The elementary gas$-$phase reaction

$($CH$3)3$COOC$($CH$3)3->$C$2$H$6+2$CH$3$COCH$3$

is carried out isothermally in a flow reactor with no pressure drop. The specific reaction rate at $50$C is $10-4$ min$-1$ and the activation energy is $85$ kJ$/$mol$.$ Pure di$-$tert$-$butyl peroxide enters the reactor at $10$ atm and $127$C and a molar flow rate of $2\mathrm{.}5$ mol$/$min$.$

a$)$ Calculate the reactor volume and space time to achieve $90\%$ conversion in a CSTR$.$

b$)$ Assume that the reaction is reversible with KC $=0\mathrm{.}025$ mol$2/$dm$6.$ Calculate the equilibrium conversion. Redo part a$)$ to achieve a conversion that $90\%$ of the equilibrium conversion. Polymath or a similar numerical solver is very useful for solving the set of equations.