Pseudo-equilibrium equations for calcium phosphate precipitation with multi-unit particles

The chemistry underlying bone mineral formation in vertebrates is the reaction of calcium phosphate precipitation. In a near-neutral solution, an amorphous phase and hydroxyapatite nanoparticles appear successively, and the reaction system containing either of the two kinds of precipitates is in a non-equilibrium state. Here, we propose a pseudo-equilibrium approach to the solution chemistry of the precipitation reactions. We employed two series of reaction systems, collected samples at various stages, and analyzed the solution chemistry data on the basis of a simplified model of reaction. We derived two types of pseudo-equilibrium equations from the two series, respectively. These equations reveal the existence of multiple structural units in a precipitate particle and correlate the ionic product with the surface proportion per structural unit ( m ). The surface proportion, in turn, is related to the whole particle through a particle-surface equation. Notably, the two types of pseudo-equilibrium constants have the common expression of " K d = ionic product" if the number of the structural units ( u ) is large enough. Together, these findings have revealed some aspects of the non-equilibrium thermodynamics of precipitation reactions, indicating the solution chemistry route to the equilibrium state. The concept of the multi-unit particle may shed new light on the study of precipitation reactions of other slightly soluble electrolytes. And the relationship between the ionic product and the surface proportion of a structural unit is not only fundamental in chemistry, but may also apply to non-equilibrium systems in nature and biology, such as marine sedimentation, human vascular calcification, and bone mineral metabolism. Using a pseudo-equilibrium approach, we have derived two types of pseudo-equilibrium equations that reveal the multiple structural units in a precipitate particle and relate the ionic product to the surface proportion of a structural unit.