The following gas phase reaction is carried out in a batch reactor, under isothermal conditions and at constant pressure Alongrightarrow$2B.$

Initially, the reactor contains $100$ moles of A$,50$ moles of B and $100$ moles of inert materials. The reaction is $2^{nd}$ order with a rate constant of $0\mathrm{.}005\frac{L}{\text{m}\text{o}\text{l}*s}$ and the initial reactor volume is $10,000L.$

a$)$ Determine the time required for $50\%$ conversion.

b$)$ Calculate the time needed for the same conversion, if the reaction was conducted in a constant volume batch reactor?