Consider the $($pseudo$)$steady state, radial diffusion of oxygen from air to the surface of a

spherical particle of coal having initial radius $r_i=1mm.$ At the surface of the particle,

oxygen gas reacts instantaneously with solid carbon $($C$)$ in the coal to form carbon

monoxide gas $(CO)$ and carbon dioxide $(C{O}_2)$ gas according to the heterogeneous gas

reaction equation

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

The product gases back diffuse through the gas film adjacent to the coal. As the coal

particle is oxidized the particle shrinks with time. Assuming that far away from the surface

$(r=)$ of the coal particle the mole fraction of oxygen is $0\mathrm{.}21$ derive expressions for radial

distribution of oxygen concentration and rate of transfer of oxygen per unit time at steady

state. The diffusion coefficient of oxygen through the gas mixture at the reaction

temperature is $1\mathrm{.}5{10}^{-4}\frac{m^2}{s}.$

How long will it take to reduce the size of the particle from initial radius of $r_i=1mm$ to final

radius $r_f=0\mathrm{.}2mm?$ The density of carbon, $=2000k\frac{g}{m^3}$