For a given elementary gas phase reaction below takes places in membrane flow reactor isothermally at constant pressure.

$A2B+C$

The membrane is permeable to $B$ but not to A and $C.$ Pure gaseous A enters the reactor at $P900$kPa and $T280C$ at a molar flow rate $($FAe$)$ of $10mo\frac{l}{m}in.$

The rate of diffusion of $B$ out of the reactor per unit volume of reactor, $R_B,$ is proportional to the concentration of $B($i$.$e$.,R_B=k_c{C}_B).$

$R=8\mathrm{.}314$kPa$dm\frac{3}{m}olK$

$k=0\mathrm{.}9mi{n}^{-1}$

$K_c=0\mathrm{.}05mo\frac{l}{d}{m}^3$

$K_c=0\mathrm{.}4$

a$)$ Show mole balances, rate laws, stoichiometry, and combination of these to provide differential mole balances on $A,B,$ and $C.$

b$)$ Plot the molar flow rates of each species as a function of reactor volume.

c$)$ By using the plot, calculate the conversion of $A$ at $V=500d{m}^3$

d$)$ Find and discuss the change of conversion when there is a $20\%$ increase in the value $k_c.$

e$)$ Find and discuss the change of conversion when the value of ${?}_{kc}$ is doubled.

f$)$ Assume that instead of membrane flow reactor, you have a PFR$.$ Calculate the conversion and reactor volume when the system reaches equilibrium.