$(4)$ A mixed solution composed of ethanol and water is in the form of a liquid film with a thickness of $3mm$ at $298K.$ The liquid film is in contact at one surface with an organic solvent in which ethanol is soluble and water is insoluble. At the interface between ethanol and the organic solvent $($position $1),$ the concentration of ethanol and the solution density are $5\mathrm{.}4wt\%$ and $982\mathrm{.}7k\frac{g}{m^3},$ respectively. At the other side of the liquid film $($position $2),$ the concentration of ethanol and the solution density are $14\mathrm{.}6wt\%$ and $968\mathrm{.}7k\frac{g}{m^3},$ respectively. The diffusivity of ethanol is $7\mathrm{.}40{10}^{-10}\frac{m^2}{s}.$

$($a$)$ Develop the general differential equation for mass transfer of ethanol in the stagnant film at steady state $($mole fraction of ethanol as a function of position$).$ State assumptions that allow for appropriate simplification of the general differential equation for mass transfer. $(6\%)$

$($b$)$ Calculate the mole fraction of ethanol at position $1$ and position $2.(4\%)$

$($c$)$ Based on the derived differential equation for mass transfer, calculate the steady$-$state flux of ethanol and water. $(4\%)$

$($d$)$ If temperature is increased to $323K,$ what do you expect the change in the steady$-$state flux of ethanol? Why? $(2\%)$

$($e$)$ Calculate the convective mass$-$transfer coefficient. $(4\%)$