Question $1.(50$ pts$)$ The following irreversible elementary liquid phase reaction is taking place in a $0\mathrm{.}01m^{3}PFR$ which is mounted with a heat exchanger.

$2C_4{H}_4,C_6{H}_{5}CH=C{H}_2$

The feed at $400C$ contains only $C_4{H}_4$ and the concentration of $C_4{H}_4$ in the feed is $1$kmo$\frac{l}{m^3}$ and the molar flow rate of it is $300mo\frac{l}{m}in.$ The heat capacities of $C_4{H}_4$ and $C_6{H}_{5}CH=C{H}_2$ are $122\frac{J}{m}$ole.$K$ and $244\frac{J}{?}$ mole.K$,$ respectively. The $"$Ua$"$ value for the heat exchanger is $5000\frac{J}{s}*d\frac{m^3}{K}$ and the heat exchanger fluid has a temperature of $425C.$ The heat of reaction $(H_{rxn})$ and the rate constant $(k)$ of the reaction are a function of temperature according to the following expressions:

$H_{rxn}=-231000-12(T-25)\frac{J}{m}$ole where $T$ is in $Ck=1\mathrm{.}48{10}^8$exp$(-\frac{19124}{T})d\frac{m^3}{?}$ mole.s where $T$ is in $K$ Determine the conversion of $C_4{H}_4$ that can be achieved in this reactor.