Determination of Gibbs Energies of Formation of Cr₃B₄, CrB₂, and CrB₄ by Electromotive Force Measurement Using Solid Electrolyte

Abstract

The standard Gibbs energies of formation, Δ_fG°, of intermediate compounds of Cr₃B₄, CrB₂, and CrB₄ in the boron-rich side of the Cr–B binary system were determined by measuring electromotive forces of the galvanic cells using ZrO₂–Y₂O₃ solid electrolyte. The phase equilibria for the Cr–B binary system were clarified to construct the cells using the solid electrolyte. The measured electromotive forces of the cells were evaluated as linear temperature functions under the conditions that the transport numbers of oxide ion in solid electrolyte were regarded as 1.0. The Δ_fG° functions determined from the electromotive forces via the Nernst equation were as follows:

Δ_fG°(Cr₃B₄)/J (mol of compd)⁻¹ = −264540 − 26.697 T ± 970 (1256–1322 K),

Δ_fG°(CrB₂)/J (mol of compd)⁻¹ = −84572 − 32.442 T ± 790 (1253–1350 K),

Δ_fG°(CrB₄)/J (mol of compd)⁻¹ = −105120 − 57.921 T ± 2200 (1280–1352 K).

The present Δ_fG° values satisfied the phase equilibria in the Cr–B binary system. Using the Δ_fG° values determined in the present study, the composition-oxygen partial pressure diagram of the Cr–B–O ternary system was constructed under the conditions of 1300 K and a total pressure of 1 bar (100 kPa). It is useful to understand the oxidation path of the boron-rich side of Cr–B binary alloys.

Fig. 4 Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂), and Δ_fG°(CrB₄) values expressed for the formulation of one mole of atoms at 1300 K compared with those in the references.^11,12)

1. Introduction

The multicomponent Ni-based alloys composed of Cr, Fe, Mo, W, Si, B, and so on, with high hardness, high melting points, high acid resistance, and high oxidation resistance have been used as corrosion-, wear-, and heat-resistant materials.^1–5) To evaluate the structure of the high-temperature oxidation films over the alloys from the view of chemical thermodynamics, it is important to understand the equilibrium oxygen partial pressures, $p_{\text{O}_{2}}$, as a function of the standard Gibbs energies of formation, Δ_fG°, of the phases that exist on the oxidation path.

In the present study, we focused on the Cr–B binary system, which is the fundamental system of the Ni-based corrosion and heat resistance alloys. The Δ_fG° values of the intermediate compounds in the Cr–B binary system were investigated to construct the composition-$p_{\text{O}_{2}}$ diagram of the oxidation path.

In the B-rich side of the Cr–B binary system, there are three stoichiometric compounds, namely, Cr₃B₄, CrB₂, and CrB₄.⁶⁾ The solubility of Cr in B is approximately 2.0 mol% at 2103 K, however, it was negligibly small at the experimental temperatures in the present study.^6,7) The thermodynamic information about the Cr–B binary compounds is limited. Brewer and Haraldsen⁸⁾ investigated the relative stabilities of the borides and nitrides in N₂ gas to estimate the standard enthalpies of formation, Δ_fH°, of CrB₂ at 298 K: Δ_fH°₂₉₈(CrB₂) > −126 kJ (mol of compd)⁻¹. Topor and Kleppa⁹⁾ used high-temperature drop solution calorimetry to determine the Δ_fH°₂₉₈(CrB₂) value by allowing the borides to react with Pt, Pd, or Ni to form liquid alloys: Δ_fH°₂₉₈(CrB₂) = −119.4 ± 3.4 kJ (mol of compd)⁻¹. Kubaschewski and Alcock,¹⁰⁾ as well as Barin,¹¹⁾ listed the Δ_fH°(CrB₂) and Δ_fH°(CrB) values: Δ_fH°(CrB₂) = −94.1 kJ (mol of compd)⁻¹; Δ_fH°(CrB) = −75.3 kJ (mol of compd)⁻¹, and the thermodynamic values of the heat capacities, the third law enthalpies, Δ_fH°, and Δ_fG° of CrB and CrB₂, respectively, in the thermodynamic tables, however, information regarding how and why the values were selected was not provided. Liao and Spear¹²⁾ calculated the phase diagram of the Cr–B binary system and optimized the Δ_fG° values of chromium borides. However, there is little experimental information for the Δ_fG° values optimized by Spear.¹²⁾

In our previous study, the Δ_fG° values of intermediate compounds in the Ni–B,¹³⁾ Mo–B,¹⁴⁾ and W–B¹⁵⁾ binary systems and in the Ni–B–O¹⁶⁾ ternary system have been determined by electromotive force measurements (EMF) using ZrO₂–Y₂O₃ solid electrolyte. From the principle of EMF using solid electrolyte, the phase equilibrium involving the target substance must be known. In the present study, the phase relationships for the Cr–B–O ternary system were clarified experimentally. The Δ_fG° values of Cr₃B₄, CrB₂, and CrB₄ in the B-rich side of the Cr–B binary system were determined by preparing EMF cells using ZrO₂–Y₂O₃ solid electrolyte on the basis of the clarified phase equilibria. Using the Δ_fG° values determined in the present study, the oxidation path was visualized as the composition-$p_{\text{O}_{2}}$ diagram.

2. Experimental Procedure

2.1 Preparation of cells

According to the clarified phase relationships for the Cr–B–O ternary system described later in section “3.1 Phase relationships for the Cr–B–O ternary system”, the following three-phase equilibria exist at around the experimental temperature of 1273 K in the vicinity of boron: CrB–Cr₃B₄–B₂O₃; Cr₃B₄–CrB₂–B₂O₃; and CrB₂–CrB₄–B₂O₃. To determine the Δ_fG° values of Cr₃B₄, CrB₂, and CrB₄, the following galvanic cells were constructed to measure the electromotive forces:   

\begin{equation*} \text{Cell(1): W $|$ CrB, Cr$_{3}$B$_{4}$, B$_{2}$O$_{3}{}|$ YSZ(8) $|$ O$_{2}$ in air $|$ Pt}, \end{equation*}

  

\begin{equation*} \text{Cell(2): W $|$ Cr$_{3}$B$_{4}$, CrB$_{2}$, B$_{2}$O$_{3}{}|$ YSZ(8) $|$ O$_{2}$ in air $|$ Pt}, \end{equation*}

  

\begin{equation*} \text{Cell(3): W $|$ CrB$_{2}$, CrB$_{4}$, B$_{2}$O$_{3}{}|$ YSZ(8) $|$ O$_{2}$ in air $|$ Pt}, \end{equation*}

where YSZ(8) is one end-closed tube composed of ZrO₂ solid electrolyte stabilized by 8 mol% Y₂O₃ (Nikkato Corp., Sakai-shi Osaka Japan). The two-phase alloys of CrB–Cr₃B₄, Cr₃B₄–CrB₂, and CrB₂–CrB₄ used for these cells were prepared by arc melting using a tungsten electrode and a water cooled copper hearth in an argon atmosphere for the compressed compacts of the mixtures of Cr (purity: 99.9 mass%, High Purity Chemical Institute Co., Ltd., Sakado-shi Saitama Japan) and B (purity: 99.0 mass%, High Purity Chemical Institute Co., Ltd., Sakado-shi Saitama Japan) powders in the compositions of 45.5 mol%Cr–54.4 mol%B, 37.0 mol%Cr–63.0 mol%B, and 25.0 mol%Cr–75.0 mol%B, respectively. The two-phase alloys were heat-treated at 1373 K for 1.2 Ms under vacuum and were crushed to fine powders in a mortar with a pestle made of tungsten carbide. These two-phase alloy powders were mixed with 2.5 mol%B₂O₃, and used as the electrode materials of the cells.

2.2 Electromotive force measurement

The details of the experimental procedure were almost the same as mentioned in our previous studies.^13–15) The electromotive force, which was derived from the difference in the equilibrium oxygen partial pressure between the cell material and the reference (O₂ gas in air), was measured every 5 minutes. When the change in electromotive force was less than 1 mV over 60 minutes at a given temperature, the average of the electromotive forces was adopted as the measured value. The measured values were corrected by subtracting the thermoelectric force generated between tungsten and platinum wires, which were used as lead wires of the cell, measured before experiments.

2.3 Calculation principle

The total cell reactions of Cell(1), Cell(2), and Cell(3) are shown as follows:   

\begin{equation} \text{Cell(1):}\ \text{(4/3) Cr$_{3}$B$_{4}$} + \text{O$_{\text{2 in air}}$} = \text{4 CrB} + \text{(2/3) B$_{2}$O$_{3}$}, \end{equation} (1)

  

\begin{equation} \text{Cell(2):}\ \text{2 CrB$_{2}$} + \text{O$_{\text{2 in air}}$} = \text{(2/3) Cr$_{3}$B$_{4}$} + \text{(2/3) B$_{2}$O$_{3}$}, \end{equation} (2)

  

\begin{equation} \text{Cell(3):}\ \text{(2/3) CrB$_{4}$} + \text{O$_{\text{2 in air}}$} = \text{(2/3) CrB$_{2}$} + \text{(2/3) B$_{2}$O$_{3}$}. \end{equation} (3)

When oxygen gas can be treated as an ideal gas and the solubility of the solid phase in the liquid B₂O₃ phase is negligibly small, the Gibbs energies of cell reactions, Δ_rG, are written as follows:   

\begin{align} \Delta_{\text{r}}G(1) & = 4\ \Delta_{\text{f}}G^{\circ}(\text{CrB}) + (2/3)\ \Delta_{\text{f}}G^{\circ} (\text{B$_{2}$O$_{3}$})\\ &\quad - (4/3)\ \Delta_{\text{f}}G^{\circ} (\text{Cr$_{3}$B$_{4}$})-RT\ln(p_{\text{O$_{2}$ in air}}/p^{\circ}), \end{align} (4)

  

\begin{align} \Delta_{\text{r}}G(2) & = (2/3)\ \Delta_{\text{f}}G^{\circ} (\text{Cr$_{3}$B$_{4}$}) + (2/3)\ \Delta_{\text{f}}G^{\circ} (\text{B$_{2}$O$_{3}$})\\ &\quad - 2\ \Delta_{\text{f}}G^{\circ} (\text{CrB$_{2}$})-RT\ln(p_{\text{O$_{2}$ in air}}/p^{\circ}), \end{align} (5)

  

\begin{align} \Delta_{\text{r}}G(3) & = (2/3)\ \Delta_{\text{f}}G^{\circ} (\text{CrB$_{2}$}) + (2/3)\ \Delta_{\text{f}}G^{\circ} (\text{B$_{2}$O$_{3}$})\\ &\quad - (2/3)\ \Delta_{\text{f}}G^{\circ} (\text{CrB$_{4}$})-RT\ln(p_{\text{O$_{2}$ in air}}/p^{\circ}) \end{align} (6)

where R, p°, and $p_{\text{O}_{2}\ \text{in}\ \text{air}}$ express the gas constant, the standard pressure (1 bar (10⁵ Pa)) and the oxygen partial pressure in air (0.21226 bar (21226 Pa)¹⁷⁾), respectively. Under the condition that the transport number of the oxide ion in the solid electrolyte is unity, Δ_rG = −4EF for eqs. (4) through (6), where E is the electromotive force and F is the Faraday constant, and the Δ_fG° values of Cr₃B₄, CrB₂, and CrB₄ are given by the following equations:   

\begin{align} \Delta_{\text{f}}G^{\circ} (\text{Cr$_{3}$B$_{4}$}) & = 3\ E(1)F + 3\ \Delta_{\text{f}}G^{\circ} (\text{CrB})\\ &\quad + (1/2)\ \Delta_{\text{f}}G^{\circ} (\text{B$_{2}$O$_{3}$})\\ &\quad-(3/4)\ RT\ln(p_{\text{O$_{2}$ in air}}/p^{\circ}), \end{align} (7)

  

\begin{align} \Delta_{\text{f}}G^{\circ} (\text{CrB$_{2}$}) & = 2\ E(2)F + (1/3)\ \Delta_{\text{f}}G^{\circ} (\text{Cr$_{3}$B$_{4}$})\\ &\quad + (1/3)\ \Delta_{\text{f}}G^{\circ} (\text{B$_{2}$O$_{3}$})\\ &\quad-(1/2)\ RT\ln(p_{\text{O$_{2}$ in air}}/p^{\circ}), \end{align} (8)

  

\begin{align} \Delta_{\text{f}}G^{\circ} (\text{CrB$_{4}$}) & = 6\ E(3)F + \Delta_{\text{f}}G^{\circ} (\text{CrB$_{2}$})\\ &\quad + \Delta_{\text{f}}G^{\circ} (\text{B$_{2}$O$_{3}$})\\ &\quad-(3/2)\ RT\ln(p_{\text{O$_{2}$ in air}}/p^{\circ}). \end{align} (9)

The values of Δ_fG°(B₂O₃)¹⁸⁾ and Δ_fG°(CrB)¹¹⁾ were taken from the thermochemical tables and are expressed as linear functions as follows:   

\begin{align} &\Delta_{\text{f}}G^{\circ} (\text{B$_{2}$O$_{3}$})/\text{J (mol of compd)$^{-1}$} \\ &\quad= - 1231400 + 213.17\ T \pm 580, \end{align} (10)

  

\begin{align} &\Delta_{\text{f}}G^{\circ} (\text{CrB})/\text{J (mol of compd)$^{-1}$} \\ &\quad= - 74953 - 6.5514\ T \pm 6. \end{align} (11)

The above errors were estimated as twice the standard deviations at the 0.95 level of confidence for the linear least-squares approximations.

First, the electromotive force of Cell(1), E(1), should be measured to determine Δ_fG°(Cr₃B₄) from eq. (7); then, using the determined Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂) can be determined from eq. (8) by measuring E(2) of Cell(2). In the same way, Δ_fG°(CrB₄) can be determined from E(3) of Cell(3).

2.4 Activities of Cr and B in the Cr–B binary system

When the elemental bcc Cr and rhombohedral B are adopted as the standard state, the chemical potential of Cr, Δμ_Cr, and the chemical potential of B, Δμ_B, are expressed as   

\begin{equation} \Delta\mu_{\text{Cr}} = \mu_{\text{Cr}} - \mu^{\circ}{}_{\text{Cr}} = RT\ln a_{\text{Cr}} \end{equation} (12)

  

\begin{equation} \Delta\mu_{\text{B}} = \mu_{\text{B}} - \mu^{\circ}{}_{\text{B}} = RT\ln a_{\text{B}} \end{equation} (13)

where a_Cr and a_B are the activities of Cr and B, respectively. The values of a_Cr and a_B are given by   

\begin{equation} a_{\text{Cr}} = \exp(\Delta\mu_{\text{Cr}}/RT) \end{equation} (14)

  

\begin{equation} a_{\text{B}} = \exp(\Delta\mu_{\text{B}}/RT) \end{equation} (15)

According to the equilibrium phase diagram of the Cr–B binary system,⁶⁾ all of the intermediate compounds are stoichiometric, and the following two-phase equilibria exist in the vicinity of B: CrB–Cr₃B₄, Cr₃B₄–CrB₂, CrB₂–CrB₄, and CrB₄–B. The values of Δμ_Cr and Δμ_B as well as a_Cr and a_B in the two-phase equilibria can be calculated on the basis of the two-phase equilibria. Note that Δμ_B is zero in the CrB₄–B two-phase region where a_B is unity.

3. Results and Discussions

3.1 Phase relationships for the Cr–B–O ternary system

The phase relationships for the Cr–B–O ternary system were investigated to construct the cell for electromotive force measurement. The starting materials were mixed in the specific composition and heat-treated at 1273 K for 605 ks. The starting materials, composition, and detected phases after heat treatment are summarized in Table 1. Starting materials 1 through 9 were heat-treated under vacuum, and starting materials 10 and 11 were heat-treated in ambient air. The phases of specimens after heat treatments were identified by an X-ray diffractometer (Rigaku, RINT-2200). No Cr–B–O ternary compounds except for CrBO₃ were detected. The phase relationships were clarified by considering the detected phases as the equilibrium phases.

Table 1 Summary of the starting materials, composition, and detected phases after heat treatment at 1273 K for 605 ks.

Figure 1 shows the phase relationships for the Cr–B–O ternary system at 1273 K determined in the present study. The compositions of starting materials shown in Table 1 are indicated as closed circle (●) symbols in Fig. 1. Using the materials in CrB–Cr₃B₄–B₂O₃, Cr₃B₄–CrB₂–B₂O₃, and CrB₂–CrB₄–B₂O₃ three-phase equilibria as cell materials, Cell(1), Cell(2), and Cell(3) were constructed to measure electromotive forces.

Fig. 1

Phase relationships for the Cr–B–O ternary system at 1273 K. The compositions of starting materials shown in Table 1 are indicated as closed circle (●) symbols.

3.2 Results of electromotive force measurements

Figure 2 shows the measured electromotive forces of Cell(1), Cell(2), and Cell(3) as a function of temperature. Linear relationships between the electromotive forces and temperature were observed. Thus, the electromotive forces of the cells, E(1), E(2), and E(3), were expressed as a function of temperature by using the linear least-squares method as follows:   

\begin{align} E(1)/\text{V} &= 1.9900-4.2595 \times 10^{-4}\ T \pm 3.2 \times 10^{-3} \\ & \quad\text{Temperature range: 1256–1322 K,} \end{align} (16)

  

\begin{align} E(2)/\text{V} &= 2.1458-5.2362 \times 10^{-4}\ T \pm 3.6 \times 10^{-3} \\ & \quad\text{Temperature range: 1253–1359 K,} \end{align} (17)

  

\begin{align} E(3)/\text{V} &= 2.0916-4.4563 \times 10^{-4}\ T \pm 3.4 \times 10^{-3} \\ & \quad\text{Temperature range: 1280–1352 K.} \end{align} (18)

The errors in the eqs. (16) through (18) were estimated as twice the standard deviations at the 0.95 level of confidence for the linear least-squares approximations. In the measurement of Cell(3), the electromotive forces measured above 1360 K were not used because the transport number of the oxide ion of Cell(3) was lower than 0.95. The details about the transport number of the oxide ion are described later in section “3.5 Oxygen Partial Pressure and Transport Number of Oxide Ion”.

Fig. 2

Electromotive forces of the following cells as a function of temperature: Cell(1): W | CrB, Cr₃B₄, B₂O₃ | YSZ(8) | O₂ in air | Pt, Cell(2): W | Cr₃B₄, CrB₂, B₂O₃ | YSZ(8) | O₂ in air | Pt, Cell(3): W | CrB₂, CrB₄, B₂O₃ | YSZ(8) | O₂ in air | Pt.

3.3 Δ_fG° values for the Cr–B binary system

Using the measured electromotive forces, Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂), and Δ_fG°(CrB₄) were determined from eqs. (7), (8), and (9).   

\begin{align} &\Delta_{\text{f}}G^{\circ} (\text{Cr$_{3}$B$_{4}$})/\text{J (mol of compd)$^{-1}$} \\ &\quad = -264540-26.697\ T \pm 970 \\ &\qquad \text{Temperature range: 1256–1322 K,} \end{align} (19)

  

\begin{align} &\Delta_{\text{f}}G^{\circ} (\text{CrB$_{2}$})/\text{J (mol of compd)$^{-1}$} \\ &\quad = -84572-32.442\ T \pm 790 \\ &\qquad \text{Temperature range: 1253–1359 K,} \end{align} (20)

  

\begin{align} &\Delta_{\text{f}}G^{\circ} (\text{CrB$_{4}$})/\text{J (mol of compd)$^{-1}$} \\ &\quad = -105120-57.921\ T \pm 2200 \\ &\qquad \text{Temperature range: 1280–1352 K.} \end{align} (21)

Figure 3 shows the Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂), and Δ_fG°(CrB₄) values as a function of temperature compared with those in the thermodynamic tables¹¹⁾ and the optimized values for phase diagram calculation.¹²⁾ The present Δ_fG°(Cr₃B₄) value was approximately 12 kJ mol⁻¹ smaller than that in Ref. 12). The Δ_fG°(CrB₂) value determined in the present study was approximately 3 kJ mol⁻¹ and 32 kJ mol⁻¹ smaller than that in Ref. 12) and that in Ref. 11), respectively. The present Δ_fG°(CrB₂) value exhibited close agreement with that in Ref. 12). However, the difference between the present Δ_fG°(CrB₄) value and that in Ref. 12) was approximately 56 kJ mol⁻¹, and this difference appeared to be too large. The methods of Δ_fG° determination are different, and the reasons for such discrepancies are unknown.

Fig. 3

Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂), and Δ_fG°(CrB₄) values expressed for the formulation of one mole of compound as a function of temperature compared with those in the references.^11,12)

The Δ_fG° value for one mole of atoms is convenient for comparison of the molar thermodynamic values in the Cr–B binary system. The Δ_fG° values for one mole of atoms of Cr₃B₄, CrB₂, and CrB₄ were calculated from eqs. (19) through (21) as follows:   

\begin{align} &\Delta_{\text{f}}G^{\circ}(\text{Cr$_{3}$B$_{4}$})/\text{J (mol of atoms)$^{-1}$} \\ &\quad = -37791 - 3.8139\ T \pm 140 \end{align} (22)

  

\begin{align} &\Delta_{\text{f}}G^{\circ}(\text{CrB$_{2}$})/\text{J (mol of atoms)$^{-1}$}\\ &\quad = -28191 - 10.814\ T \pm 260 \end{align} (23)

  

\begin{align} &\Delta_{\text{f}}G^{\circ}(\text{CrB$_{4}$})/\text{J (mol of atoms)$^{-1}$} \\ &\quad = -21024 - 11.584\ T \pm 440 \end{align} (24)

Figure 4 shows the Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂), and Δ_fG°(CrB₄) values for one mole of atoms at 1300 K compared with those in Refs. 11) and 12). The Δ_fG°(CrB) value¹¹⁾ for one mole of atoms is also shown in this figure. The Δ_fG°(Cr₃B₄) and Δ_fG°(CrB₂) values were close to those in Ref. 12), whereas the present Δ_fG°(CrB₄) value was smaller than that in Ref. 12).

Fig. 4

Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂), and Δ_fG°(CrB₄) values expressed for the formulation of one mole of atoms at 1300 K compared with those in the references.^11,12)

3.4 Activities of Cr and B

Figure 5 shows a_Cr and a_B in the Cr–B binary system at 1300 K calculated on the basis of the two-phase equilibria. The activities exhibit monotonic and stepwise change, that is, the Δ_fG° values determined in the present study satisfy the phase equilibria for the Cr–B binary system. In the range of 0.667 ≤ X_B ≤ 0.8, a_Cr decreases and a_B increases significantly, suggesting that there is strong attractive interaction between Cr and B.

Fig. 5

Activity of Cr, a_Cr, and B, a_B, in the Cr–B binary system at 1300 K.

3.5 Oxygen partial pressure and transport number of oxide ion

Figure 6 shows the equilibrium oxygen partial pressures, $p_{\text{O}_{2}}$, of Cell(1), Cell(2), and Cell(3) as a function of temperature, namely, the $p_{\text{O}_{2}}$ values for the three-phase equilibria of CrB–Cr₃B₄–B₂O₃, Cr₃B₄–CrB₂–B₂O₃, and CrB₂–CrB₄–B₂O₃. These values were calculated by using the Δ_fG° values determined in the present study. In the experimental temperature range, the $p_{\text{O}_{2}}$ values for the cells increased with temperature, and the following relationship was confirmed: $p_{\text{O}_{2}}$ for Cell(1) > $p_{\text{O}_{2}}$ for Cell(2) > $p_{\text{O}_{2}}$ for Cell(3). This relationship corresponds with the information of oxidation path of the Cr–B binary alloy understood from Fig. 1.

Fig. 6

Equilibrium oxygen partial pressures of Cell(1), Cell(2), and Cell(3) as a function of temperature. The oxygen partial pressure that gives the transport number of oxide ion of 0.95 for ZrO₂–Y₂O₃ solid electrolyte is indicated as a dashed line.

The electromotive force measurement to determine Δ_fG° must be conducted under the condition that the transport number of oxide ion, t_ion, of solid oxide electrolyte can be regarded as unity. The t_ion value is given by¹⁹⁾   

\begin{equation} t_{\text{ion}} = 1/[1 + (p\theta/p_{\text{O${_{2}}$}})^{1/4}], \end{equation} (25)

where pθ is the partial pressure where t_ion becomes 0.5, owing to the formation of excess electrons, and is given by the following equation for YSZ(8).²⁰⁾   

\begin{equation} \log(p\theta/\text{Pa}) - \log(101325) = 18.08 - 6.24 \times 10^{4}\ T^{-1} \end{equation} (26)

The oxygen partial pressure that results in the transport number of oxide ion of 0.95 for the ZrO₂–Y₂O₃ solid electrolyte is also indicated as a dashed line in Fig. 6, which is calculated from eqs. (25) and (26). In the conditions above the dashed line, the t_ion values of the cells are not unity in a precise sense but can be regarded as 1.0. Almost all of the electromotive force measurements in the present study were conducted under the conditions above the dashed line. However, the t_ion was less than 0.95 for the measurements of Cell(3) above 1360 K. As described in our previous paper, the electromotive force measurements with t_ion less than 0.95 were unstable due to electronic conduction.¹⁴⁾ This is the reason why the measurement results for Cell(3) above 1360 K, which appeared to exhibit a linear relationship with temperature, were not used for the E(3) estimation in eq. (18). Therefore, the electromotive force values adopted in the present study were all measured under the condition that the t_ion values were regarded as 1.0.

3.6 Composition-oxygen partial pressure diagram

Figure 7 shows the composition-oxygen partial pressure diagram of the Cr–B–O ternary system at 1300 K under the total pressure of 1 bar (100 kPa). This figure was constructed by considering the relationships between the oxidation paths of Cr–B binary alloys as understood from Fig. 1 and the $p_{\text{O}_{2}}$ values of three-phase equilibria on the oxidation path.

Fig. 7

Composition-oxygen partial pressure diagram of the Cr–B–O ternary system at 1300 K. The horizontal lines show the following three-phase equilibria: (1) CrB₄–B–B₂O₃; (2) CrB₂–CrB₄–B₂O₃; (3) Cr₃B₄–CrB₂–B₂O₃; (4) CrB–Cr₃B₄–B₂O₃.

The $p_{\text{O}_{2}}$ values of (1) CrB₄–B–B₂O₃, (2) CrB₂–CrB₄–B₂O₃, (3) Cr₃B₄–CrB₂–B₂O₃, and (4) CrB–Cr₃B₄–B₂O₃ three-phase equilibria were calculated by using the Δ_fG° values of Cr₃B₄, CrB₂, and CrB₄ determined in the present study and the literature values of CrB and B₂O₃.^11,18) The $p_{\text{O}_{2}}$ values of these three-phase equilibria are shown as the horizontal lines in Fig. 7. This diagram is useful for understanding the relationship between the composition of Cr–B binary alloy and the oxidation product at any specific oxygen partial pressure.

4. Conclusion

The Δ_fG° values of Cr₃B₄, CrB₂, and CrB₄ in the B-rich side of the Cr–B binary system were determined by preparing cells for electromotive force measurement using ZrO₂–Y₂O₃ solid electrolyte on the basis of the phase relationships for the Cr–B–O ternary system clarified in the present study. The obtained Δ_fG°(Cr₃B₄), Δ_fG°(CrB₂), and Δ_fG°(CrB₄) values satisfied the phase equilibria in the Cr–B binary system. Using the Δ_fG° values determined in the present study, the composition-oxygen partial pressure diagram of the oxidation path was constructed.

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