The irreversible liquid phase reaction

$AB+C$

is carried out in a batch reactor at $400C.$ The reaction is first$-$order and the time required to achieve $60\%$ conversion was found to be $50$ mins.

a$)$ Determine the rate constant by using the data given for the batch reactor.

b$)$ Suppose that this reaction is to be carried out in a plug$-$flow reactor $($PFR$)$ which operates isothermally at $400C.$ The volumetric flow rate of the feed entering the reactor is $0\mathrm{.}05m^3/$h and it consists of pure A$.$ Calculate the volume of the reactor required to achieve $80\%$ conversion.