Relation of total volume to total surface area for a suspension of particles undergoing dissolution

Abstract

For a suspension of spherical particles with initial radii distributed either normally or lognormally, undergoing dissolution at rates proportional to their surface areas, expressions are derived for the total number (N) of particles, the sum (R) of their radii, their total surface area (A) and total volume (V), in terms of a parameter θ which is a linear function of time and also involves the initial mean value (r₀) and initial standard deviation (σ) of the radii, as well as the dissolution rate constant. Simple approximate expressions are also derived, valid for a range of θ corresponding to dissolution of up to 95% of the particles when r₀/σ is assigned the plausible value 4. For this range and r₀/σ value, it is shown that V∝A^1·343(error in V within 1·9%) for the normal distribution, compared with V∝A^1·5 for a monodisperse suspension. For the initial lognormal distribution, and the same r₀/σ and range of θ, a similar relation holds provided the the skewness coefficient γ is not too large. For γ= 0·778 and 1·75, the exponent of A is 1·284 and 1·219 respectively (error in V within 3·6 and 4·3% respectively). It is suggested that the applicability of these results is not limited to spherical particles.