Amorphous calcium carbonate (ACC) precipitation is modeled using particle nucleation, growth, and aggregation. The particles are tracked in terms of their radial size and particle density using direct quadrature method of moments (DQMOM). Four separate nucleation models are implemented and are compared to experimental data. In discord with a recent study, it is shown that classical nucleation, coupled with equilibrium chemistry, is in good agreement with experimental data. Novel nucleation mechanisms are presented which fit the experimental data with slightly greater accuracy. Using equilibrium chemistry it is shown that the equilibrium value of ACC is pKeq = 7.74 at 24C, which is a factor of two smaller than the originally published equilibrium constant. Additionally, legacy equilibrium chemistry expressions are shown to accurately capture the fraction of calcium carbonate ions formed into ACC nano-clusters. The density, solubility, and water content of ACC are discussed in a brief review, finding that a wide variety of properties are reported in the literature. Based on literature findings, it is proposed that the broad variety of reported properties may be due to ACC having several unique thermodynamic states. Compelling evidence is presented exposing errors made by experimentalists studying the calcium carbonate system. The errors correct for mistakes of experimental kinetic data of the chemical-potential cascade of calcium carbonate due to the formation of meta-stable phases. Correlations are presented which correct for these mistakes. A time-scale analysis shows the overlapping of kinetic scales and mixing scales within the calcium carbonate system. The kinetic scales are based on classical nucleation theory, coupled with diffusion limited growth. The mixing scales were computed using one-dimensional turbulence (ODT).