Consider a heterogeneous reaction R $+$ S $->$ P occurring at a solid surface located at z $=$ W$.$ Species R

is a reactant that moves freely in a stagnant liquid solution between z $=0$ and z $=$ W with a volumetric

concentration CR$($z$,$ t$)[=]$ amt$/$vol and diffusivity D$.$ S is a surface$-$bound substrate species that is completely

immobile at the solid surface, with an area concentration CS$($t$)[=]$ amt$/$area$.$ P is a surface$-$bound product

species that is completely immobile at the solid surface, with an area concentration CP $($t$)[=]$ amt$/$area$.$ The

reaction mechanism is first$-$order in R and first$-$order in S so that the rate of formation of P is dP$/$dt $=$

kCR$($W$,$ t$)$CS$($t$),$ where k $[=]$ vol$/($amt $$ time$)$ is the reaction rate constant.

Initially, the liquid solution is filled with R at constant concentration CR$($x$,0)=$ CR$0$ and the substrate

is at concentration CS$(0)=$ CS$0.$ Species R is supplied at the z $=0$ edge of the liquid film to keep its

concentration there constant at CR$(0,$ t$)=$ CR$0.$

Figure $1$

$($a$)$ Assume diffusion of R is significantly faster than the surface reaction. Show the steps to deriving the

pseudosteady concentration profile CR$($z$,$ t$)$ of R in the liquid. The final result is

CeR$($z$,$ e et$)=1$

Da CeS$($et$)$

$1+$ Da CeS$($et$)$

ze $(1)$

where the dimensionless variables are

CeR $=$

CR

CR$0$

$,$ CeS $=$

CS

CS$0$

$,$ ze $=$

z

W

$,$ Da $=$ kW CS$0$

D

$,$ et $=$ kCR$0$t $(2)$

$($b$)$ Using the result of part $($a$),$ derive the ODE and initial condition governing CeS$($et$).$ Do not solve for

CeS$($et$).$