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

15 September 2023, Version 2
This content is a preprint and has not undergone peer review at the time of posting.


In the reaction of calcium phosphate precipitation from 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 report an extension of our previous study on the relationship between a precipitate and the corresponding ionic product in pseudo-equilibrium states. 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 multiple structural units in a precipitate particle and correlate the ionic product with the surface proportion of a structural unit (m). By means of a particle-surface equation, we further related the surface proportion to the whole particle. Notably, the two types of pseudo-equilibrium constants have the common expression of “Kd = ionic product” if the number of the structural units (u) gets large enough. The concept of the multi-unit particle may shed new light on the study of precipitation reactions of other slightly soluble salts. Moreover, 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 the non-equilibrium systems in nature and biology, such as marine sedimentation and human vascular calcification.


surface proportion
structural unit
ionic product
non-equilibrium thermodynamics
amorphous calcium phosphate
bond-type transition
solution chemistry.


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Comment number 1, Tian-Lan ZHANG: Jan 03, 2024, 07:06

This paper has been published: Zhang, T.-L., Pseudo-equilibrium equations for calcium phosphate precipitation with multi-unit particles. Physical Chemistry Chemical Physics 2024, 26 (1), 581-593.