Determination of Equilibrium Constant of Calcium Oxidation by Slag-Metal Equilibrium Distribution Method

Abstract

The equilibrium constant of calcium oxidation is important to evaluate the ability of deoxidation of the dissolved oxygen in metals and the reduction of metal-oxide. Therefore, in this study, the equilibrium constant for calcium oxidation has been determined by a slag-metal equilibrium distribution method by using molten iron and/or titanium-based alloys. In general, this method can be used to determine the equilibrium constant for solved elements, e.g., as expressed by “CaO_(s) = Ca_{in metal} + O_{in metal}”. When the oxygen partial pressure and the activity of calcium in the surroundings are established, the equilibrium constant of “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)” is determined to be log[K] = −17.50 for titanium-based alloys at 1473 K and log[K] = −12.91 for molten iron at 1873 K. These values are essentially medians obtained using an experimental measurement of the oxygen partial pressure and are based on the definition of the activity of calcium in the slag as given in literature. The temperature dependence of these results has been determined as follows:

log[K] = 4.814 − 32960 (1/T) (1473–1873 K)

or

ΔG° = 631 − 0.0922T [kJ/mol] (1473–1873 K)

Fig. 7 Temperature dependence of the equilibrium constant, “CaO = Ca + 1/2O₂”, combination of Fig. 4 (literature and present study), Fig. 5 (the titanium and titanium-based alloys) and Fig. 6 (molten iron).

1. Introduction

In theory, metallic calcium presents high oxygen affinity; it is not only used as a strong reducer or deoxidizer for the manufacturing of iron,^1–5) but also for the reduction process of numerous elements, e.g., titanium,^6–12) zirconium,^13,14) vanadium,¹⁵⁾ tantalum,¹⁶⁾ thorium.¹⁷⁾ Indeed, it presents a higher oxygen affinity than that of iron. However, metallic calcium presents several problems that hinder its use as the reducer or deoxidizer of metals. The first of these problems is a high vapor pressure¹⁸⁾ leading to the spread into the atmosphere as calcium vapor. Moreover, the evaporated calcium becomes difficult to solve again into the metals. The second problem is the low solubility^19,20) for molten iron and titanium, which decreases the probability of contact between the deoxidizer and the solved oxygen. Consequently, the deoxidizer that reacts with solved oxygen in the metal decreases. The affinity of calcium for oxygen can be simply interpreted from the equilibrium constant (i.e., Gibbs free energy) of the reaction “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)”, which is summarized in the relevant thermodynamic literature.^21–26) The equilibrium constant of this reaction is widely required for the estimation of many metallurgical and/or electrochemical processes using the deoxidizer of calcium. However, it is also necessary to consider dissolution reactions such as “Ca_{in molten metal} = Ca_{(l, g)}” and “O_{in molten metal} = 1/2O_2(g)” in some metallurgical cases. In spite of the importance of the equilibrium constant of the reaction “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)” and the above-mentioned problems of calcium, the thermodynamic literature^21,22) still refers to the equilibrium constant that was experimentally determined by the combustion reaction of calcium²⁷⁾ over 60 years ago without any justification. In the present paper, I have investigated the equilibrium constant of the reaction “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)”, and its temperature dependence using experimental data of previous research and original data obtained in this study. I have also compared my results with the ones obtained in recent experimental investigations.

2. Theoretical Background

2.1 Thermodynamic relation

The general reactions of calcium oxidation or CaO deoxidation are expressed by activities of CaO and Ca, and an oxygen partial pressure as shown in the following eq. (1). The equilibrium constant, K, of the eq. (1) is expressed by eq. (1*).   

\begin{equation} \text{CaO$_{(\text{s})}$} = \text{Ca$_{(\text{l})}$}+\text{1/2O$_{2(\text{g})}$} \end{equation} (1)

  

\begin{equation} K_{(1)} = a_{\text{Ca}}\cdot P_{\text{O}_{2}}{}^{1/2}/a_{\text{CaO}} \end{equation} (1*)

For the slag-metal equilibrium distribution method, the reaction of calcium oxidation and the deoxidation of CaO incorporate the dissolved calcium and oxygen into a molten metal (eq. (2)). Equation (2) shows the equilibrium constant of this reaction.   

\begin{equation} \text{CaO$_{(\text{s})}$} = \underline{\text{Ca}}_{\text{in metal}} + \underline{\text{O}}_{\text{in metal}} \end{equation} (2)

  

\begin{equation} K_{(2)} = a_{\text{Ca}}[\text{mass%Ca$_{\text{in metal}}$}] \cdot a_{\text{O}}[\text{mass%O$_{\text{in metal}}$}]/a_{\text{CaO}} \end{equation} (2*)

Herein, activities of the solved calcium and oxygen are conveniently determined by the analytical concentration of these elements in the molten metal. Conversely, the activity of CaO is thermodynamically defined by the slag composition and the crucible that was employed. To determine the equilibrium constant of eq. (1) from the reaction of eq. (2), the relation of calcium and the dissolved calcium in the molten metal must be clear; this is expressed in eq. (3), and its corresponding equilibrium constant is expressed by eq. (3*). Moreover, a relation of oxygen gas in the atmosphere and dissolved oxygen in the molten metal is also elucidated by eq. (4), and the corresponding equilibrium constant is expressed by eq. (4*).   

\begin{equation} \underline{\text{Ca}} = \text{Ca$_{(\text{l})}$} \end{equation} (3)

  

\begin{equation} K_{(3)} = a_{\text{Ca}}/a_{\text{Ca}}[\text{mass%Ca$_{\text{in metal}}$}] \end{equation} (3*)

  

\begin{equation} \underline{\text{O}} = \text{1/2O$_{\text{2(g)}}$} \end{equation} (4)

  

\begin{equation} K_{(4)} = P_{\text{O}_{2}}{}^{1/2}/a_{\text{O}}[\text{mass%O$_{\text{in metal}}$}] \end{equation} (4*)

In the present study, the activity of calcium is defined from experimental conditions; this is achieved by adding calcium into the slag to attain a specific slag amount as stated in the literature. Although the partial pressure of oxygen pressure is also defined in the literature, in the present study, it is experimentally determined by an oxygen sensor to investigate the reliability of these defined values.

2.2 Measurement of oxygen partial pressure

The oxygen partial pressure in the surroundings of the specimens in the furnace was experimentally measured using an oxygen sensor composed of tubular CaO-stabilized ZrO₂ solid solution. A chromium powder obtained with the equilibrium oxygen partial pressure of “Cr/Cr₂O₃” was incorporated in the solid solution tube and was further enclosed by an alumina cement. To avoid any damage induced by heat in the oxygen sensor, it was kept at the top of the furnace and in the direction of the specimen; further, it continuously measured the electric voltage during these experiments. It is important to emphasize that the measured oxygen partial pressure at the furnace top corresponds to the actual oxygen partial pressure around the specimen. Although, temperature changes led to significant differences in the mean free path of non-reactive atoms that were predominantly filled by Argon gas, the effect of the different temperatures on the oxygen partial pressures was interpreted as negligible.

Moreover, for the measurement of the oxygen partial pressure, it is important that the application range of the oxygen sensor dominates the ionic conduction.^28,29) The ionic conducting oxygen partial pressure region of the CaO-stabilized ZrO₂ solid solution is higher than 10⁻³⁰ atm at 1273 K, and this value decreases to 10⁻⁴⁰ atm at 1073 K. In the literature, the equilibrated oxygen partial pressure of the reaction “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)” is determined as 10⁻³⁴ to 10⁻²⁴ at 1473 K to 1873 K.²¹⁾ Because the ionic conducting region of the oxygen sensor is 10⁻⁴⁰ atm at 1073 K, the equilibrated oxygen partial pressure region performed by the reaction, “CaO_(s) = Ca_(s) + 1/2O_2(g)”, is satisfied.

An additional issue is the transportation rate of oxygen iron of the CaO-stabilized ZrO₂ solid solution. The measurement region of the oxygen partial pressure is expected to be 10⁻²³, as indicated in the literature²¹⁾ for the reaction, “CaO_(s) = Ca_(g) + 1/2O_2(g)”, at 1873 K. In the present study, because the oxygen sensor is placed at the furnace top, the temperature is lower than that where the actual specimen is; this temperature is about 1100 K. At this temperature, the transportation rate, i.e., the transportation rate of oxygen iron becomes almost “1”.³⁰⁾ A previous study, in which the oxygen pressure is measured to be 10⁻³⁶ for C21, and the measuring temperature is 843 K,³¹⁾ seems to corroborate the ion transportation number of “1”. This means that the measured oxygen partial pressure is not required to correct the transportation rate in the region of study being considered at present.

Another issue is that the following cell reaction is expected to occur on the surface of the oxygen sensor; this is because the furnace is filled up with a calcium vapor at the experiment.   

\begin{align*} &\text{O$_{\text{2$\,$equilibrated by}}\,$CaO/Ca$_{\text{ on surface of oxygen sensor}}$}\\ &\quad \mid\text{O$^{2-}{}_{\text{electrolyte}}$}\mid\text{O$_{\text{2 equilibrated by}}\,$Cr$_{2}$O$_{3}$/Cr}. \end{align*}

Here, the oxygen partial pressure of CaO/Ca equilibration, shown on the left side, is about 10⁻⁵⁵ atm at 1000 K, and that of Cr₂O₃/Cr is about 10⁻³⁰ atm at 1000 K.²²⁾ Conversely, the equilibrated oxygen partial pressure of CaO/Ca for the sample is 10⁻²³ atm at 1873 K for the experiment using molten iron. In spite of the higher oxygen partial pressure around the sample in the furnace, it is difficult to consider that the oxygen partial pressure is equilibrated to the lower side of the same furnace. Therefore, it is reasonable to conclude that the oxygen partial pressure is equilibrated to the higher side. Moreover, the vapor pressure of calcium is higher than that of CaO³²⁾; this is because its equilibrated oxygen partial pressure at 1873 K is considerably high at 1000 K; however, the equilibrated calcium at 1873 K is immediately oxidized at 1000 K. It is, therefore, reasonable to conclude that the vaporized element is predominantly CaO. Therefore, the previous cell reaction is expressed as the following reaction,   

\begin{align*} &\text{O$_{\text{2$\,$of atmosphere in the furnace}}$}\mid\text{O$^{2-}{}_{\text{electrolyte}}$}\\ &\quad \mid\text{O$^{2}{}_{\text{equilibrated by}}\,$Cr$_{2}$O$_{3}$/Cr}. \end{align*}

In addition, although the condensation of CaO on the surface of the oxygen sensor by the vaporized and deposited Ca or CaO may be affected by the measured oxygen partial pressure, which in turn is affected by the electric conductivity and not by the electromotive force (EMF).³⁰⁾ Moreover, a thermal expansion of the CaO stabilized ZrO₂ solid solution is slightly smaller than other typical stabilized zirconia solid solutions, without drastic mass changes causing a phase transformation.³³⁾ Indeed, because the workability of the CaO stabilized ZrO₂ solid solution as the oxygen sensor is high, it was used in our previous research.^34,35) In the above-mentioned studies, the inside of the furnace was actually coated by calcium oxide (seen as white powder); this is because the coated white powder was easily wiped. However, the effect of the calcium vapor for the measurement of the oxygen partial pressure is evaluated to be considerably small in the case of this study.

3. Experimental Procedure

The equilibrium constants of the reaction, “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)”, were determined for the two temperature regions of 1473 K and 1873 K. The experiment at 1473 K was performed using a molten titanium–copper based alloy with CaO–MgO–CaCl₂ melts in CaO crucible heated by an electric resistant vertical furnace, which has an alumina tube protected by a carbon tube.^31,35) The experiment at 1873 K was performed using a molten iron with CaO–Al₂O₃–ZrO₂ melts in CaO crucible.³⁶⁾ For both experiments, the oxygen partial pressure in the furnace was experimentally measured using an oxygen sensor. Details of the experiments, analytical methods and its inspections have been thoroughly described in the literature.

4. Results

Figure 1 shows the measured oxygen partial pressure and the temperature of the oxygen sensor against the experimental time for one of the samples using molten iron (sample C10). The time shown in Fig. 1 starts from the heating of the sample (the sample was hanged from the top of the furnace at the start of the experiment, and was slowly descended to the experimental position). Although the measured partial oxygen pressure and temperature have not reached a stationary state at an isothermal heating and a calcium addition, because the behaviors seem to be stabilized to the stationary state before quenching, the stabilized oxygen partial pressure just before the quenching is used for the thermodynamic investigation. This problem requires a prolonged stabilizing time for the stationary state and is expected to be caused by a water-cooling rod settled nearby the oxygen sensor and used to hang the sample inside of the furnace. However, because the oxygen partial pressure and temperature seem to reach the stationary state, this problem should not influence the following thermodynamic investigation. Analytical concentrations of the solved elements for the experiments using the molten iron and titanium alloy are listed in our previous papers,^31,36) and are also summarized in Table 1 with the measured oxygen partial pressure and the experimental conditions.

Fig. 1

Measured oxygen partial pressure and temperature of the oxygen sensor for the reaction time (Sample C10, which is performed for equilibrium distribution method using molten iron and CaO–Al₂O₃–ZrO₂ flux).

Table 1 Experimental conditions and estimated equilibrium constant, “CaO = Ca + 1/2O₂”, for titanium and titanium-based alloys and molten iron.

5. Discussion

5.1 Equilibrium constant of “O = 1/2O_2(g)” and “Ca_(g) = Ca” for original measurements

Figure 2 shows a concentration dependence of the solved oxygen in molten iron for an equilibrium constant of “O = 1/2O_2(g)”, which is determined by analytically determined oxygen contents in the molten iron and pressured oxygen partial pressure. Here, the equilibrium constant of the reaction is determined to be −9.95 (= log K at 1873 K), as an intercept of the fitting line by a least squares method, and is smaller than that presented by Han et al. (log K = −0.38 at 1873 K)³⁷⁾ or G.K. Sigworth and J.F. Elliot (log K = −3.40 at 1873 K).³⁸⁾

Fig. 2

Relationship between the experimentally determined equilibrium constant for the reaction, “O = 1/2O₂”, at 1873 K for analytically determined oxygen concentration.

On the other hand, Fig. 3 shows a concentration dependence of the solved oxygen (a) and calcium (b) in the molten iron for an equilibrium constant of “Ca_(g) = Ca” which is determined by the activity of calcium solved or mixed with molten salt and the analytically measured calcium content in the molten iron. For the determination of equilibrium constants, the activities of the solved elements are used analytically determined apparent values and are not corrected by interaction parameters, because the solved contents are quite small and its influences are also expected to be quite small. Although the activity of calcium in the molten slag (CaO–Al₂O₃–ZrO₂) must be thermodynamically determined, because the metallic calcium seems to be hardly solved in oxygen-based salt,^39,40) the activity of calcium, a_Ca, in the molten salt is defined to be “1”. Then, the equilibrium constant of the reaction is conveniently determined to be −3.4 (= log K), as the intercept of the fitting line for the oxygen concentration by the least squares method. This was done because the solubility of calcium is small for the molten iron, and it is difficult to estimate the equilibrium constant from the fitting line. The determined equilibrium constant was fitted to a thermodynamic recommended value (log K = −3.0 at 1873 K).³⁴⁾

Fig. 3

Comparison of the relationship between the experimentally determined equilibrium constant for the reaction, “Ca = Ca”, at 1873 K for analytically determined (a) oxygen and (b) calcium concentrations.

The equilibrium constants of the reactions, “O = 1/2O_2(g)” and “Ca_(l) = Ca”, for titanium-based alloy were determined to be −15.82 and −2.29 (= log K at 1473 K), respectively, in our previous study.³¹⁾

5.2 Equilibrium constant of “CaO = Ca + 1/2O₂” for original measurements

Figure 4 shows the equilibrium constant of “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)” experimentally investigated in the present study and those determined in previous studies,^21,23–26) and is also listed in Table 1. Here, the CaO activity, a_CaO, in the case of titanium-based alloy using CaO saturated composition flux melted in CaO crucible, is defined to be “1”.³¹⁾ For the case of molten iron, a_CaO varies from 0.025 to 0.03, and is determined from flux compositions, which are Al₂O₃ or ZrO₂ crucible saturated composition.³⁶⁾

Fig. 4

Temperature dependence of the equilibrium constant, “CaO = Ca + 1/2O₂”, and that found in the literature.^21,23–26)

From the equilibrium constants of “O = 1/2O_2(g)” and “Ca_{(l, g)} = Ca”, the equilibrium constant of “CaO_(s) = Ca_{(l, g)} + 1/2O_2(g)” is determined from that of the reaction, “CaO_(s) = Ca_{in metal} + O_{in metal}”, as listed in Table 1 and shown in Fig. 4 for the molten titanium based alloy (at 1473 K) and iron (at 1873 K), respectively. Within the temperature range of 1473–1873 K, although a phase change of calcium from liquid to gas can happen, for the thermodynamic investigation this is ignored. Indeed, although J.F. Elliott and M. Gleiser²³⁾ distinguish between the solid and liquid phases of the metallic calcium, the difference is not significant. The equilibrium constant and that of temperature dependence of the reaction determined in the present study are in good agreement with that of Barin²¹⁾ and J.F. Elliott and M. Gleiser,²³⁾ and also with that of Turkdogan²⁴⁾ at 1473 K.

5.3 Comparison of the equilibrium constant in recent researches

Table 1 summarizes experimental conditions and analytical data of typical metallurgic papers including original experimental data. T.H. Okabe et al.^41–43) perform the deoxidation of titanium using a titanium crucible in a closed metallic container with metallic calcium. From the experimental condition, the calcium activity, a_Ca, is defined to be “1” because pure (solid) calcium is still in the reaction field. Although the activity of CaO is not explained in the papers, because factors responsible for the decrease of the activity, such as the use of multi composition slag, are not found, the CaO activity, a_CaO, is estimated to be “1”. In these studies, although the equilibrated oxygen partial pressure is not reported, it can be estimated by a relation between the oxygen partial pressure and solved oxygen in titanium experimentally investigated by Ono et al.⁴⁴⁾ referring Barin’s typical literatures.²¹⁾ Since the equilibrium constant of “O = 1/2O_2(g)” and “CaO_(s) = Ca + O_{in metal}” are established, “CaO_{(s, l)} = Ca_{(l, g)} + 1/2O_2(g)” is also established as shown in Table 1.

F. Tsukihashi et al.⁴⁵⁾ and Kobayashi et al.⁴⁶⁾ also perform the deoxidation of titanium and titanium-based alloys by the slag/metal equilibrium distribution method at various temperatures. The metals are melted in CaO crucible, then the activity of CaO, a_CaO, is defined to be “1”. The activity of calcium, a_Ca, is difficult to be determined from information in the literature. However, the activity, a_Ca, of the metallic calcium for CaO–CaCl₂ melts was investigated by some researchers^39,40) and found to be “1” for a content of calcium of 1 to 4 mol% in the melts. Therefore, it is reasonable that the calcium activity, a_Ca, of most cases in these studies is expected to be “1”. Although the equilibrated oxygen partial pressure is also missing in the papers, that is also estimated using the relation reported by Ono et al.⁴⁴⁾ From the definition, the equilibrium constant of “Ca = Ca_{(l, g)}”, “O = 1/2O_2(g)”, “CaO_{(s, l)} = Ca_{in metal} + O_{in metal}” and “CaO_{(s, l)} = Ca_{(l, g)} + 1/2O_2(g)” are determined as shown in Table 1.

R.O. Suzuki et al.⁸⁾ performed the precipitation of titanium particles by an electrochemical method using CaCl₂ melts. The activity of CaO, a_CaO, is determined from a ratio of CaO/CaCl₂, and is reported by Morita et al.⁴⁷⁾ From a concept of the research, because CaO is locally and continuously generated by the reduction of TiO₂, the CaO activity, a_CaO, is defined to be “1”. From the metallic calcium, also temporarily and electrochemically precipitated from CaO solved in the melts, the calcium activity, a_Ca, is also defined to be “1”. For the equilibrium constant of “O = 1/2O_2(g)”, because the solved oxygen content in the titanium particles is reported in the paper, that is determined as shown in Table 1 by the relation reported by Ono et al.⁴⁴⁾

S. Kobayashi et al.,⁴⁸⁾ Q. Han et al.,³⁷⁾ T. Kimura and H. Suito,⁴⁹⁾ H. Fujiwara et al.,⁵⁰⁾ H. Ohta and H. Suito⁵¹⁾ and H. Itoh et al.⁵²⁾ performed the slag/metal equilibrium distribution method for a molten iron. K. Mineura et al.⁵³⁾ and H. Ohta and H. Suito⁵⁴⁾ used a stainless-steel (Fe–Ni–Cr alloy) for their studies. Among these studies, Q. Han et al.³⁷⁾ used a metallic crucible made of molybdenum, while other researchers used ceramic crucibles. For experimental conditions, the CaO activity, a_CaO, is defined to be selected values influenced by flux compositions referred from the literature.^51,55) For the studies by Q. Han et al., because the flux composition is not particular mentioned and is interpreted as applying no-correction, the CaO activity, a_CaO, is defined to be “1”. The partial pressure of calcium, P_Ca, is roughly interpreted as total pressure to be “1”, because the oxygen partial pressure is significantly low for the total pressure mostly filled up by calcium vapor generated from the sufficient deoxidizer into the samples. For the equilibrium constant of “Ca = Ca_(g)”, the recommended value by M. Hino and K. Ito³⁴⁾ is used, although the equilibrium constant determined in the present study also matches well with the recommended value, because a temperature dependence of the equilibrium constant is not experimentally cleared. For the equilibrium constant of “O = 1/2O_2(g)”, because the suitable value having temperature dependence has not been found, the determined value in the present study is used ignoring the temperature dependence. The determined equilibrium constants are shown in Table 1.

Figure 5 shows the temperature dependence of the equilibrium constants of “CaO_{(s, l)} = Ca_{(l, g)} + 1/2O_2(g)” for the titanium-based alloys, and Fig. 6 shows the same relation for the molten iron from the recent experimental studies previously mentioned. Figure 7 shows a combination of Fig. 4–6. The values for the equilibrium constant determined from the titanium-based alloys fluctuate around the typical thermodynamic values by Barin,²¹⁾ being mostly lower than the latter. Moreover, the equilibrium constant determined in the present study is slightly smaller than that of Barin²¹⁾ at the temperature range from 1473 K to 1873 K. The recent experimental values may be dispersing around an extrapolation line estimated from the presented values. Regarding the relations for the molten iron, although it is difficult to assess temperature dependence from the relation, the value of equilibrium constant at 1873 K (including at 1823 K) fluctuates around the typical thermodynamic values by Barin and Elliott^21,23) and the presented value. In fact, the temperature dependence of the equilibrium constant includes a phase change from solid to liquid calcium in the temperature range similar to what is described in Elliot.²³⁾ However, although the problem due to the temperature dependence of the present study is ignored, the effect of the phase change seems to be small as shown in the Fig. 7. The temperature dependence of the present value is only preliminary thermodynamic information. At this time, it is possible to say that, from the results presented here, there is positive information. For the accurate determination of the equilibrium constant of the reaction, including its temperature dependence, more extensive instrumentation and many more experiments are necessary. This includes the experimental determination for a shorter temperature span, with more temperature regions for complete clarification.

Fig. 5

Temperature dependence of the equilibrium constant, “CaO = Ca + 1/2O₂”, for the titanium and titanium-based alloys experimentally investigated in recently published studies.

Fig. 6

Temperature dependence of the equilibrium constant, “CaO = Ca + 1/2O₂”, for the molten iron experimentally investigated in recent papers.

Fig. 7

Temperature dependence of the equilibrium constant, “CaO = Ca + 1/2O₂”, combination of Fig. 4 (literature and present study), Fig. 5 (the titanium and titanium-based alloys) and Fig. 6 (molten iron).

6. Conclusions

In the present study, the temperature dependence of the equilibrium constant for the reaction, “CaO_{(s, l)} = Ca_{(l, g)} + 1/2O_2(g)”, was experimentally determined by the slag/metal equilibrium distribution method and the measurement of the oxygen partial pressure using CaO stabilized ZrO₂ solid solution. The determined equation is expressed as follows:   

\begin{equation*} \log[K] = 4.814-32960\ (1/T)\quad (1473\unicode{x2013}1873\,\text{K}) \end{equation*}

From the equation, the Gibbs free energy of the reaction is expressed as follows:   

\begin{equation*} \varDelta G^{\circ} = 631-0.0922T\ [\text{kJ/mol}]\quad (1473\unicode{x2013}1873\,\text{K}) \end{equation*}

The equilibrium constant and the Gibbs free energy determined in the present study agree well with the literature. Such as the Barin thermodynamic data, log[K] = 5.470 − 33172 (1/T), ΔG° = 635 − 0.1047T [kJ/mol] for 1473–1873 K, differs only by 14 to 19 kJ/mol in the Gibbs free energy.

REFERENCES

• 1)   Y.  Miyashita and  K.  Nishikawa: Tetsu-to-Hagané 57 (1971) 1969–1975.
• 2)   S.  Gustafsson and  P.-O.  Melberg: Scand. J. Metall. 9 (1980) 111–116.
• 3)   F.  Ishii,  M.  Hino and  H.  Itoh: Tetsu-to-Hagané 82 (1996) 465–470.
• 4)   T.  Ototani and  Y.  Kataura: Tetsu-to-Hagané 57 (1971) 1753–1763.
• 5)   T.  Ototani,  Y.  Kataura and  T.  Degawa: Tetsu-to-Hagané 61 (1975) 2167–2181.
• 6)   M.  Okamoto: Kinzoku no Kenkyu 14 (1937) 46–59.
• 7)   G.Z.  Chen,  D.J.  Fray and  T.W.  Farthing: Nature 407 (2000) 361–364.
• 8)   R.O.  Suzuki,  K.  Teranuma and  K.  Ono: Metall. Mater. Trans. B 34 (2003) 287–295.
• 9)   R.O.  Suzuki and  S.  Fukui: Mater. Trans. 45 (2004) 1665–1671.
• 10)   O.  Takeda and  T.H.  Okabe: Mater. Trans. 47 (2006) 1145–1154.
• 11)   H.  Zheng,  H.  Ito and  T.H.  Okabe: Mater. Trans. 48 (2007) 2244–2251.
• 12)   P.K.  Tripathy,  M.  Gauthier and  D.J.  Fray: Metall. Mater. Trans. B 38 (2007) 893–900.
• 13)   A.M.  Abdelkader,  A.  Daher,  R.A.  Abdelkareem and  E.  El-Lashif: Metall. Mater. Trans. B 38 (2007) 35–44.
• 14)   K.S.  Mohandas and  D.J.  Fray: Metall. Mater. Trans. B 40 (2009) 685–699.
• 15)   T.  Matsuzaki,  S.  Natsui,  T.  Kikuchi and  R.O.  Suzuki: Mater. Trans. 58 (2017) 371–376.
• 16)   R.O.  Suzuki,  M.  Baba,  Y.  Ono and  K.  Yamamoto: J. Alloy. Compd. 389 (2005) 310–316.
• 17)   S.  Takeuchi,  T.  Honma,  T.  Satow,  S.  Sonoda,  H.  Kawai and  K.  Koyama: J. Japan Inst. Metals 31 (1967) 42–48.
• 18)   C.B.  Alcock,  V.  PItkin and  M.K.  Horrigan: Can. Metall. Quart. 23 (1984) 309–313.
• 19)  O. Kubaschewski: Iron Binary Phase Diagram, (Springer-Verlag Stahleisen m. b. H., Dusseldorf, 1982) p. 26.
• 20)  J.L. Murray: Phase Diagrams of Binary Titanium Alloys, (ASM International, Metals Park, Ohio, 1987) pp. 52–58.
• 21)  I. Barin and G. Platzki: Thermochemical Data of Pure Substances, third edition, (VCH Verlagsgesellschaft mbH., D-69451 Weinheim, Germany, 1995) pp. 416–417, 440.
• 22)  M.W. Chase, Jr., C.A. Davies, J.R. Downey, Jr., D.J. Frurip, R.A. McDonald and A.N. Syverud: JANAF Thermochemical Tables, third edition, (American Institute of Physics Inc., New York, 1985) pp. 685–690 and 711–714.
• 23)  J.F. Elliott and M. Gleiser: Thermochemistry for Steelmaking, Vol. 1, (Addison-Wesley Publishing Company, Inc., Boston, 1960) p. 266.
• 24)  E.T. Turkdogan: Fundamentals of Steelmaking, (The Institute of Materials, London, 1996) p. 10.
• 25)   T.  Wakasugi and  N.  Sano: Metall. Mater. Trans. B 20 (1989) 431.
• 26)   H.  Ono,  A.  Kobayashi,  F.  Tsukihashi and  N.  Sano: Metall. Mater. Trans. B 23 (1992) 313.
• 27)   E.J.  Huber, Jr. and  C.E.  Holly, Jr.: J. Phys. Chem. 60 (1956) 498–499.
• 28)   J.W.  Patterson,  E.C.  Bogren and  R.A.  Rapp: J. Electrochem. Soc. 114 (1967) 752–758.
• 29)   K.  Suzuki and  H.  Inoue: Bull. Res. Inst. Mineral Dressing Metall., Tohoku Univ. 25 (1970) 129–146.
• 30)   S.  Komatsu,  M.  Yonehara and  Z.  Kozuka: J. Japan Inst. Metals 36 (1972) 674–679.
• 31)   I.  Seki,  K.  Nagata,  J.  Tanabe and  T.  Ashino: J. Japan Inst. Met. Mater. 79 (2015) 475–484.
• 32)   T.  Sata,  T.  Sasamoto and  K.  Matsumoto: Gypsum & Lime 175 (1981) 261–267.
• 33)   R.F.  Geller and  P.J.  Yavorsky: J. Res. Natl. Bur. Stand. 35 (1945) 87–110.
• 34)  M. Hino and K. Ito: Thermodynamic Data for Steelmaking, (Tohoku University Press, Sendai, 2010) p. 43.
• 35)   I.  Seki,  D.V.  Louzguine-Luzgin,  T.  Takahashi,  H.  Kimura and  A.  Inoue: Mater. Trans. 52 (2011) 458–463.
• 36)   I.  Seki,  K.  Nagata,  T.  Ashino and  J.  Tanabe: ISIJ Int. 51 (2011) 1369–1374.
• 37)   Q.  Han,  X.  Zhang,  D.  Chen and  P.  Wang: Metall. Mater. Trans. B 19 (1988) 617–622.
• 38)   G.K.  Sigworth and  J.F.  Elliot: Met. Sci. 8 (1974) 298–310.
• 39)   R.O.  Suzuki and  S.  Inoue: Metall. Mater. Trans. B 34 (2003) 277–285.
• 40)   T.  Ogasawara,  K.  Mori,  T.  Tominaga,  F.  Tsukihashi and  N.  Sano: ISIJ Int. 36 Suppl. (1996) S30–S33.
• 41)   T.H.  Okabe,  R.O.  Suzuki,  T.  Oishi and  K.  Ono: Tetsu-to-Hagané 77 (1991) 93–99.
• 42)   T.H.  Okabe,  R.O.  Suzuki,  T.  Oishi and  K.  Ono: Mater. Trans. JIM 32 (1991) 485–488.
• 43)   T.H.  Okabe,  M.  Nakamura,  T.  Oishi and  K.  Ono: Metall. Mater. Trans. B 24 (1993) 449–455.
• 44)   K.  Ono and  S.  Miyazaki: J. Japan Inst. Metals 49 (1985) 871–875.
• 45)   F.  Tsukihashi,  E.  Tawara and  T.  Hatta: Metall. Mater. Trans. B 27 (1996) 967–972.
• 46)   Y.  Kobayashi and  F.  Tsukihashi: Metall. Mater. Trans. B 28 (1997) 956–959.
• 47)   Y.  Tago,  Y.  Endo,  K.  Morita,  F.  Tsukihashi and  N.  Sano: ISIJ Int. 35 (1995) 127–131.
• 48)   S.  Kobayashi,  Y.  Omori and  K.  Sanbongi: Tetsu-to-Hagané 56 (1970) 998–1013.
• 49)   T.  Kimura and  H.  Suito: Metall. Mater. Trans. B 25 (1994) 33–42.
• 50)   H.  Fujiwara,  M.  Tano,  K.  Yamamoto and  E.  Ichise: ISIJ Int. 35 (1995) 1063–1071.
• 51)   H.  Ohta and  H.  Suito: ISIJ Int. 36 (1996) 983–990.
• 52)   H.  Itoh,  M.  Hino and  S.  Banya: Tetsu-to-Hagané 83 (1997) 695–700.
• 53)   K.  Mineura,  I.  Takahashi and  K.  Tanaka: ISIJ Int. 30 (1990) 192–198.
• 54)   H.  Ohta and  H.  Suito: ISIJ Int. 43 (2003) 1293–1300.
• 55)   T.  Fujisawa,  C.  Yamauchi and  H.  Sakao: Tetsu-to-Hagané 76 (1990) 368–375.