A solid sphere $($radius $R)$ of substance $A$ is suspended in a liquid $B$ in which it is slightly soluble $($solubility is $c_{A0}),$ and with which it undergoes a first$-$order chemical reaction with rate constant $k_1.$ At steady state, the diffusion is exactly balanced by the chemical reaction.

$($a$)$ Derive an expression for the concentration profile of $A$ in the diffusion boundary layer around the solid sphere $($i$.$e$.,$ find $c_A$ as a function of $r).$

$($b$)$ Derive expressions for the molar flux at the sphere surface, $N_{AR},$ as well as the molar transfer rate, $W_{AR}.$

$($c$)$ Show by quasi$-$steady$-$state arguments how to calculate the gradual decrease in diameter of the sphere as A dissolves and reacts. Show that the radius of the sphere is given by:

$\sqrt[\phantom{2}]{\frac{k_1}{D_{AB}}}(R-R_0)-ln(\frac{1+R\sqrt[\phantom{2}]{\frac{k_1}{D_{AB}}}}{1+R_0\sqrt[\phantom{2}]{\frac{k_1}{D_{AB}}}})=-\frac{k_1{M}_A{C}_{A0}}{{}_s}(t-t_0).$

$1$

where $R_0$ is the sphere radius at time $t_0,M_A$ is the molecular mass of A$,$ and ${}_S$ is the density of the sphere.