x O 4 compounds with different Fe/V molar ratios were prepared through the conventional solid-state reaction method under CO-CO 2 gas mixture atmosphere at 1473 K, where CO-CO 2 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, Fe 2 O 3 , and V 2 O 3 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.After the pure Fe 3-x V x O 4 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-x V x O 4 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-x V x O 4 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-x V x O 4 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-x V x O 4 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-x V x O 4 sample can be deduced by using the following method.As the Fe 3-x V x O 4 sample was equilibrated with the liquid copper chemically, the chemical potential of Fe in the Fe 3-x V x O 4 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).
where µ θ Fe is the chemical potential of pure iron, γ Fe(Cu) is the activity coefficient of Fe in liquid copper, and x Fe(Cu) 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 γ Fe(Cu) can be replaced by the Henry constant of Fe in liquid copper γ 0 Fe(Cu) , 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).
After µ Fe in the Fe 3-x V x O 4 sample with different Fe/V molar ratio was determined, the dependence of µ Fe on y can be deduced.When the relationship between µ Fe and y was obtained, the molar Gibbs energy of Fe 3-x V x O 4 can then be obtained by the approach presented in the previous section.

Chemical Potential of Fe in the Fe 3-x V x O 4 under Different Oxygen Partial Pressures
In our previous publication [21], the phase diagram of the "FeO"-V 2 O 3 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-x V x O 4 is summarized in Table 2.Where y L and y s correspond to the maximal and minimum Fe/V molar ratio of Fe 3-x V x O 4 single-phase area.When the Fe/V molar ratio in the system is larger than y L , Fe 3-x V x O 4 will coexist with the FeO phase.And when the Fe/V in the system is less than y s , Fe 3-x V x O 4 will coexist with the V 2 O 3 phase.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-x V x O 4 .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-x V x O 4 are present.For the sake of clarity, the Fe/V molar ratio of the equilibrating Fe 3-x V x O 4 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-x V x O 4 was calculated according to Equation (12), and the calculated values are shown in the form of µ Fe − µ θ Fe in the last column of Table 3.

From the calculated results of µ Fe − µ θ
Fe shown in the last column of Table 3, it can be found that both the Fe/V molar ratio of Fe 3-x V x O 4 and the equilibrating oxygen partial pressure have a significant effect on the chemical potential of Fe in the Fe 3-x V x O 4 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 µ Fe − µ θ Fe for Fe 3-x V x O 4 (Fe/V = 1.6) decreased from −23,861.3 J/mol to −28,362.0J/mol and then to −37,547.4J/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 µ Fe and y separately at each experimental oxygen partial pressure.It was assumed that the relations between µ Fe and y could be expressed by the relation shown in Equation (13).
To determine the model parameters a and b, exp µ Fe − µ θ Fe /RT was plotted against y, and the results are shown in Figure 1.
In view of the above-indicated situation, it is an expedient approach to deduce the functional relations between  and y separately at each experimental oxygen partial pressure.It was assumed that the relations between  and y could be expressed by the relation shown in Equation (13).
To determine the model parameters  and  , exp[( −  )/RT] was plotted against y, and the results are shown in Figure 1.From Figure 1, it can be seen that satisfactory linear relationships exist between exp[( −  )/RT] and y.This implies that the assumed functional relation by Equation From Figure 1, it can be seen that satisfactory linear relationships exist between exp µ Fe − µ θ Fe /RT 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 µ Fe and y at different oxygen partial pressures are summarized and presented in Table 4.After obtaining the functional relationships between the µ Fe and y, the chemical potential of V in Fe 3-x V x O 4 under different oxygen partial pressures were derived.The detailed procedure is as follows.Firstly, the functional relationship between µ Fe and y, as shown in Equation ( 13), was inserted into Equation ( 8), yielding Equation (14).
Equation ( 16) was rearranged and integrated again, and Equation ( 17) was obtained.
Then, inserting Equation (10) into Equation ( 17) yielded Equation ( 18) where ∆ f G θ V 2 O 3 is the standard Gibbs energy of formation of V 2 O 3 , and its reported value at 1473 K is −864,690.7J/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 µ V − µ θ V for Fe 3-x V x O 4 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 µ V − µ θ
V as well as the measured µ Fe − µ θ Fe in Fe 3-x V x O 4 compounds.

Equilibrating Experimental Conditions Results
No.

Standard Gibbs Energy of Formation of Fe 3-x V x O 4 at 1473 K
According to the definition of standard Gibbs energy of formation, the standard Gibbs energy of Fe 3-x V x O 4 can be determined by the following relation.
where µ θ Fe 3−x V x O 4 , the chemical potential of pure Fe 3-x V x O 4 , is equal to the molar Gibbs energy of pure Fe 3-x V x O 4 , so Equation (1) can be inserted into Equation (19) to replace the µ θ Fe 3−x V x O 4 .Then, a relation shown as Equation ( 20) can be obtained.
Furthermore, according to the relationship between µ O 2 and oxygen partial pressure, Equation ( 20) can be changed to Equation (21).
Now, by inserting the values of µ Fe − µ θ Fe and µ V − µ θ V presented in Table 4 into Equation (21), the standard Gibbs energy of formation of Fe 3-x V x O 4 with different Fe/V ratios under experimental oxygen partial pressures was calculated, and the calculated results are listed in Table 6.To further understand the influence of Fe/V ratio and oxygen partial pressure on the standard Gibbs energy of formation of Fe 3-x V x O 4 , the calculated values presented in Table 5 were demonstrated in Figure 2 by plotting the ∆ f G θ Fe 3−x V x O 4 against the Fe/V molar ratio.From Figure 2, it can be seen apparently that the data of ∆ f G θ Fe 3−x V x O 4 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-x V x O 4 is mainly influenced by the Fe/V molar ratio, and the effect of oxygen partial pressure is almost negligible.To further understand the influence of Fe/V ratio and oxygen partial pressure on the standard Gibbs energy of formation of Fe3-xVxO4, the calculated values presented in Table 5 were demonstrated in Figure 2 by plotting the ∆  against the Fe/V molar ratio.From Figure 2, it can be seen apparently that the data of ∆  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 Fe3-xVxO4 is mainly influenced by the Fe/V molar ratio, and the effect of oxygen partial pressure is almost negligible.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 ∆   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 Fe3-xVxO4 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 Fe3-xVxO4 can be considered The changing trend of ∆ f G θ Fe 3−x V x O 4 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 FeV 2 O 4 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 ∆ f G θ FeV 2 O 4 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-x V x O 4 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 In Figure 3, the standard Gibbs energies of the formation of FeV 2 O 4 and Fe 3 O 4 were plotted together with that of Fe 3-x V x O 4 against x/2.Where x/2 can be regarded as the molar fraction of the end member FeV 2 O 4 .The adopted data of the standard Gibbs energies of the formation of Fe 3 O 4 were -648765.3J/mol [26].
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 Fe3-xVxO4 against /2.Where /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 -648765.3J/mol [26].
From Figure 3, it can be seen that the present determined standard Gibbs energy of formation of Fe3-xVxO4 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 Fe3-xVxO4.According to the regular solution model, the Gibbs energy of a solution composed of A and B can be expressed by Equation (22).
To evaluate the estimating effect of the derived model, Equation ( 23) was used to calculate the ∆  with the  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 Fe3-xVxO4 satisfactorily.

Conclusions
In view of the lack of primary thermodynamic data for the solid-solution compound Fe3-xVxO4, an approach of determining the standard Gibbs energy of formation of Fe3-xVxO4 was proposed, and its applicability was confirmed.The detailed procedure of the approach and the obtained results are as follows: From Figure 3, it can be seen that the present determined standard Gibbs energy of formation of Fe 3-x V x O 4 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-x V x O 4 .
According to the regular solution model, the Gibbs energy of a solution composed of A and B can be expressed by Equation (22).
where G θ A and G θ B are Gibbs energy of pure components A and B, x A and x B are molar fractions of A and B, and Ω is the model parameter obtained from experimental data.In the present study, by using Fe 3 O 4 to replace A, FeV 2 O 4 to replace B, and fitting the model to the determined values of ∆ f G θ Fe 3−x V x O 4 , the following expression of calculating the standard Gibbs energy of formation of Fe 3-x V x O 4 was obtained.
To evaluate the estimating effect of the derived model, Equation (23) was used to calculate the ∆ f G θ Fe 3−x V x O 4 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-x V x O 4 satisfactorily.

Conclusions
In view of the lack of primary thermodynamic data for the solid-solution compound Fe 3-x V x O 4 , an approach of determining the standard Gibbs energy of formation of

Figure 1 .
Figure 1.Plots of exp µ Fe − µ θ Fe /RT against Fe/V ratio of Fe 3-x V x O 4 and the linear fitting results.

Figure 2 .
Figure 2. The changing trend of ∆  with Fe/V ratio under different oxygen partial pressures.

Figure 2 .
Figure 2.The changing trend of ∆ f G θ Fe 3−x V x O 4 with Fe/V ratio under different oxygen partial pressures.
the formation of FeV 2 O 4 and Fe 3 O 4 were considered because Fe 3-x V x O 4 can be considered as a complete solid solution formed with FeV 2 O 4 and Fe 3 O 4 FeV 2 O 4 and Fe 3 O 4 as the end members.

Figure 3 .
Figure 3. Comparing the present determined ∆  with the model calculating results.

Figure 3 .
Figure 3. Comparing the present determined ∆ f G θ Fe 3−x V x O 4 with the model calculating results.

Table 1 .
Composition and preparing conditions of Fe 3-x V x O 4 compounds.

Table 2 .
Compositional limitations of Fe 3-x V x O 4 single-phase area.

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

Table 4 .
The functional relations between µ Fe and y at different oxygen partial pressure.

p O 2 /atm Functional Relations between µ Fe and y
y s + [y s RTln(a + by s ) − yRTln(a + by)] + RT

Table 5 .
Cont. is the molar ratio of Fe to V in Fe 3-x V x O 4 . y

Table 6 .
The standard Gibbs energy of formation of Fe 3-x V x O 4 .

Table 6 .
Cont. is the molar ratio of Fe to V in Fe 3-x V x O 4 . y is the molar ratio of Fe to V in Fe3-xVxO4.