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.

$2C_4{H}_{4}longrightarrow{C}_6{H}_{5}CH=C{H}_2$

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 $244\frac{J}{m}$ole.$K,$ 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:

$H_n=-231000-12(T-25)\frac{J}{m}$ole where $T$ is in $C$

$k=1\mathrm{.}48{10}^8$exp$(-\frac{19124}{T})d\frac{m^3}{m}ol{e}_{.}.$ where $T$ is in $K$

Determine the conversion of $C_4{H}_4$ that can be achieved in this reactor.