# Diffusion-controlled leaching

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The reactants are diffusing through a porous layer.
Concentration of reactant at surface = Ci

When the chemical reaction on the surface is much faster than the diffusion, leaching becomes diffusion-controlled. In this case the reagent concentration at the surface becomes zero, i.e. Ci=0. The leaching mechanism might become diffusion-controlled when, during leaching, a porous product layer forms on the surface of the particle to be leached. This can for example happen in the case of leaching of sulfides where a layer of elemental sulfur can be deposited on the sulfide surface. The mechanism of diffusion-controlled leaching of a spherical particle is often called the shrinking core model.

With the same assumptions as for the shrinking particle model i.e. that the reagent concentration is constant and that spherical and equal sized particles are leached, an expression for diffusion controlled leaching can be arrived at by applying Fick’s law.

$1-2/3\alpha-(1-\alpha)^{2/3} = \frac {2 \cdot M \cdot D \cdot C}{\beta \cdot \rho \cdot r_0^2}\cdot t$
Where:
α = fraction leached
β = stoichiometric factor
M = molecular weight of leached mineral
ρ = density of particle
t = time of leaching
C = concentration of reagent
D = diffusion constant (gram/cm2 or mole/cm2)
r0 = initial radius of particle at time zero

As is evident from the equation, the leach rate is inversely proportional to the square of the radius of the particle. The diffusion constant can be determined by plotting the left hand side against time in a diagram. Given the assumptions that C is constant and that volume changes has not been taken into account, this model is accurate until 80-90% has been leached out.

Schematic image of leaching according to the shrinking core model.