Problem $4(4$ pts$).$ A reactive gas, "gas A$,"$ diffuses across a distance in the forward z$-$direction. The gas reacts at the surface layer, releasing a gas $B$ via the following reaction:

gas$)2B($gas$)+C($solid$)$

$($eq$.1)$

The distance $z$ does not change with the changing thickness of the solid, $,$ which grows over time. This is a good assumption because $dz>.$ The system is at steady state.

$($a$)$ State the general three$-$dimensional continuity equation in molar units $($the form with grad $)$

$($b$)$ Write out the $x,y,$ and $z$ components of the continuity equation $($from part a$)$ and indicate which terms are zero

$($c$)$ By combining $($eq$.1)$ and $($eq$.2)($the definition of total flux, below$),$ eliminate the net flux of the gas $B$ from $($eq$.2)$

$N_A=-c{D}_{AB}\frac{d{y}_A}{dz}+y_A(N_A+N_B)$

$($d$)$ Solve for $N_A$

$($e$)$ What is the concentration of $A$ at the surface, if the reaction is 'fast?$\text{'}$