$2\mathrm{.}16{?}^c$ Mass transfer inside a circular pipe

Water flows through a thin tube, the walls of which are lightly coated with benzoic acid $(C_7{H}_6{O}_2).$ The water flows slowly, at $298K$ and $0\mathrm{.}1c\frac{m}{s}.$ The pipe is $1cm$ in diameter. Under these conditions, equation $(2-98)$ applies.

$($a$)$ Show that a material balance on a length of pipe $Z$ leads to

$c_{\text{A}\text{o}\text{u}\text{t}}=c_A**\{1-$exp$[\frac{-k_{L}aZ}{v}]\}$

where $v$ is the average fluid velocity, and $c_A^{**}$ is the equilibrium solubility concentration.

$($b$)$ What is the average concentration of benzoic acid in the water after a length of pipe of $2m.$ The solubility of benzoic acid in water at $298K$ is $0\mathrm{.}003\frac{g}{c}{m}^3,$ and the mass diffusivity is $1\mathrm{.}0{10}^{-5}c\frac{m^2}{s}($Cussler$,1997).$

Answer: $1\mathrm{.}14m\frac{g}{c}{m}^3$