Assessment of the thermodynamic activity and solid solubility for solute element in multi-component alloy is critical in thermodynamics, but an effective first-principles estimation of the parameter has been an unsolved issue for many years. By taking the excess Mg in Mg₃(Sb_1-xBi_x)₂ alloy as an example, Mg atoms could occupy the regular lattice sites and the interstitial sites as well, which makes the estimation of thermodynamic activity of Mg solute not easy. In this work, we develop a model to estimate the activity and solid solubility of Mg in alloys such as Mg₃(Sb_1-xBi_x)₂ by considering the formation energy contribution from both Mg interstitials and Mg vacancies, and the associated entropies to final Gibbs energy. Accurate assessment of the parameter by using first-principles thermodynamics approach is also realized. The estimated activity of Mg in Mg₃Bi₂ and Mg₃Sb₂ reproduce well the available experimental results. The solubility limits of Mg and equilibrium concentrations of Mg interstitials and Mg vacancies in Mg₃(Sb_1-xBi_x)₂ are also estimated. The results could be used to understand Mg–Mg₃Sb₂–Mg₃Bi₂ phase diagram as well as the Mg effect on thermoelectric performance of Mg₃(Sb_1-xBi_x)₂ alloy.

评估多组分合金中溶质元素的热力学活性和固溶度在热力学中是至关重要的，但多年来对该参数的有效第一性原理估计一直是一个悬而未决的问题。以Mg ₃ (Sb _{1- x} Bi _x ) ₂合金中过量的Mg为例，Mg原子既可以占据规则的晶格位置，也可以占据间隙位置，这使得对Mg溶质的热力学活性的估计并不容易。在这项工作中，我们开发了一个模型来估计镁在合金中的活性和固溶度，例如 Mg ₃ (Sb _{1- x} Bi _x ) ₂通过考虑 Mg 间隙和 Mg 空位的形成能贡献，以及相关的熵对最终吉布斯能量的影响。还实现了使用第一性原理热力学方法对参数的准确评估。Mg ₃ Bi ₂和Mg ₃ Sb ₂中Mg的估计活性很好地再现了可用的实验结果。还估计了 Mg 的溶解度极限以及 Mg ₃ (Sb _{1- x} Bi _x ) ₂中Mg 间隙和 Mg 空位的平衡浓度。结果可用于理解 Mg-Mg ₃ Sb ₂ -Mg ₃ Bi图₂相图以及Mg对Mg ₃ (Sb _{1- x} Bi _x ) ₂合金热电性能的影响。