An isothermal continuous stirred tank reactor is used to form component B from component $A$ by the reaction $2AB.$ The rate of the reaction is second order with respect to component A and is given as

$r=k{C}_A^2$

where $r$ is the rate of reaction of A per unit volume of the reactor, $k$ is the rate constant given as $0\mathrm{.}025$ liters$/($mol$.$ min$).$

The volume of the reactor $($V$)$ and the flow rates of the feed and the product streams $($q$)$ are held constant with $V=5$ liters and $q=1$ liter $\frac{?}{\text{m}\text{i}\text{n}}.$

The CSTR is perfectly mixed, the mass densities of the feed and the product streams are equal and constant.

$($a$)$ Initially the reactor is operating at steady state with feed composition $C_{Ai}$ of $2\mathrm{.}5$ mol$/$liter$.$ Calculate the product composition $C_A$ at steady state.

$($b$)$ Write an unsteady state component balance for species A that gives the variation of $C_A$ with the variation of feed composition $C_{Ai}.$ Linearize the non$-$linear differential equation and obtain the transfer function

$C_A^{\text{'}}\frac{s}{C_{Ai}}(s)=\frac{K_A}{s+1}$

Calculate $K_A$ and $$ for your linear model.

$($c$)C_{Ai}$ is changed by a step increase from the initial value of $2\mathrm{.}5mo\frac{l}{l}$iter to $3\mathrm{.}0mo\frac{l}{?}$ liter. Find an expression for variation of $C_A$ with time. What is $C_{A}2\mathrm{.}5$ minutes after the step change? What is the final steady state value of $C_A?$