Estimation of missing third-law standard entropy of apatite supergroup minerals using the optimized Volume-based Thermodynamics

The thermodynamic characterization of apatite minerals, critical for understanding geological processes and material applications, faces significant challenges due to the scarcity of experimental data, particularly standard entropy (S°) values. In this study, we address this gap by optimization of predictive method based on Volume-based Thermodynamics. In the proposed method, the optimization of the widely used Volume-based Thermodynamics is based on breaking down a single linear functional relationship of formula unit volume (V_m) with S° into a set of linear equations. The apatite supergroup splits into distinct subgroups (populations) formed by Me₁₀(AO₄)₆X₂ with the same Me²⁺ cations and tetrahedral AO₄³⁻ anions but with different anions at the X position. Our approach leverages empirical correlations between V_m and S° within specific apatite subgroups. By analyzing the correlations within the subgroups, we established the system of precise linear relationships between S° and V_m, facilitating accurate S° predictions for a wide range of apatite compositions. The proposed approach represents a significant advancement over existing predictive methods offering unparalleled accuracy in estimating S° values for apatite minerals. Through rigorous regression analysis and validation against experimental data, we demonstrate the reliability and robustness of our predictive model across various apatite subgroups. Our findings provide crucial thermodynamic data for understudied apatite compositions and shed light on fundamental relationships between crystal structure and thermodynamic properties in apatite minerals. The precise estimation of S° values enables more accurate modeling of phase equilibria, reaction kinetics, and geological processes involving apatite minerals, facilitating advancements in diverse fields ranging from environmental geochemistry to material science.