Problem $2($Based on the July $08$ class problem, although $V$ is no longer arbitrary, and the CVBR is no longer adiabatic.$)$

The oxidation of carbon monoxide, in the presence of water vapour, is described by the following irreversible reaction:

$CO(g)+\frac{1}{2}{O}_2(g)C{O}_2(g)$

The rate law for this reaction is

$-r_{CO}=1\mathrm{.}26{10}^{10}$exp$(\frac{-20131}{T})C_{CO}{C}_{O_2}^{0\mathrm{.}25}{C}_{H_{2}O}^{0\mathrm{.}5}$

where $T$ is in K $,$ the concentrations are in $mo\frac{l}{m^3},$ and $r_{CO}$ is in $mo\frac{l}{m^3}*s.$

The reaction occurs in a CVBR $($constant volume batch reactor$)$ with the following composition at time $t=0$ :

$COat2$mol$\%,O_{2}at1$mol$\%,H_{2}Oat7$mol$\%,$ and $N_{2}at90$mol$\%$

All components, including the product $C{O}_2,$ behave as ideal gases. $($Note that water vapour and nitrogen gas do not appear in the overall reaction.$)$

Initialy, at $t=0,$ the $P,T,$ and $V$ of the CVBR are: $1$ bar, $700$ K $,$ and $0\mathrm{.}2m^3.$

The CVBR is not adiabatic; it is exposed to external heat transfer given by

$q=U_{HT}A(T_{}-T)$

where $U_{HT}$ is the overall heat transfer coefficient, $A$ is the area over which heat transfer occurs, $T$ is the temperature inside the reactor, and $T_{}$ is some constant temperature $($usually that of a temperature bath$).$

This heat transfer mechanism is an attempt to regulate the reactor temperature at $700$ K $($although you will show that it is not perfect$).$ Suppose a heat exchanger is installed with

$U_{HT}=2\frac{W}{m^2}*K$;$,A=1\mathrm{.}5m^2$;$,T_{}=700K$

$($a$)$ Show plots of $T$ vs $t$ and $x_{CO}$ vs $t$ for a reaction with heat transfer as described. What is the conversion of CO at $t=10$min $?$

Your time $($horizontal$)$ axes should be from $0$ to $600$ s $(10$ min$).$ The temperature axis should begin at $700$ K $.$

$($b$)$ What is the maximum temperature experienced by the reactor? At what time $($to within $+-1s$ does this maximum temperature occur?

Note: Since this problem involves the same reaction as in the class problem on July $08,$ to save you some time, the enthalpy of reaction $($as shown in class$)$ is

${}_{r}H(T)=-2\mathrm{.}7959{10}^5-18\mathrm{.}61T$

$+(2\mathrm{.}5224{10}^{-2})T^2-(1\mathrm{.}2247{10}^{-5})T^3+(2\mathrm{.}255{10}^{-9})T^4$ all $SI$ units