Rate Equations and an Ion-pair Mechanism for Batch Dissolution of Gypsum: Repositioning the Shrinking Object Model at the Core of Hydrodynamic Modelling

Abstract

Recent improvements to experiments and modelling of batch dissolution in a turbulent reactor, based upon the shrinking object model, are extended to middle loadings of gypsum, that is, in the region between low and high loadings, which lead, respectively, to high under-saturation or saturation with a great excess of solid left undissolved. Dissolved calcium sulphate concentration was monitored by change in electrical conductivity. This investigation uses an improved, ion-pair model for CaSO ⁰₄ to allow for the presence of calcium or sulphate added as common ions. The study demonstrates that the full dissolution curve for 5.82 mM loadings of 106-μm particles of gypsum (~1.00 g L⁻¹) in de-ionised water barely changed in the presence of either 4.64 or 8.09 mM calcium chloride, or 4.39 mM sodium sulphate. However, this masked a doubling of dissolution rate imposed by comparable increases in ionic strength from sodium chloride. The results are consistent with the ion pair, CaSO ⁰₄ , being the key species in the rate-determining step of the back-reaction, and perhaps all salt dissolutions, including calcium carbonate. In this case, the rate equation is as follows: ${\frac{{{\text{d}}c}}{{{\text{d}}t}}} = \frac{S}{V} \cdot (k_{1} - k_{2}^{\prime } \cdot [{\text{CaSO}}_{ 4}^{0} ])$, where k ₁ and k ₂′ are rate constants. The reported observations are interpreted as effects of ionic strength and common ion concentrations upon the formation equilibrium for the ion pair. This rate equation readily transforms mathematically to one involving the product of [Ca²⁺] and [SO₄ ²⁻] in the back-reaction. The parallel of this with the well-known PWP equation used in calcium carbonate dissolution is discussed, with the CaHCO₃ ⁺ ion pair of the equation being replaced by that of CaCO ⁰₃ . Meanwhile, the earlier use of the product, [Ca²⁺]^½ × [CO₃ ²⁻]^½, in the back-reaction term of another dissolution rate equation for calcite is shown to be incorrect. Finally, it is argued that the shrinking object model should be repositioned as a logical derivative of the hydrodynamical approach to dissolution.