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

21 June 2023, Version 1
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 ion product in pseudo-equilibrium states. We adopted two reaction series, collected samples at various stages, and analyzed the solution chemistry data on the basis of a simplified reaction model. We derived two types of pseudo-equilibrium equations from the two series, respectively. These equations reveal the presence of multiple structural units in a precipitate particle and correlate the ion product with the average value of the surface proportion per structural unit (m). The m value decreased as the number of structural units in a particle (u) increased. Besides, we determined the free energy differences (ΔG) from the pseudo-equilibrium constants (Kd), which explains the observations in the crystal ripening process well. Notably, the two types of pseudo-equilibrium equations take the common form of “Kd = ion product” if u→∞. Overall, our findings provide new insights into the study of precipitation reactions. The relationship between the ion product and the structural unit of a precipitate particle is not only fundamental in chemistry, but may also be applicable to certain natural or biological systems at non-equilibrium, such as marine sedimentation and human vascular calcification.


pseudo-equilibrium equation
amorphous calcium phosphate
structural unit
bond-type transition
solution chemistry
non-equilibrium thermodynamics


Comments are not moderated before they are posted, but they can be removed by the site moderators if they are found to be in contravention of our Commenting Policy [opens in a new tab] - please read this policy before you post. Comments should be used for scholarly discussion of the content in question. You can find more information about how to use the commenting feature here [opens in a new tab] .
This site is protected by reCAPTCHA and the Google Privacy Policy [opens in a new tab] and Terms of Service [opens in a new tab] apply.