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Critical assessment of three kinds of activity coefficients of carbon and related mixing thermodynamic functions of Fe–C binary melts based on atom–molecule coexistence theory

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Abstract

Raoultian activity coefficients \({\gamma}_{\text{C}}^{0}\) of C in infinitely dilute Fe–C binary melts at temperatures of 1833, 1873, 1923, and 1973 K have been determined from the converted mass action concentrations \(N_{\text{C}}^{\prime}\) of C in Fe–C binary melts by the developed AMCT-N i model based on the atom–molecule coexistence theory (AMCT). The obtained expression of \(\gamma_{\text{C}}^{0}\) by the developed AMCT-N i model has been evaluated to be accurate based on the reported ones from the literature. Meanwhile, three activity coefficients \(\gamma_{\text{C}}\), \(f_{{\%,{\text{ C}}}}\), and \(f_{\text{H, C}}\) of C coupled with activity \(a_{\text{R, C}}\) or \(a_{{\%,{\text{ C}}}}\) or \(a_{\text{H, C}}\) have been obtained by the developed AMCT-N i model and assessed through comparing with the predicted ones by other models from the literature. The first-order activity interaction coefficients \(\varepsilon_{\text{C}}^{\text{C}}\), \(e_{\text{C}}^{\text{C}}\), and \(h_{\text{C}}^{\text{C}}\) related to \(\gamma_{\text{C}}\), \(f_{{\%,{\text{ C}}}}\), and \(f_{\text{H, C}}\) are also determined and assessed in comparison with the reported ones from the literature. Furthermore, the integral molar mixing thermodynamic functions such as \(\Delta_{\text{mix}} H_{{{\text{m, Fe}} - {\text{C}}}}\), \(\Delta_{\text{mix}} S_{{{\text{m, Fe}} - {\text{C}}}}\), and \(\Delta_{\text{mix}} G_{{{\text{m, Fe}} - {\text{C}}}}\) of Fe–C binary melts over a temperature range from 1833 to 1973 K have been determined and evaluated to be valid based on the determined ones from the literature.

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Abbreviations

a i :

Activity of element i;

a R,i :

Activity of element i or compound i relative to pure liquid i, saturated liquid or solid i, pure solid i or diatomic gas i2 as standard state with mole fraction x i as concentration unit and following Raoult’s law under condition of taking ideal solution as reference state, i.e., \(a_{{{\text{R, }}i}} = \gamma_{i} x_{i}\);

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

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

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

Activity of element i relative to hypothetical pure liquid i as standard state with mole fraction x i as concentration unit and following Henry’s law under condition of taking infinitely dilute ideal solution as reference state, i.e., \(a_{{{\text{H, }}i}} = f_{{{\text{H, }}i}} x_{i}\);

e i i :

First-order activity interaction coefficient of element i in metallic melts related to activity \(a_{{\%, i}}\);

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

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

\(f_{{{\text{H, }}i}}\) :

Activity coefficient of element i in metallic melts coupled with activity \(a_{{{\text{H, }}i}}\);

\(f_{{\%,i}}^{0}\) :

Henrian activity coefficient of element i in infinitely dilute metallic melts related to activity \(a_{{\%,i}}\), equal to value of \(f_{{\%,i ,w_{{\text [} i {\text ]}} \to 0.0}}\);

\(f_{{{\text{H, }}i}}^{0}\) :

Henrian activity coefficient of element i in infinitely dilute metallic melts coupled with activity \(a_{{{\text{H, }}i}}\), equal to value of \(f_{{{\text {H}}, i,} {x_{i} \to 0.0}}\);

h i i :

First-order activity interaction coefficient of element i in metallic melts related to activity \(a_{{{\text{H, }}i}}\);

\(\Delta _{\text{mix}} G_{{{\text{m, }}i}}\) :

Partial molar mixing Gibbs free energy change of element i in a real solution (J/mol);

\(\Delta_{\text{mix}} G_{{{\text{m, Fe}} - {\text{C}}}}\) :

Molar mixing Gibbs free energy change of Fe–C binary melts (J/mol);

\( \Delta_{\text{sol}} G_{{{\text{m}},i ( {\text{l)}} \to [ {{i]}}_{{ \, w_{[i]} = 1.0}} }}^{{\Theta , {\text{ \% }}}} \) :

Standard molar Gibbs free energy change of dissolving pure liquid element i for forming w[i] as 1.0 in metallic melts based on activity \({a_{{\%,}{i}}}\) referred to 1 mass% of element i as standard state and obeying Henry’s law (J/mol);

\(\Delta_{\text{mix}} H_{{{\text{m, }}i}}\) :

Partial molar mixing enthalpy change of element i in a real solution (J/mol);

\(\Delta_{\text{mix}} H_{{{\text{m, Fe}} - {\text{C}}}}\) :

Molar mixing enthalpy change of Fe–C binary melts (J/mol);

i(l):

Element i in liquid state;

M i :

Relative atomic mass of element i or relative molecular mass of component i;

N i :

Mass action concentrations of structural unit i in metallic melts based on AMCT;

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

Converted mass action concentrations of structural unit i in metallic melts based on AMCT;

R :

Gas constant (8.314 J/(mol K));

\(\Delta_{\text{mix}} S_{{{\text{m, }}i}}\) :

Partial molar mixing entropy change of element i in a real solution (J/(mol K));

\(\Delta_{\text{mix}} S_{{{\text{m, Fe}} - {\text{C}}}}\) :

Molar mixing entropy change of Fe–C binary melts (J/(mol K));

T :

Absolute temperature (K);

x i :

Mole fraction of element i in metallic melts;

w [i] :

Mass percentage of element i in metallic melts (%);

γ i :

Activity coefficient of element i in metallic melts related to activity \(a_{{{\text{R, }}i}}\);

γ 0 i :

Raoultian activity coefficient of element i in infinitely dilute metallic melts related to activity \(a_{{{\text{R, }}i}}\), equal to value of \(\gamma_{{i, x_{i} \to 0.0}}\);

ɛ i i :

First-order activity interaction coefficient of element i in metallic melts related to activity \(a_{{{\text{R, }}i}}\);

ci :

Molecule i or compound i.

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Acknowledgements

This work is supported by the Beijing Natural Science Foundation under Grant No. 2182069 and the National Natural Science Foundation of China under Grant No. 51174186.

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Authors and Affiliations

  1. CAS Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences, Beijing, 100190, China

    Xue-min Yang, Fang-jia Yan & Dong-ping Duan

  2. Department of Metallurgy and Raw Materials, China Metallurgical Industry Planning and Research Institute, Beijing, 100711, China

    Jin-yan Li

  3. School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Jian Zhang

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  1. Xue-min Yang
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Correspondence to Xue-min Yang.

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Yang, Xm., Li, Jy., Yan, Fj. et al. Critical assessment of three kinds of activity coefficients of carbon and related mixing thermodynamic functions of Fe–C binary melts based on atom–molecule coexistence theory. J. Iron Steel Res. Int. 25, 181–199 (2018). https://doi.org/10.1007/s42243-018-0019-1

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