Example of a simultaneous diffusion and a heterogeneous reaction process is the diffusion controlled combustion of coal in a fluidized bed to produce energy by the heat of combustion. Pulverized coal particles are fluidized within a hot combustion chamber, where oxygen in the air reacts with coal to produce carbon monoxide and$/$or carbon dioxide gas.

Consider the steady$-$state, one$-$dimensional diffusion of oxygen to the surface of a spherical particle of coal along the $r$ coordinate. At the surface of the particle, oxygen gas $(O_2)$ reacts with solid carbon $(C)$ in the coal to form carbon monoxide gas $(CO)$ and carbon dioxide $(C{O}_2)$ gas according to the heterogeneous reaction equation:

$3C+2\mathrm{.}5O_2(g)2C{O}_2(g)+CO(g)$

as illustrated in the figure shown below

Diffusion through a spherical film

As the coal particle is oxidized, the particle shrinks with time as the carbon is converted to carbon monoxide and carbon dioxide.

Required:

Derive an expression to predict the moles of oxygen transferred per unit time in terms of the diffusion coefficient and particle radius