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 solution compositions 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 containing multiple structural units, only a fraction of the units was able to reach pseudo-equilibrium with the solution composition. This fraction, with its value being determined experimentally, represents the average number of surficial clusters per unit. These findings enabled us to propose a general form of equilibrium constant equation. The equation fits the pseudo-equilibrium data well, and it converts to the “solubility product (Ksp)” and conventional “reaction quotient” in two limit cases, respectively. Further, using a cube model, we derived a “particle equation” that reveals the connection between certain particle parameters and the equilibrium constant equation.