Consider the reversible, isothermal, liquid$-$phase reaction:

$A+B2C$

where $k_1$ is the rate constant for the forward reaction, $k_2$ is the rate constant for the backward reaction and $K_c=\frac{k_1}{k_2}$ is the equilibrium constant. The forward and backward reactions are elementary. The reaction is carried out using equimolar concentrations of species A and $B.$ The equilibrium conversion for this reaction is $80\%.$

$($a$)$ Determine the value of the equilibrium constant $K_c$ and express the rate of disappearance of species $A$ as function of $K_c.$

$[9$ marks$]$

$($b$)$ Express the rate of disappearance of species A exclusively as a function of the conversion of species $A,$ the rate constant for the forward reaction $k_1$ and the concentration of species $A$ in the feed stream to the reactor.

$[4$ marks$]$

$($c$)$ When the same reaction is carried out in a CSTR the conversion of A is measured to be $60\%.$ The equilibrium conversion under the same conditions is known to be $80\%.$ Determine the value of the Damk$$hler number. Further, determine the conversion of species $A$ if the volumetric flow rate into the CSTR is to be increased by a factor of $1\mathrm{.}5$ with the inlet concentrations of species A and $B$ and the temperature remaining constant.

$[12$ marks$]$