Recall the etching reaction mechanism discussed in class:

$A(g)+S,{}^{k_1},A(a)$

$A(a),{?}^{k_2},B(a)$

$B(a),longrightarro{w}^{k_3},B(g)+S$

$S$ is a "surface site" and $A(a)$ is an adsorbed molecule of $A,$ the total concentration of surface sites is

$C_S,$ etc. Consider the case where species B does not quickly desorb from the surface. Under

conditions of constant etch rate, there will therefore be two steady$-$state surface coverages, ${}_A$ and ${}_B.$

$($a$)$ Write out expressions for $d\frac{{}_A}{d}t$ and $d\frac{{}_B}{d}t.$ Set each to zero, and derive expressions for

$\frac{{}_A}{{}_B}$ and ${}_A$ as a function of the gas concentration $C_A.$

$($b$)$ Supposed $k_{-1}$ is small enough that desorption of $A(a)$ can be ignored, and that $k_3{k}_2($step $3$

is rate determining.$)$ Use these approximations to simplify the steady$-$state expression to obtain

estimates of both ${}_A$ and ${}_B.$

$($c$)$ If the surface should "fill $up"so$ that ${}_A+{}_{B}1.$ Obtain estimates for ${}_A$ and ${}_{B}in$

this case, and compare them with part $(b).$