Assume only one reaction takes place with a stoichiometry of:

$A+\frac{1}{2}BC$

Assume the reaction kinetics can be given as a reversible reaction that is first order in the forward direction, with respect to $A,$ and first order in the reverse direction with respect to $C$ :

$-\frac{d[A]}{dt}=k_1[A]-k_2[C]$

We will assume our reaction takes place isobarically and isothermally at $1$ atm and $26\mathrm{.}85$ degC respectively, with an equimolar ratio of A and $B$ and no $C$ present in the reactor feed. This is a single pass reactor.

The rate constants are:

$k_1=0\mathrm{.}010se{c}^{-1}$

$k_2=0\mathrm{.}004se{c}^{-1}$

The reaction takes place in a PFR with a volume of $80m^3$ and the inlet gas flowrate of $1\mathrm{.}5\frac{m^3}{sec}.$

Question:

Use the reaction stoichiometry and rate equation above to derive an analytic solution to the rate equation $($ie using calculus and integrating the rate equation$).$ Once you have solved the rate equation, plot your solution as concentration of A $($y$-$axis$)$ against time $($x$-$axis$).$