Problem $(3)$: Consider the system shown in the figure. Here gas A dissolves in liquid B in a beaker and diffuses isothermally into the liquid phase. As it diffuses, A also undergoes an irreversible third$-$order with respect to A and zero$-$order with respect to $B$ homogeneous reaction. Assume dilute solution and the liquid phase is sufficiently deep that $L$ can be taken to be infinite. Derive the following expression for the concentration profile for $A$ :

$\frac{C_{A0}}{C_A}=1+z\sqrt[\phantom{2}]{\frac{k_3^{\text{'}\text{'}\text{'}}{C}_{A0}^2}{2D_{AB}}}$;