$2.$ A gas feedstock $($species A$)$ undergoes a rapidly and irreversibly chemical reaction on a heated metal plate $($z $=0)$ at the bottom of a cylindrical gas phase reactor with a height of $30$ cm and a diameter of $10$ cm$.$ The temperature and pressure of the gas in the vessel is maintained at $305$ K and $1$ atm, and DAB is $0\mathrm{.}15$ cm$2/$s at these conditions. Pure A is fed in at the top of the reactor $($z $=$ L$).$ The metal plate yields a first$-$order reaction that generates B by reducing the molecular weight of A by an average factor of $3($A $3$B$).$ You may assume the reaction is fast $($the value of the second Damk$$hler number for this system is $21).$

a$)$ Derive an equation for the flux of A $($NA$).$ Then, calculate the explicit flux of A in moles per cm$2$ per second.

b$)$ Use the flux of A to calculate the rate of production of B$.$