Problem $2(30/100)$: A wild$-$type enzyme $($following the Michaelis$-$Menten kinetics with a

maximum rate of $0\mathrm{.}3m\frac{M}{m}in)$ was used for catalyzing a chemical reaction in a batch reactor with

an appropriate enzyme concentration and an initial substrate concentration of $0\mathrm{.}5mM.$ It was

observed that the half$-$life for the substrate $($the time required to consume $50\%$ of substrate$)$ was $5$

min. A biochemical engineer derived a mutant version of the enzyme to improve substrate binding

affinity so that the Michaelis$-$Menten constant became half of that for the wild$-$type enzyme

without affecting the maximum rate.

a$)$ What would be the half$-$life $($in $h)$ for the substrate under the abovementioned batch reaction

conditions $($with the initial substrate concentration of $0\mathrm{.}5mM)$ but using the mutant enzyme?

b$)$ What is the operating inlet flow rate $($in $\frac{L}{m}in)$ if the mutant enzyme $($with the same enzyme

concentration as that in part a$))$ is used to catalyze an enzyme reaction in a continuous stirred

tank reactor with a working volume of $10L,$ inlet substrate concentration of $0\mathrm{.}8mM$ and

conversion of $90\%?$