Thermodynamic models are employed to describe the equilibrium between nonstoichiometric carbonitride and the Fe-based solid solution. The solubility of fcc Ti and fcc V in the Fe-based solid solution was developed separately. Values for four thermodynamic interaction parameters of the nonstoichiometric carbonitride were determined using equilibrium equations, by comparing the calculated results with previously published experimental data. Specific solubility product expressions for \({\text{TiC}}_{x}{\text{N}}_{y}\) and \({\text{VC}}_{x}{\text{N}}_{y}\) in austenite (containing Mn, Ni, Cr, and Mo as solid solution elements) and ferrite (containing Mn and Ni as solid solution elements) were then developed. The calculated results in this study show good agreement with both calculated results from ThermoCalc and experimental data from earlier references, demonstrating their reliability. The developed solubility products of \({\text{TiC}}_{x}{\text{N}}_{y}\) and \({\text{VC}}_{x}{\text{N}}_{y}\) are as follows (solubility products of \(\text{TiC}\) , \(\text{TiN}\) , \(\text{VC}\) , and \(\text{VN}\) can be referenced from our previous work).
\(\text{log}{^{\alpha /\upgamma }K}_{{\text{TiC}}_{ x}{\text{N}}_{y}}=x\text{log}{^{\alpha /\upgamma }K}_{\text{TiC}}+y\text{log}{^{\alpha /\upgamma }K}_{\text{TiN}}+\left(1-x-y\right)\text{log}{^{\alpha /\upgamma }K}_{\text{Ti}}+x\text{log}x+y\text{log}y+\left(1-x-y\right)\text{log}\left(1-x-y\right)+xy\frac{121}{T}-x\left(1-x-y\right)\frac{4709}{T}-y\left(1-x-y\right)\frac{2093}{T},\)
\(\text{log}{^{\alpha /\upgamma }K}_{{\text{VC}}_{ x}{\text{N}}_{y}}=x\text{log}{^{\alpha /\upgamma }K}_{\text{VC}}+y\text{log}{^{\alpha /\upgamma }K}_{\text{VN}}+\left(1-x-y\right)\text{log}{^{\alpha /\upgamma }K}_{\text{V}}+x\text{log}x+y\text{log}y+\left(1-x-y\right)\text{log}\left(1-x-y\right)+xy\frac{13}{T}-x\left(1-x-y\right)\frac{3663}{T}-y\left(1-x-y\right)\frac{2093}{T}\) .
