Consider cylindrical$-$shaped absorbent material designed to selectively remove solute A from a solution. The uptake of solute A through the homogeneous material is limited by molecular diffusion. Furthermore, the absorbent has a higher affinity for solute A relative to the surrounding solution as described by the linear equilibrium relationship

$c_A=\frac{O}{c_A^{\text{'}}}$

Where $c_A$ is the molar concentration of $A$ in the absorbent$,c_A^{\text{'}}$ is the molar concentration of $A$ in the surrounding fluid, and $$ is a partition coefficient for solute A between the fluid and the absorbent material. Assume that $(1)$ the surrounding fluid is very well mixed and has a constant concentration $c_A^{\text{'}}=2\mathrm{.}00$gmo$\frac{l}{m^3}$; $(2)$ the diffusion coefficient of $A$ in the homogeneous absorbent material is $4{10}^{-7}c\frac{m^2}{s}$ and $K=1\mathrm{.}5c{m}^3$ fluid $\frac{?}{c{m}^3}$ absorbent; $(3)$ the cylindrical absorbent pellet is $1\mathrm{.}0cm$ in diameter and $5\mathrm{.}0cm$ long; and $(4)$ there is no solute A initially in the absorbent material.

If the "edge effects" associated with the ends of the cylinder can be neglected, how long will take for solute A concentration to reach $2\mathrm{.}94$gmo$\frac{l}{m^3}$ at a depth of $0\mathrm{.}4cm$ from the surface of cylinder?