$\frac{m_{\text{c}\text{a}\text{t}}}{n_{A,0}^{}}={\displaystyle \underset{0}{\overset{x_A}{}}}\frac{d{x}_A}{r_{\text{e}\text{f}\text{f}}}$

The kinetics of a second$-$order heterogeneously catalysed gas$-$phase reaction $AR$ is investigated in a differential

catalytic reactor. Under isothermal conditions with a reactor feed stream of $V_0=1\frac{L}{h}$ and $C_{A,0}=4mo\frac{l}{L}$ and a

catalyst quantity of $2\mathrm{.}8gm,$ an outlet concentration of $C_A=1\mathrm{.}5mo\frac{l}{L}$ was obtained.

$($a$)$ What type of reactor is it$?$

$($b$)$ Calculate the rate constant for the reaction and indicate the value and the unit.

$($c$)$ What quantity of catalyst would be required in an ideal plug flow reactor, in which a conversion of $90\%$ is

to be achieved for a feed stream of $1000\frac{L}{h}$ with a concentration of $C_{A,0}=2mo\frac{l}{L}?$

$($d$)$ What will happen to conversion if you reduce the feed stream to $500\frac{L}{h}$ but use the same quantity of

catalyst calculated in $($c$)$ above

$($e$)$ The same reaction is carried out in a reactor with complete recirculation. Calculate the quantity of

catalyst required assuming same conditions as in $($b$)$ above.

$($f$)$ Discuss the reasons for the difference in results in $($b$)$ and $($c$).$