Shrinking Sphere Kinetics for Batch Dissolution of Mixed Particles of a Single Substance at High Under-Saturation: Validation with Sodium Chloride, but with Biogenic Silica in Mind

The cubic equation recently derived for the increase in concentration of a solute with time, as the solid dissolves in batch according to the shrinking sphere model at high under-saturation, is extended to dissolutions of mixtures of differently sized particles. This problem needs to be solved if batch dissolutions are to play their part in the proposed amelioration of global warming and associated climate change by accelerated ‘re-burial’ of excess CO₂ in ocean sediment. The upgraded model was tested using sodium chloride dissolved in 50% aqueous propanone, whence the model fitted two separate runs with 500 and 212 μm, and 212 and 38 μm, diameter crystals, respectively. The key to simulating dissolution in this way lies in the dissolutions being independent of each other. It is further shown that although this condition was implicit in the recent derivation of the cubic equation, it was not recognised at the time. The work should be applicable to any batch dissolution of mixed particles at high under-saturation, and hence, may find use in many industrial and laboratory dissolutions. Simulations show how agglomerated mixtures can yield a straight line on the plot of ln(1 − C/C _T) versus time, as was reported to occur recently with sodium chloride taken ‘straight from the bottle’. It is shown that this probably explains why exponential dissolutions may have seemed appropriate to the dissolution of biogenic silica in earlier literature. This study suggests that a new round of biogenic silica dissolutions, but with sieved samples, would be worthwhile, with the likelihood that shrinking sphere behaviour might well be found to characterise the kinetics. The opportunity is taken to investigate a number of aspects of the shrinking sphere model not generally discussed before, e.g. the graph for the change in surface area with time. The limitations of using cubic salt crystals with the shrinking sphere model are discussed.     

Sucrose Dissolution Studies Leading to a Generic Shrinking Object Model for Batch Dissolution of Regular-Shaped Particles

The dissolution of sieved sucrose crystals has been studied spectrophotometrically by observing the increase in dissolved sucrose concentration with time. Equations recently derived from the shrinking sphere model for the batch dissolution of a solid in under-saturated conditions tested successfully on both single crystal-size and mixtures of two sizes of sucrose crystals. Single-sized crystals provided a straight line for the plot of the fraction of un-dissolved solid to the power one-third, versus time ( vs. t). The dissolution of mixtures of two crystal sizes fitted the non-linear equation tested earlier on sodium chloride in water-propanone mixtures. Together, these two sets of tests on ionic and covalent substances verify that many simple dissolutions will be easily modelled using this physical model based on shrinkage, where the chemical composition of the solids is very much of secondary importance. Consequently, there is an increased chance that the equations will describe the dissolution of biogenic silica in seawater, the problem which originally inspired this study. More than this, though, the equations are discovered to be mathematically generic; very many geometries other than the sphere satisfy the same equations, and the “shrinking object dissolution model” is thereby defined. The approach should also apply even to non-aqueous dissolutions. A prototype plot of shrinking object rate constant (obtained from numerical fitting of the model to sucrose) versus particle size is presented, and it is shown how analogous treatments for other substances will be central to collection and use of much dissolution data in the future. The study is placed in context with much earlier solid phase decomposition studies, concluding that the key characteristic of the simplest of all dissolutions is that the interface between solid and liquid should advance at a uniform linear rate. It is shown how this approach leads to equations of the same mathematical forms already discussed above.     

Silica Gel as a Surrogate for Biogenic Silica in Batch Dissolution Experiments at pH 9.2: Further Testing of the Shrinking Object Model and a Novel Approach to the Dissolution of a Population of Particles

Recently, the increase in dissolved concentration in the batch dissolution of various salts or sucrose has been successfully modelled with three equations, one a cubic in time. However, from three separate earlier investigations with ocean sediments and phytoplankton frustules, there is residual suspicion that biogenic silica does not follow this behaviour. This paper shows that the Shrinking Object Model applies to the dissolution of sieved silica gel particles, as well as to a sample of unsieved, freeze-dried frustules of Odentella sp. Silica gel, being readily available in quantities that can be sieved, is a useful surrogate for biogenic silica in allowing problems of experimental design to be overcome. The dissolutions covered three possible analytic integrations that arise from the model: an exponential for approach to saturation with excess solid silica; the approach to near saturation with either a slight excess or deficiency of silica; dissolution at high under-saturation. Good agreement was found between experimental results and mathematical modelling. The paper provides template calculations by which future raw results can be parameterized. Nevertheless, the reasons for non-linear kinetics reported in earlier work have not been identified, and so controversy over non-linear dissolution kinetics is enhanced. Stirring regime was found to be important with silica gel dissolution, and so biogenic silica dissolution is therefore likely to be ‘transport limited’ at low stirring rate. Accordingly, all archived and future data should be scrutinized for stirring effects before being applied to the oceanic environment. A rigorous test for determining whether a substance’s dissolution deviates from the model is recommended as a preliminary to any future dissolutions, whether in batch or with the chemo-stat. A fixed amount of frustule sample is added to a series of buffered mixtures containing increasing background silicic acid concentrations. Absence of any problem is marked by a linear plot between the increase in silicic concentration accruing over a fixed reaction period and that of the background silicic acid. A novel mathematical proof is provided to justify the test’s use. The reasons for the earlier deviations from expected behaviour of, for example, oceanic sediments, are discussed. Lastly, the paper provides a novel approach to the dissolution of a population of particles of mixed sizes which will probably find ready future application in oceanography.