Modification of the Activities of Components in Ca₂SiO₄–Ca₃P₂O₈ Solid Solution at 1573 K

Abstract

The activities of components in Ca₂SiO₄–Ca₃P₂O₈ solid solutions have been reported by preceding studies as the basic thermodynamic data for phosphorus removal in the steelmaking process. Since the reported activities contained the uncertainties accumulated from thermodynamic data, this study aimed to reevaluate the experimental results in the preceding studies. Firstly, the equilibrium constant of the following reaction used to calculate the P₂O₅ activities was measured at 1573 K through a gas equilibrium method.

2P_{in Cu} + 5H₂O(gas) = P₂O_{5 in oxide} + 5H₂(gas)

It was discussed that the use of the equilibrium constant determined in the present study could reduce the uncertainty in the P₂O₅ activities. Subsequently, the recalculated P₂O₅ activities at 1573 K and available thermodynamic data for the Ca₂SiO₄–Ca₃P₂O₈ solid solution-containing system were compared in terms of the activities of components in the Ca₂SiO₄–Ca_1.5PO₄ (= (1/2)Ca₃P₂O₈) pseudo-binary system.

1. Introduction

Phosphorus is a typical harmful element in steel and is removed from molten iron by the following oxidation reaction in steelmaking processes.

  

$$2{\text{P}}_{\text{in}\mathrm{\hspace{0.17em} }\text{Fe}}+5\mathrm{F}\mathrm{e}{\text{O}}_{\text{in}\mathrm{\hspace{0.17em} }\text{slag}}={\text{P}}_2{\text{O}}_{5\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }\text{slag}}+5\mathrm{F}\mathrm{e}(\text{liquid})$$(1)

  

$$\begin{array}{l}log{K}_1=log{a}_{{\text{P}}_2{\text{O}}_5}-5log{a}_{\text{FeO}}-2log\left({[f_{\text{P}}]}_{\text{Fe}}\cdot {[mass\%\text{P}]}_{\text{Fe}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}=12\mathrm{\hspace{0.17em} }730/(T/\text{K})-20.0,\end{array}$$(2) ¹⁾

where [mass%P]_Fe and [f_P]_Fe denote the mass% content and Henrian activity coefficient of phosphorus in molten iron, respectively, while a_i represents the Raoultian activity of component i in the slag. The standard states of a_P2O5 and a_FeO are taken to be pure hypothetical liquid P₂O₅ and pure liquid FeO in equilibrium with metallic iron, respectively. It is known that P₂O₅ reacts with CaO and SiO₂ in the slag and is enriched in solid solutions between di-calcium silicate, Ca₂SiO₄, and tri-calcium phosphate, Ca₃P₂O₈,^{2,3,4,5,6,7,8,9,10,11,12)} which have an important role in the dephosphorization reaction.¹³⁾ Figure 1 gives a part of the CaO–SiO₂–P₂O₅ ternary phase diagram at 1573 K^14,15,16) and the Ca₂SiO₄–Ca₃P₂O₈ pseudo-binary phase diagram.¹⁷⁾ It can be seen that the solid solution between α-Ca₂SiO₄ and α-Ca₃P₂O₈ can coexist with CaSiO₃ or CaO when the Ca₃P₂O₈ content is 10–40 mass% at 1573 K. In Fig. 1 and hereafter, the following abbreviations are used.

  

$$C3S =3\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2=\mathrm{C}{\mathrm{a}}_3\mathrm{S}\mathrm{i}{\mathrm{O}}_5$$

  

$$C2S=2\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2=\mathrm{C}{\mathrm{a}}_2\mathrm{S}\mathrm{i}{\mathrm{O}}_4$$

  

$$C3S2=3\mathrm{C}\mathrm{a}\mathrm{O}\cdot 2\mathrm{S}\mathrm{i}{\mathrm{O}}_2=\mathrm{C}{\mathrm{a}}_3\mathrm{S}{\mathrm{i}}_2{\mathrm{O}}_7$$

  

$$CS=\mathrm{C}\mathrm{a}\mathrm{O}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2=\mathrm{C}\mathrm{a}\mathrm{S}\mathrm{i}{\mathrm{O}}_3$$

  

$$C4P=4\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{P}}_2{\mathrm{O}}_5=\mathrm{C}{\mathrm{a}}_4{\mathrm{P}}_2{\mathrm{O}}_9$$

  

$$C3P=3\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{P}}_2{\mathrm{O}}_5=\mathrm{C}{\mathrm{a}}_3{\mathrm{P}}_2{\mathrm{O}}_8$$

  

$$C5PS=5\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{P}}_2{\mathrm{O}}_5\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_2=\mathrm{C}{\mathrm{a}}_5{\mathrm{P}}_2\mathrm{S}\mathrm{i}{\mathrm{O}}_{12}$$

  

$$C7PS2 =7\mathrm{C}\mathrm{a}\mathrm{O}\cdot {\mathrm{P}}_2{\mathrm{O}}_5\cdot 2\mathrm{S}\mathrm{i}{\mathrm{O}}_2=\mathrm{C}{\mathrm{a}}_7{\mathrm{P}}_2\mathrm{S}{\mathrm{i}}_2{\mathrm{O}}_{16}$$

  

$$<C2S-C3P{>}_{SS}=\mathrm{C}{\mathrm{a}}_2\mathrm{S}\mathrm{i}{\mathrm{O}}_4-\mathrm{C}{\mathrm{a}}_3{\mathrm{P}}_2{\mathrm{O}}_8\mathrm{\hspace{0.17em} }\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{\hspace{0.17em} }\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$

Fig. 1. (a) CaO–SiO₂–P₂O₅ ternary phase diagram at 1573 K.^14,15,16) The grey-shaded areas are unknown and estimated. (b) Ca₂SiO₄–Ca₃P₂O₈ pseudo-binary phase diagram.¹⁷⁾

According to Le Chatelier’s principle, Reaction (1) can proceed effectively in conditions of low temperature, high FeO activity, low P₂O₅ activity, and high P activity coefficient. If the molten iron is regarded as a Fe–C–P liquid alloy, [f_P]_Fe can be expressed by using interaction parameters, e_i ^j.

  

$$\begin{array}{l}log{[f_{\text{P}}]}_{\text{Fe}}=e_{\text{P}}^{\text{C}}{[mass\%\text{C}]}_{\text{Fe}}+e_{\text{P}}^{\text{P}}{[mass\%\text{P}]}_{\text{Fe}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=0.126{[mass\%\text{C}]}_{\text{Fe}}+0.054{[mass\%\text{P}]}_{\text{Fe}}\end{array}$$(3) ¹⁾

When iron ore is reduced by carbon, the dissolving carbon can decrease the liquidus temperature as well as increase [f_P]_Fe in the iron alloy, both of which are advantageous for the phosphorus removal. On the other hand, concerning hydrogen reduction aiming at carbon neutrality, there is a possibility that phosphorus exists as an impurity in hydrogen-reduced iron to the same extent as the carbon-reduced one.¹⁸⁾ Now, the equilibrium phosphorus content after dephosphorization from the carbon- or hydrogen-reduced molten iron can be estimated as follows. The refining temperatures were assumed to be 1573 K for the carbon-reduced iron and 1873 K for the hydrogen-reduced one. Compositions of the liquid slags saturated with C2S were determined as given in Table 1 based on the FeO–CaO–SiO₂ ternary phase diagram,¹⁹⁾ Fig. 2, with the assumption that CaO/SiO₂ mole ratios were 2 and mole fractions of P₂O₅ were 0.02. The activities of FeO and P₂O₅ were calculated with a regular solution model,²⁰⁾ and the phosphorus concentrations in the iron could be estimated by solving Eqs. (2) and (3); the calculated values are shown in Table 1. The equilibrium phosphorus content in the hydrogen-reduced iron was found to be 16 times higher than that in the carbon-reduced one. In order to overcome such a disadvantageous situation, it is essential to know accurately the basic thermochemical properties of the slags containing <C2S-C3P>_SS, which provides new insights into slag design for refining the hydrogen-reduced iron.

Table 1. Comparison of the phosphorus contents in the C-reduced and H-reduced iron.

Reduced by	T/K	Mole fraction in liquid slag				Activity20)		[mass% i]Fe
		FeO	CaO	SiO2	P2O5	FeO	P2O5	C	P
Carbon	1573	0.73	0.17	0.08	0.02	0.90	6.2×10−17	4.5	0.0024
Hydrogen	1873	0.41	0.38	0.19	0.02	0.55	5.3×10−18	0	0.039

Fig. 2. Liquidus line coexisting with C2S in the FeO–CaO–SiO₂ ternary system.

From the consideration above, the present authors have reported the P₂O₅ activities in the CaO–SiO₂–P₂O₅ ternary system with bulk compositions shown in Fig. 1(a) at 1573 K through a gas equilibrium method, in which copper-phosphorus alloys were equilibrated with oxides under a stream of Ar + H₂ + H₂O gas mixture.^21,22,23) The P₂O₅ activities were derived from the equilibrium phosphorus concentrations in the copper alloys and H₂/H₂O partial pressure ratios, p_H2/p_H2O, by using the equilibrium constant of Reaction (4).

  

$$2{\text{P}}_{\text{in}\mathrm{\hspace{0.17em} }\text{Cu}}+5{\text{H}}_2\mathrm{O}(\text{gas})={\text{P}}_2{\text{O}}_{5\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }\text{oxide}}+5{\text{H}}_2(\text{gas})$$(4)

  

$$\begin{array}{l}log{K}_4=log{a}_{{\text{P}}_2{\text{O}}_5}-2log\left({[f_{\text{P}}]}_{\text{Cu}}\cdot {[mass\%\text{P}]}_{\text{Cu}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+5log(p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O}})\end{array}$$(5)

However, inaccuracies in thermal data used to calculate K₄ were involved in the measured P₂O₅ activities, which makes it difficult to compare the experimental results with other literature data. In this study, therefore, K₄ at 1573 K was measured directly with the same gas equilibrium method using the copper alloys as reference metal and oxid mixtures of MgO + Mg₃P₂O₈, in which a_P2O5 has been reported. Subsequently, the activities of components in the CaO–SiO₂–P₂O₅ ternary system were recalculated using the determined value for K₄. The advantage of this calculating method is that the systematic errors in the experiments conducted on the MgO + Mg₃P₂O₈ two-phase mixture and the CaO–SiO₂–P₂O₅ ternary oxide would be canceled out in deriving the activities in the ternary system.

2. Materials and Procedure

In the present experiments, liquid Cu–P alloys were brought into equilibrium with the MgO + Mg₃P₂O₈ mixtures at 1573 K under a stream of Ar + H₂ + H₂O gas mixture, in which p_H2/p_H2O was fixed. Mg₃P₂O₈ is hereafter abbreviated as M3P. The underlying reaction can be expressed as Eq. (4). The phosphorus in the molten copper was reported to obey Henry’s law in the composition range of [mass%P]_Cu < 2,^24,25) i.e., [f_P]_Cu was unity. As the value for a_P2O5 in MgO + M3P can be calculated from thermal data,^26,27) Eq. (5) indicates that K₄ is obtainable by analyzing the phosphorus content in the copper alloy equilibrated with MgO + M3P under the fixed p_H2/p_H2O.

Regent grade MgO and Mg(H₂PO₄)₂·3H₂O obtained from Nacalai Tesque Inc. Kyoto, Japan were mixed at a mole ratio of 2:1 and heated slowly to 1273 K in air to prepare M3P. The resulting compounds were submitted to powder X-ray diffraction analysis to confirm the expected phases only. Oxide mixtures of MgO + M3P were pressed in a steel die to form crucible shapes (15 mm o.d., 8 mm i.d., and 8 mm in height). The starting materials for the metallic phase were copper shavings obtained from Nakalai Tesque Inc., Kyoto, Japan and Cu₃P from Hirano Seizaemon Co. Ltd., Tokyo, Japan.

The crucibles made of MgO + M3P were charged with Cu and Cu₃P and held at 1573 K in a mullite reaction tube equipped with a SiC resistance furnace; details of the experimental apparatus were described in the previous work.²⁸⁾ The Ar + H₂ + H₂O gas mixture prepared by passing the Ar + 12%H₂ gas mixture through distilled water kept at 283 K–304 K in a thermostat bath was introduced into the reaction tube. The partial pressure of H₂O in the gas mixture can be calculated as follows.

  

$$\begin{array}{l}log(p_{{\text{H}}_2\text{O}}/\text{atm})=19.732-2\mathrm{\hspace{0.17em} }900/(T_{\text{bath}}/\text{K})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-4.65log(T_{\text{bath}}/\text{K}),\end{array}$$(6) ²⁹⁾

where T_bath is the temperature of the thermostat bath. The experimental conditions are listed in Table 2.

Table 2. Experimental conditions and results.

pH2/pH2O	Holding time/hour	[mass%P]Cu
		initial	equilibrium	average
2.53	42.0	0	0.0225	0.0248 ± 0.0016
		0.11	0.0264
		0.26	0.0254
3.61	44.2	0	0.0543	0.0539 ± 0.0004
		0.59	0.0535
4.93	42.3	0	0.0765	0.0731 ± 0.0076
		0.20	0.0802
		0.57	0.0625
6.80	45.6	0	0.154	0.159 ± 0.011
		0.22	0.149
		0.64	0.174
9.49	49.1	0	0.395	0.395 ± 0.001
		0.68	0.394

The metallic phase then melted to form the Cu–P molten alloys coexisting with the oxides. After being held at 1573 K, the quenched alloys were submitted to ICP-OES to determine the phosphorus contents with an indirect spectrometric method involving solvent extraction.³⁰⁾ The alloy samples with different initial phosphorus contents were simultaneously held in the reaction tube. When the phosphorus concentration was higher or lower than the equilibrium value, Reaction (4) proceeded toward the right or left hand, respectively. The phosphorus concentrations in such alloys that matched each other within experimental uncertainties were able to be confirmed as the equilibrium value.

3. Result and Discussion

3.1. Equilibrium Constant of Reaction (4) and the P₂O₅ Activities

Figure 3 shows a typical relationship between the compositions of the copper alloys and duration time. The phosphorus concentrations in the alloys with different initial compositions agreed well after 42.0 hours, and the equilibrium value for [mass%P]_Cu was thus determined to be 0.0248 ± 0.0016. Table 2 summarizes all experimental results and Fig. 4 shows log[mass%P]_Cu plotted against log(p_H2/p_H2O). Equation (7) given by rewriting Eq. (5) indicates that the logarithmic relationship between [mass%P]_Cu and p_H2/p_H2O should be linear with a slope of 5/2 and an intercept of [(1/2)loga_P2O5 - (1/2)logK₄].

  

$$\begin{array}{l}log{[mass\%\text{P}]}_{\text{Cu}}=(5/2)log(p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O}})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+(1/2)log{a}_{{\text{P}}_2{\text{O}}_5}-(1/2)log{K}_4\end{array}$$(7)

Such a linear relation can be observed in Fig. 4. Since the thermodynamic property of M3P has been well assessed based on measured heat capacity, standard enthalpy of formation, and Gibbs energy,²⁶⁾ the P₂O₅ activity in the MgO + M3P two-phase mixture at 1573 K can be obtained from the following formulae.

  

$$3\mathrm{M}\mathrm{g}\mathrm{O}(\text{solid})+{\text{P}}_2{\text{O}}_{5\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }M3P}=M3P(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d})$$(8)

  

$$log{K}_8=-log{a}_{{\text{P}}_2{\text{O}}_5\left(\text{MgO}+M3P\right)}=13.60\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}$$(9) ^26,27)

K₄ can be determined from the intercept of the regression line for MgO + M3P in Fig. 4.

  

$$log{K}_4=-8.04\pm 0.24\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \mathrm{K}$$(10)

Fig. 3. Typical changes in the compositions of alloys over duration time.

Fig. 4. Relationship between log[mass%P]_Cu and log(p_H2/p_H2O) at 1573 K.

Figure 4 also shows experimental results reported by the preceding studies for <C2S-C3P>_SS + CS^22,23) and <C2S-C3P>_SS + CaO²²⁾ two-phase regions (31 mass%C3P in <C2S-C3P>_SS) and <C2S-C3P>_SS + C2S + C3S three-phase region²¹⁾ in the CaO–SiO₂–P₂O₅ ternary system in Fig. 1(a). Intercepts of regression lines with slopes of 5/2 correspond to [(1/2)loga_P2O5 - (1/2)logK₄], and thus give the P₂O₅ activities by using the determined value for K₄ at 1573 K. The obtained activities for the three oxide mixtures are summarized in Table 3 with the original values in the preceding studies. The modification of K₄ resulted in about two-digit increases in a_P2O5 from original values. Table 3 also shows that the P₂O₅ activities vary drastically depending on the second phase coexisting with <C2S-C3P>_SS, i.e., CaO or CS, even at the same C3P concentrations in the solid solutions.

Table 3. Comparison between the modified and original values at 1573 K.

	Modified	Original	Ref.
logK4	−8.04 ± 0.24	−10.34	24,29,31)
$log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+CS\right)}$	−17.25 ± 0.13	−19.60 ± 0.13	23)
$log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+\text{CaO}\right)}$	−24.08 ± 0.05	−26.43 ± 0.05	22)
$log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+C2S+C3S\right)}$	−24.44 ± 0.31	−26.53 ± 0.31	21)

3.2. Experimental Errors

The equilibrium constant of Reaction (4) in the experiments with the four types of oxide mixtures is expressed as

  

$$\begin{array}{l}log{K}_4=log{a}_{{\text{P}}_2{\text{O}}_5(i)}-2log{[mass\%\text{P}]}_{\text{Cu(}i\text{)}}\\ \hspace{1em}\hspace{1em}\hspace{1em}+5log(p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O(}i\text{)}}),\end{array}$$(11)

where i means the oxide mixture of MgO + M3P, <C2S-C3P>_SS + CS, <C2S-C3P>_SS + CaO, or <C2S-C3P>_SS + C2S + C3S. When the regression lines in Fig. 4 are compared with fixed values for log[mass%P]_Cu on the vertical axis, the differences in loga_P2O5(i) are written as

  

$$\begin{array}{l}log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+CS\right)}-log{a}_{{\text{P}}_2{\text{O}}_5(\text{MgO}+M3P)}=\\ 5log\left({\displaystyle \frac{p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O(MgO}+M3P\text{)}}}{p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O}\left({⟨C2S-C3P⟩}_{SS}+CS\right)}}}\right)\end{array}$$(12)

  

$$\begin{array}{l}log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+\text{CaO}\right)}-log{a}_{{\text{P}}_2{\text{O}}_5(\text{MgO}+M3P)}=\\ 5log\left({\displaystyle \frac{p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O(MgO}+M3P\text{)}}}{p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O}\left({⟨C2S-C3P⟩}_{SS}+\text{CaO}\right)}}}\right)\end{array}$$(13)

  

$$\begin{array}{l}log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+C2S+C3S\right)}-log{a}_{{\text{P}}_2{\text{O}}_5(\text{MgO}+M3P)}=\\ 5log\left({\displaystyle \frac{p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O(MgO}+M3P\text{)}}}{p_{{\text{H}}_2}/p_{{\text{H}}_2\text{O}\left({⟨C2S-C3P⟩}_{SS}+C2S+C3S\right)}}}\right)\end{array}$$(14)

Equations (12), (13), (14) indicate that the P₂O₅ activities in <C2S-C3P>_SS + CS, <C2S-C3P>_SS + CaO, and <C2S-C3P>_SS + C2S + C3S can be derived from the ratios of p_H2/p_H2O(i). Systematic experimental errors in these experiments would be canceled out in calculating a_P2O5(i) since they would be involved in both experimental procedures for MgO + M3P and <C2S-C3P>_SS + CS, <C2S-C3P>_SS + CaO, or <C2S-C3P>_SS + C2S + C3S, which makes the present results more reliable.

3.3. Activities of Ca₂SiO₄ and Ca_1.5PO₄

Zhong et al. measured a_P2O5 at higher temperatures and wide composition range in <C2S-C3P>_SS coexisting with CaO through a chemical equilibrium method using iron alloys as reference metals.³²⁾ Figure 5 shows a_P2O5 by the present study and Zhong et al. plotted against the C3P content in the solid solution. In this figure, the results for the high basicity regions, <C2S-C3P>_SS + CaO and <C2S-C3P>_SS + C2S + C3S, modified in the present study are plotted to compare with the literature data for <C2S-C3P>_SS + CaO. At all the temperatures, it can be seen that the P₂O₅ activities in <C2S-C3P>_SS coexisting with CaO or C3S increase as the C3P contents increase. Although the temperature dependence of a_P2O5 seems very small, it is difficult to understand the thermochemical properties of <C2S-C3P>_SS from the P₂O₅ activities since a_P2O5 varies depending on the second phase coexisting with the solid solution even at the same composition of <C2S-C3P>_SS.

Fig. 5. P₂O₅ activities in the Ca₂SiO₄–Ca₃P₂O₈ solid solutions. The present results are for the CaO coexisting and C3S coexisting regions (high basicity regions).

The C2S and C3P activities are determined depending on temperature and the composition of <C2S-C3P>_SS but not on the second phase. Thus, the activities of C2S and C3P represent the thermochemical properties of the solid solution better than that of P₂O₅. In the <C2S-C3P>_SS + CS and <C2S-C3P>_SS + CaO two-phase regions, Reactions (15) and (17) achieve the equilibrium states, respectively, and the P₂O₅ activities are fixed through the C2S and C3P activities.

<C2S-C3P>_SS + CS region (31 mass%C3P in <C2S-C3P>_SS)

  

$${\text{P}}_2{\text{O}}_{5\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }SS}+3C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}=C3P_{\text{in}\mathrm{\hspace{0.17em} }SS}+3CS(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d})$$(15)

  

$$\begin{array}{l}log{K}_{15}=log{a}_{C3P}-3log{a}_{C2S}-log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+CS\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}=16.51\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}\end{array}$$(16) ^33,34,35)

<C2S-C3P>_SS + CaO region (31 mass%C3P in <C2S-C3P>_SS)

  

$${\text{P}}_2{\text{O}}_{5\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }SS}+3\mathrm{C}\mathrm{a}\mathrm{O}(\text{solid})=C3P_{\text{in}\mathrm{\hspace{0.17em} }SS}$$(17)

  

$$\begin{array}{l}log{K}_{17}=log{a}_{C3P}-log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+\text{CaO}\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}=21.98\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}\end{array}$$(18) ³⁵⁾

The standard states of a_C2S and a_C3P are taken to be the pure supercooled solid α-C2S and α-C3P. By solving the Eqs. (16) and (18) with the measured results for $a_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+CS\right)}$ and $a_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+\text{CaO}\right)}$, the C2S and C3P activities can be obtained at 31 mass%C3P and 1573 K.

  

$$log{a}_{C3P}=-2.10\pm 0.05$$(19)

  

$$log{a}_{C2S}=-0.45\pm 0.06$$(20)

On the other hand, a_C2S and a_C3P in <C2S-C3P>_SS doubly saturated with C2S and C3S can be derived as follows.

<C2S-C3P>_SS + C2S + C3S region

  

$${\text{P}}_2{\text{O}}_{5\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }SS}+3\mathrm{\hspace{0.17em} }C3S(\text{solid})=C3P_{\text{in}\mathrm{\hspace{0.17em} }SS}+3C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}$$(21)

  

$$\begin{array}{l}log{K}_{21}=log{a}_{C3P}+3\mathrm{\hspace{0.17em} }log{a}_{C2S}-log{a}_{{\text{P}}_2{\text{O}}_5\left({⟨C2S-C3P⟩}_{SS}+C2S+C3S\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}=21.80\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}\end{array}$$(22) ^33,34,35)

  

$$C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}={\alpha }^{\prime }–C2S$$(23)

  

$$G{°}_{\alpha –C2S}+RT\mathrm{l}\mathrm{n}{a}_{C2S}=G{°}_{{\alpha }^{\prime }–C2S}$$(24)

  

$$log{a}_{C2S}=-0.04$$(25) ^33,34)

  

$$log{a}_{C3P}=-2.52\pm 0.31$$(26)

In Eq. (24), the solubility of C3P in α′-C2S was neglected.

In the solid solutions, the anion PO₄³⁻ is thought to replace SiO₄⁴⁻.^36,37) Even though tri-calcium phosphate has conventionally been treated as Ca₃P₂O₈, it should be better to consider the chemical compound containing a single PO₄³⁻, Ca_1.5PO₄ (= (1/2)C3P); the relationship between the activities of Ca_1.5PO₄ and Ca₃P₂O₈ can be formulated as

  

$$log{a}_{(1/2)C3P}=(1/2)log{a}_{C3P}$$(27)

Figure 6 illustrates a_(1/2)C3P and a_C2S plotted against the mole fraction of (1/2)C3P, X_(1/2)C3P, in the C2S-(1/2)C3P pseudo-binary system. This figure also shows the literature data at higher temperatures³²⁾ and calculated values based on phase relations in the ternary system at 1573 K expressed by Eqs. (28), (29), (30), (31), (32), (33).

Fig. 6. Activities of components in the Ca₂SiO₄–Ca_1.5PO₄ pseudo-binary system. Square plots at 1573 K represent the calculated values based on the phase relations.

<C2S-C3P>_SS + C3S + CaO region

  

$$C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}+\mathrm{C}\mathrm{a}\mathrm{O}(\text{solid})=C3S(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d})$$(28)

  

$$log{K}_{28}=-log{a}_{C2S}=0.06\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}$$(29) ^32,33)

<C2S-C3P>_SS + CS + C3S2 region

  

$$C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}+CS(\text{solid})=C3S2(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d})$$(30)

  

$$log{K}_{30}=-log{a}_{C2S}=0.24\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}$$(31) ^32,33)

C5PS + C4P + CaO region

  

$$\begin{array}{c}C3P_{\text{in}\mathrm{\hspace{0.17em} }C5PS}+\mathrm{C}\mathrm{a}\mathrm{O}(\text{solid})=C4P(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d})\end{array}$$(32)

  

$$log{K}_{32}=-log{a}_{C3P}=-2log{a}_{(1/2)C3P}=0.61\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}$$(33) ³⁵⁾

Figure 6 reveals the temperature dependence of the activities in the solid solution, which was unclear in Fig. 5; the activities of (1/2)C3P at higher temperatures are much lower than the present results at 1573 K. The temperature dependence shown in Fig. 6 differs from the ordinary tendency of a solution, i.e., the higher the temperature, the closer the activity approaches Raoult’s law. For a detailed discussion, the accuracy of the C3P activities at 1823 K and 1873 K, e.g., effects of inevitable FeO contamination from the reference metal or the temperature dependence of logK₁₇, should be investigated in the future. Nevertheless, even at the low temperature, the activities in this system exhibit negative deviations from Raoult’s law. This result was not surprising since the substitution of PO₄³⁻ for SiO₄⁴⁻ would not change much the crystal structure of α-C2S^36,38) and C2S can incorporate C3P in the whole composition range as shown in Fig. 1(b).

3.4. Activities of CaO and SiO₂

Finally, the thermodynamic consistency is discussed based on the comparison of the CaO and SiO₂ activities in the CaO–SiO₂–P₂O₅ ternary system and the CaO–SiO₂ binary system using the same manner described in the previous paper.²²⁾ The activities of components in the binary system at 1573 K are shown in Fig. 7; thermodynamic properties in this binary system have been assessed in several studies^34,39) and they were in good agreement.

Fig. 7. Activities of components in the CaO–SiO₂ binary system at 1573 K.

Once the thermochemical properties of <C2S-C3P>_SS are clarified, the CaO and SiO₂ activities can be calculated in the solid solutions coexisting with CS or CaO from the following reactions, whose equilibrium constants were obtainable from the thermal data.^33,34)

<C2S-C3P>_SS + CS region (31 mass%C3P in <C2S-C3P>_SS)

  

$${\text{CaO}}_{\text{in}\mathrm{\hspace{0.17em} }SS}+CS(\text{solid})=C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}$$(34)

  

$$\begin{array}{l}log{a}_{\text{CaO}\left({⟨C2S-C3P⟩}_{SS}+CS\right)}=-log{K}_{34}+log{a}_{C2S}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=-2.28\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \mathrm{K}\end{array}$$(35)

  

$$\begin{array}{c}{\text{SiO}}_{2\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }SS}+C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}=2CS(\text{solid})\end{array}$$(36)

  

$$\begin{array}{l}log{a}_{{\text{SiO}}_2\left({⟨C2S-C3P⟩}_{SS}+CS\right)}=-log{K}_{36}-log{a}_{C2S}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=-0.64\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \mathrm{K}\end{array}$$(37)

<C2S-C3P>_SS + CaO region (31 mass%C3P in <C2S-C3P>_SS)

  

$${\text{SiO}}_{2\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }SS}+2\mathrm{C}\mathrm{a}\mathrm{O}(\text{solid})=C2S_{\text{in}\mathrm{\hspace{0.17em} }SS}$$(38)

  

$$\begin{array}{l}log{a}_{{\text{SiO}}_2\left({⟨C2S-C3P⟩}_{SS}+\text{CaO}\right)}=-log{K}_{38}+log{a}_{C2S}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-5.20\mathrm{\hspace{0.17em} } \mathrm{a}\mathrm{t}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \mathrm{K}\end{array}$$(39)

The calculated results are superimposed in Fig. 7. While the modification of the value for K₄ leads to the increasing P₂O₅ activities as already mentioned, it has no effect on the CaO and SiO₂ activities, which are consistent with the thermal data for the CaO–SiO₂ binary system.

4. Conclusions

The reported values for the activities of components in the CaO–SiO₂–P₂O₅ ternary system by the preceding studies included the uncertainties accumulated from thermodynamic data. To reevaluate the reported activities, the equilibrium constant of the following reaction at 1573 K was measured by the gas equilibrium method, in which the molten copper-phosphorus alloys were equilibrated with the MgO + Mg₃P₂O₈ mixtures under fixed H₂/H₂O partial pressure ratios.

  

$$\begin{array}{l}2{\text{P}}_{\text{in}\mathrm{\hspace{0.17em} }\text{Cu}}+5{\text{H}}_2\text{O(gas)}={\text{P}}_2{\text{O}}_{5\mathrm{\hspace{0.17em} }\text{in}\mathrm{\hspace{0.17em} }\text{oxide}}+5{\text{H}}_2(\text{gas})\\ logK=-8.04\pm 0.24\mathrm{\hspace{0.17em} } \text{at}\mathrm{\hspace{0.17em} } 1\mathrm{\hspace{0.17em} }573\mathrm{\hspace{0.17em} } \text{K}\end{array}$$

Applying the same experimental method to determine both logK and the P₂O₅ activities in the CaO–SiO₂–P₂O₅ ternary system can prevent error propagation in the thermodynamic data. Subsequently, the present results and the available data in the Ca₂SiO₄–Ca₃P₂O₈ solid solution-containing system were summarized in terms of the activities of Ca₂SiO₄ and Ca_1.5PO₄ (= (1/2)Ca₃P₂O₈). The activities of components in the Ca₂SiO₄–Ca_1.5PO₄ binary system exhibited negative deviations from Raoult’s law.

Statement for Conflict of Interest

The authors declare that they have no conflict of interest.

Acknowledgment

This work was supported by JSPS KAKENHI Grant Numbers JP22KJ1689, JP21K04737, and JP24K08126. Helpful comments and discussion were given by Professor Kazuki Morita, Department of Materials Engineering, Graduate School of Engineering, the University of Tokyo, and these are gratefully acknowledged.

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