A slab of silicon is exposed to gaseous oxygen $($Species A$)$ at pressure $p,$ producing a layer

of silicon dioxide $($Species B$).$ The layer extends from the surface $z=0,$ to the surface at

$z=(t)$ where the oxygen and silicon undergo a first$-$order reaction with rate coefficient $k_1".$

Assume that there is also a gas film above the $Si{O}_2$ layer and the bulk concentration of $A$

is $c_{Ab}.$ The thickness $(t)$ of the growing oxide layer is to be predicted. A quasi$-$steady$-$state

method is useful here, in as much as the advancement of the reaction front is very slow.

$($a$)$ Write an expression for the molar flow rate of oxygen $W_A^{}$ in the gas film, in $Si{O}_2$ layer,

and also for the reaction on the surface. Assume that $($:$k_m$:$)$ is the average mass transfer

coefficient and diffusivity $D_{AB}$ are known.

$($b$)$ Using macroscopic analysis, show that the molar flux of oxygen to the surface of $Si$ is

given by

$N_A=\frac{c_{Ab}}{\frac{(t)}{D_{AB}}+\frac{1}{k^{\text{'}\text{'}}}+\frac{1}{(\text{:}{k}_m\text{:})}}$

$($c$)$ Draw a mass circuit and indicate all the resistances and mark all nodes.

$($d$)$ Write an unsteady state mass balance to describe how the thickness of $Si{O}_2$ layer depends

on time. Show that the time to grow a layer of thickness $\frac{?}{\text{b}\text{a}\text{r}}()$ is given as a function of

$($:$k_m$:$),D_{AB},k^{\text{'}\text{'}},$tilde$(V{)}_{\text{S}\text{i}\text{O}2},C_{Ab}.$