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A Thermodynamic Model for Representation Reaction Abilities of Structural Units in Fe-S Binary Melts Based on the Atom-Molecule Coexistence Theory

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Abstract

A thermodynamic model for calculating the mass action concentrations of structural units in Fe-S binary melts based on the atom-molecule coexistence theory, i.e., AMCT-N i model, has been developed and verified through a comparison with the reported activities of both S and Fe in Fe-S binary melts with changing mole fraction \( x_{\text{S}} \) of S from 0.0 to 0.095 at temperatures of 1773 K, 1823 K, and 1873 K (1500 °C, 1550 °C, and 1600 °C) from the literature. The calculated mass action concentration \( N_{\text{S}} \) of S is much smaller than the reported activity \( a_{\text{R, S}} \) of S in Fe-S binary melts with changing mole fraction \( x_{\text{S}} \) of S from 0.0 to 0.095. The calculated mass action concentration \( N_{\text{S}} \) of S can correlate the reliable 1:1 corresponding relationship with the reported activity \( a_{\text{R, S}} \) or \( a_{\%,\text {S}} \) of S through the introduced transformation coefficients with absolutely mathematical meaning or through the defined comprehensive mass action concentration of total S with explicitly physicochemical meaning. The calculated mass action concentrations \( N_{i} \) of structural units from the developed AMCT-N i thermodynamic model can be applied to describe or predict the reaction abilities of structural units in Fe-S binary melts. The reaction abilities of Fe and S show a competitive relationship each other in Fe-S binary melts in a temperature range from 1773 K to 1873 K (1500 °C to 1600 °C). The calculated mass action concentration \( N_{{{\text{FeS}}_{ 2} }} \) of FeS2 is very small and can be ignored because FeS2 can be incongruently decomposed above 1016 K (743 °C). The very small values for the calculated mass action concentrations \( N_{{{\text{FeS}}_{ 2} }} \) of FeS2 in a range of mole fraction \( x_{\text{S}} \) of S from 0.0 to 1.0 as well as a maximum value for the calculated mass action concentration \( N_{\text{FeS}} \) of FeS with mole fraction \( x_{\text{S}} \) of S as 0.5 are coincident with diagram phase of Fe-S binary melts. A spindle-type relationship between the calculated mass action concentration \( N_{i} \) and the calculated equilibrium mole number \( n_{i} \) can be found for FeS and FeS2 in Fe-S binary melts. The Raoultian activity coefficient \( \gamma_{S}^{0} \) of S relative to pure liquid S(l) as standard state and the infinitely dilute solution as reference state in Fe-S binary melts can be determined as 1.0045 in a temperature range from 1773 K to 1873 K (1500 °C to 1600 °C). The standard molar Gibbs free energy change \( \Updelta_{\text{sol}} G_{{{\text{m, S }}({\text{l}}) \to [{\text{S}}]_{{ \, [{\text{pct \, S}}] = 1.0}} }}^{{\Uptheta,\%}} \) of dissolving liquid S for forming [pct S] as 1.0 in Fe-S binary melts relative to 1 mass percentage of S as standard state can be formulated as \( \Updelta_{\text{sol}} G_{{{\text{m, S }}({\text{l}}) \to [{\text{S}}]_{{ \, [{\text{pct \, S] }} = \, 1.0}} }}^{{\Uptheta,\, \%}} \,\, = -0.219\,-\,33.70T\,\,\left( {\text{J/mol}} \right).\)

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Abbreviations

\( a_{i} \) :

Activity of element i or compound i (–)

\( a_{{{\text{R, }}i}} \) :

Activity of element i or compound i relative to pure liquid i as standard state with mole fraction \( x_{i} \) as concentration unit and following the Raoult’s law, i.e., \( a_{{{\text{R, }}i}} \, = x_{i} \, \gamma_{i}\)

\( a_{{\%,i}} \) :

Activity of element i referred to 1 mass percentage of element i as standard state with mass percentage \( [{\text{pct}} \, i] \) as concentration unit and obeying the Henry’s law under the condition of taking the infinitely dilute ideal solution as reference state, i.e., \( a_{\%,i}\,=\,[{\text{pct}} \,i]\, f_{\%, i} \)

\( b_{i} \) :

Mole number of element i in 100-g metallic melts, having the same meaning of \( n_{i}^{0} \) (mol)

\( f_{\%,i}\) :

Activity coefficient of element i in metallic melts related with activity \( a_{\%,i}\) (–)

\( \Updelta_{\text{fus}} G_{{\text {m}}, i({\text {s}})}^{\Uptheta} \) :

Standard molar Gibbs free energy change of fusing or melting element i or compound i without standard state (J/mol)

\( \Updelta_{\text{r}} G_{{\text{m}}, i}^{{\Uptheta ,{\text{ R}}}} \) :

Standard molar Gibbs free energy change of reaction for forming compound i based on activity \( a_{{{\text{R, }}i}} \) relative to pure liquid element i or compound i as standard state and following the Raoult’s law for reactants and products (J/mol)

\( \Updelta_{\text{r}} G_{{\text{m}}, i}^{{\Uptheta ,\%}} \) :

Standard molar Gibbs free energy change of reaction for forming compound i based on activity \( a_{\%,i} \) referred to 1 mass percentage of element i as standard state and obeying the Henry’s law for reactants and products (J/mol)

\( \Updelta_{\text{sol}} G_{{{\text{m}}, \, i \, ({\text{l}}) \to [i]}}^{{\Uptheta ,\%}} \) :

Standard molar Gibbs free energy change of dissolving liquid element i or compound i into metallic melts based on activity \( a_{\%,i} \) referred to 1 mass percentage of element i as standard state and obeying the Henry’s law (J/mol)

\( \Updelta_{\text {r}} G_{{{\text{m}}, \, i \, ({\text{l}})}}^{{\Uptheta ,{\text{ R}}({\text{l}})}} \) :

Standard molar Gibbs free energy change of reaction for forming liquid compound i from reactants in liquid based on pure matters as standard states for reactants and products (J/mol)

\( \Updelta_{\text {r}} G_{{{\text{m}}, \, i \, ({\text{l}})}}^{{\Uptheta ,{\text{ R(s)}}}} \) :

Standard molar Gibbs free energy change of reaction for forming liquid compound i from reactants in solid based on pure matters as standard states for reactants and products (J/mol)

\( \Updelta_{\text {r}} G_{{{\text{m}}, \, i \, ({\text{s}})}}^{{\Uptheta ,{\text{ R}}({\text{l}})}} \) :

Standard molar Gibbs free energy change of reaction for forming solid compound i from reactants in liquid based on pure matters as standard states for reactants and products (J/mol)

\( \Updelta_{\text {r}} G_{{{\text{m}}, \, i \, ({\text{s}})}}^{{\Uptheta ,{\text{ R(s)}}}} \) :

Standard molar Gibbs free energy change of reaction for forming solid compound i from reactants in solid based on pure matters as standard states for reactants and products (J/mol)

\( \Updelta_{\text{sol}} G_{{{\text{m}}, \, i \, ({\text{l}}) \to [i]}}^{{\Uptheta ,{\text{ R(l}})}} \) :

Standard molar Gibbs free energy change of dissolving liquid element i or compound i into metallic melts based on activity \( a_{{{\text{R, }}i}} \) relative to pure liquid element i or compound i as standard state and following the Raoult’s law (J/mol)

\( \Updelta_{\text{sol}} G_{{{\text{m}}, \, i \, ({\text{l}}) \to [i]}}^{{\Uptheta ,{\text{ R(s}})}} \) :

Standard molar Gibbs free energy change of dissolving liquid element i or compound i into metallic melts based on activity \( a_{{{\text{R, }}i}} \) relative to pure solid element i or compound i as standard state and following the Raoult’s law (J/mol)

\( \Updelta_{\text{sol}} G_{{{\text{m}}, \, i \, ({\text{s}}) \to [i]}}^{{\Uptheta ,{\text{ R(l}})}} \) :

Standard molar Gibbs free energy change of dissolving solid element i or compound i into metallic melts based on activity \( a_{{{\text{R, }}i}} \) relative to pure liquid element i or compound i as standard state and following the Raoult’s law (J/mol)

\( \Updelta_{\text{sol}} G_{{{\text{m}}, \, i \, ({\text{s}}) \to [i]}}^{{\Uptheta ,{\text{ R(s}})}} \) :

Standard molar Gibbs free energy change of dissolving solid element i or compound i into metallic melts based on activity \( a_{{{\text{R, }}i}} \) relative to pure solid element i or compound i as standard state and following the Raoult’s law (J/mol)

\( \Updelta_{\text{fus}} H_{{{\text{m}}, \, i \, ({\text{s}})}}^{\Uptheta } \) :

Standard molar enthalpy change of fusing or melting solid element i or compound i (J/mol)

\( K_{i}^{{\Uptheta ,{\text{ R}}}} \) :

Standard equilibrium constant of chemical reaction for forming compound i based on activity \( a_{{{\text{R, }}i}} \) relative to pure liquid or solid as standard state with mole fraction \( x_{i}^{{}} \) as concentration unit and following the Raoult’s law (–)

\( K_{i}^{{\Uptheta ,\%}} \) :

Standard equilibrium constant of chemical reaction for forming component i or compound i based on activity \( a_{\%,i} \) referred to 1 mass percentage of element i as standard state with mass percentage \( [{\text{pct}} \, i] \) as concentration unit and obeying the Henry’s law (–)

\( L_{{i, \, a_{{{\text{R}}, \, i}} \to N_{i} }}^{'} \) :

Transformation coefficient between \( a_{{{\text{R, }}i}} \) and \( N_{i}^{{}} \), defined as \( L_{{i, \, a_{{{\text{R}}, \, i}} \to N_{i} }}^{'} = a_{{{\text{R}}, \, i}} /N_{i} \,( - ) \)

\( L_{{i, \, a_{\%,i} \to N_{i} }}^{''} \) :

Transformation coefficient between \( a_{\%,i} \) and \( N_{i}^{{}} \), defined as \( L_{{i, \, a_{\%,i} \to N_{i} }}^{''} = a_{\%,i} /N_{i} \,( - ) \)

\( \overline{{L^{'} }}_{{i \, , \, a_{{{\text{R}}, \, i}} \to N_{i} }} \) :

Average value of transformation coefficient \( \overline{{L^{'} }}_{{i \, , \, a_{{{\text{R}}, \, i}} \to N_{i} }} \) at a fixed temperature (–)

\( \overline{{L^{''} }}_{{i \, , \, a_{\%,i} \to N_{i} }} \) :

Average value of transformation coefficient \( L_{{i, \,a_{\%,i} \to N_{i} }}^{''} \) at a fixed temperature (–)

\( n_{i}^{0} \) :

Mole number of element i in a 100-g metallic melts before reaction equilibrium, having the same meaning of \( b_{i} \) (mol)

\( n_{i} \) :

Equilibrium mole number of structural unit i in 100-g metallic melts based on the AMCT (mol)

\( \sum n_{i} \) :

Total equilibrium mole number of all structural units in 100-g metallic melts based on the AMCT (mol)

\( N_{i} \) :

Mass action concentrations of structural unit i in metallic melts based on the AMCT (–)

\( N_{ci} \) :

Mass action concentrations of compound ci in metallic melts based on the AMCT (–)

\( N_{i}^{'} \) :

Converted mass action concentration of structural unit i from \( \overline{{L^{'} }}_{{i \, , \, a_{{{\text{R}}, \, i}} \to N_{i} }} \), defined as \( N_{i}^{'} = \overline{{L^{'} }}_{{i \, , \, a_{{{\text{R}}, \, i}} \to N_{i} }} N_{i}^{{}} \,( - ) \)

\( N_{i}^{''} \) :

Converted mass action concentration of structural unit i from \( \overline{{L^{''} }}_{{i \, , \, a_{\%,i} \to N_{i} }} \), defined as \( N_{i}^{''} = \overline{{L^{''} }}_{{i \, , \, a_{\%,i} \to N_{i} }} N_{i}^{{}} \,( - ) \)

\( N_{{{\text{total\,}}i}}^{'} \) :

Defined comprehensive mass action concentration of the integrated structural units containing i or of total element i (–)

\( N_{{{\text{total\,}}i}}^{''} \) :

Converted mass action concentration of the integrated structural units containing element i, similar to activity \( a_{\%,i} \) of element i referred to 1 mass percentage of element i as standard state with mass percentage \( [{\text{pct}} \, i] \) as concentration unit and obeying the Henry’s law (–)

\( p_{i} \) :

Partial pressure of component i in gaseous phase (Pa)

\( p^{\Uptheta } \) :

Standard pressure of gas at sea level and 273 K (0 °C) as 101,325 Pa (Pa)

\( R \) :

Gas constant (8.314 J/(mol·K))

\( T \) :

Absolute temperature (K)

\( T_{{{\text{f}}, \, i \, ({\text{s}})}}^{ * } \) :

Melting point of element i or compound i (K)

\( x_{i} \) :

Mole fraction of element i or compound i in metallic melts (–)

\( x_{i}^{\Uptheta } \) :

Mole fraction of element i with mass percentage of element i as 1.0, having the same meaning with \( x_{{i, \, [{\text{pct }}\,i] = 1.0}}^{{}} \,( - ) \)

\( x_{{i, \, [{\text{pct }}\,i] = 1.0}}^{{}} \) :

Mole fraction of element i with mass percentage of element i as 1.0, having the same meaning with \( x_{i}^{\Uptheta } \,( - ) \)

[pct i]:

Mass percentage of element i or compound i in metallic melts (–)

i(g):

Element i or compound i in gaseous state (–)

i(l):

Element i or compound i in liquid state (–)

i(s):

Element i or compound i in solid state (–)

\( \gamma_{i} \) :

Activity coefficient of element i or component i related with activity \( a_{{\text{R}},i} \) (–)

\( \gamma_{i}^{0} \) :

Raoultian activity coefficient \( \gamma_{i}^{0} \) of element i or compound i in infinitely dilute metallic melts related with activity \( a_{{\text{R}},i}, \) equal to value \( \gamma_{i, x_i \to 0.0} \)(–)

\( \mu_{i} \) :

Standard chemical potential of element i or compound i (J/mol)

\( \mu_{{i \, ({\text{g}})}}^{ * } \) :

Chemical potential of element i or compound i as gas (J/mol)

\( \mu_{{i \, ({\text{l}})}}^{ * } \) :

Chemical potential of element i or compound i as liquid (J/mol)

\( \mu_{{i \, ({\text{s}})}}^{ * } \) :

Chemical potential of element i or compound i as solid (J/mol)

\( \mu_{[i]}^{\Uptheta } \) :

Standard chemical potential of dissolved element i or component i (J/mol)

\( \mu_{{i \, ({\text{l}})}}^{\Uptheta} \) :

Standard chemical potential of element i or component i relative to pure liquid i as standard state (J/mol)

\( \mu_{{i \, ({\text{s}})}}^{\Uptheta} \) :

Standard chemical potential of element i or component i relative to pure solid i as standard state (J/mol)

\( \mu_{{{\text{R}}, \, i}}^{\Uptheta } \) :

Standard chemical potential of element i or compound i based on activity \( a_{{{\text{R, }}i}} \) relative to pure liquid i as standard state and following the Raoult’s law (J/mol)

\( \mu_{{\%,{i}}}^{\Uptheta } \) :

Standard chemical potential of element i or compound i based on activity \( a_{{\%, \, i}} \) referred to 1 mass percentage of element i as standard state with mass percentage \( [{\text{pct}} \, i] \) as concentration unit and obeying the Henry’s law (J/mol)

ci :

Compound i or molecule i (–)

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Acknowledgments

This work is supported by a grant from the National Natural Science Foundation of China (No. 51174186). The sincere thanks are also extended to Prof. Chang-xiang Xiang, on metallurgical physicochemistry at the School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, for valuable discussion and helps for preparing Section III–C on deciding the standard molar Gibbs free energy change of related reactions.

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Author notes

  1. Meng Zhang (Master Degree Student)

    Present address: Beijing Metallurgical Equipment Research Design Institute Company Limited, China Metallurgical Group Corporation, Beijing, 100029, P.R. China

Authors and Affiliations

  1. State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing, 100190, P.R. China

    Xue-Min Yang (Research Professor)

  2. Institute of Process Engineering, Chinese Academy of Sciences, Beijing, P.R. China

    Meng Zhang (Master Degree Student), Peng-Cheng Li (Ph.D. Candidate and Joint-Training Student) & Jin-Yan Li (Master Degree Student and Joint-Training Student)

  3. State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing, 100083, P.R. China

    Peng-Cheng Li (Ph.D. Candidate and Joint-Training Student) & Jin-Yan Li (Master Degree Student and Joint-Training Student)

  4. School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, Beijing, P.R. China

    Jian-Liang Zhang (Professor) & Jian Zhang (Professor Emeritus)

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Manuscript submitted April 10, 2012.

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Yang, XM., Zhang, M., Li, PC. et al. A Thermodynamic Model for Representation Reaction Abilities of Structural Units in Fe-S Binary Melts Based on the Atom-Molecule Coexistence Theory. Metall Mater Trans B 43, 1358–1387 (2012). https://doi.org/10.1007/s11663-012-9707-6

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  • DOI: https://doi.org/10.1007/s11663-012-9707-6

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