We are running a continuous culture in a $1,000L$ fermenter and want to have only a one in one thousand chance of spore contamination. We will run the fermentation for $4$ weeks at a dilution rate $($i$.$e$.\frac{1}{?}$ residence time$)$ of $0\mathrm{.}1h{r}^{-1}.$ We believe that the medium initially has ${10}^5$spore$\frac{s}{L}.$ We will use a sterilization temperature of $140C.)=1*{10}^{36}mi{n}^{-1}$;$E_{0d}=(67$kca$\frac{l}{m}ol.$

a$)$ How long do we need to sterilize the medium $($Note: we also need to sterilize the initial volume at the start of the run$)$ at $140C?$

b$)$ We have a key temperature$-$sensitive vitamin in the broth at a concentration of $10m\frac{g}{?}$ liter that is heat inactivated with $E_{0d}=10\frac{\text{k}\text{c}\text{a}\text{l}}{\text{m}\text{o}\text{l}}$ and $=1*{10}^{4}mi{n}^{-1}.$ What is the concentration of active vitamin after sterilization?