Pseudo-equilibrium equation of calcium phosphate precipitation from aqueous solution

An X-ray amorphous phase is frequently present at the early stage of calcium phosphate crystallization, and the relevant solution chemistry is essential for understanding the mechanism of reaction. Here, we report a quantitative study of a series of reaction systems at pseudo-equilibrium states. We determined the composition of solutions and the quantities of the precipitate samples, and characterized the long- and short-range order of the precipitate using X-ray diffraction and synchrotron X-ray absorption near-edge structure spectroscopy, respectively. We found that, in a particle with multiple structural units, only a fraction of the units was able to reach pseudo-equilibrium with the solution composition, which represents the average number of surficial clusters per unit. These findings enabled us to propose a general form of the equilibrium constant equation. The equation fits the pseudo-equilibrium data well, and it converts to the "solubility product (Ksp)" and the conventional "reaction quotient" in two limit cases, respectively. Further, using a cube model, we derived a "particle equation" that reveals the connection between the particle structure and the form of equilibrium constant equation. The dependency of the form of pseudo-equilibrium equation on the structure and size of the precipitate reveals a fundamental relation in chemistry, and its applicability remains to be examined in other reaction systems, such as those involving nanocrystals and porous materials.