Consider a liquid$-$phase reaction $(A+BC+D).$ The standard heat of reaction $(H_{rxn})$ is $-7k\frac{J}{m}ol.$ The reaction is elementary, has a rate constant $k_1=31\frac{L}{\text{m}\text{o}\text{l}\text{h}}$ at $T=360K,$ and rate follows an Arrhenius form with the activation energy $E_1=65k\frac{J}{m}ol.$ There is also its backward reaction $(C+DA+B).$ The equilibrium constant $(K_C)$ is $3$ at $T=360K.$ Assume that heat capacity $(C_P)$ of the reactants and products are $120\frac{J}{\text{m}\text{o}\text{l}\text{K}}$ and constant. The feed contains the same moles of A and $B$ with the total molar flow rate $(F_{To})$ of $100$kmo$\frac{l}{h}.$ The feed concentration of $A(C_{A0})$ is $9mo\frac{l}{L}$ and the feed temperature is $T_0=300K.$ Answer the following questions with any necessary assumptions.

$($a$)$ Find the relationship between the equilibrium constant $)$ and the equilibrium conversion $(x_e).$

$($b$)$ Find the equilibrium conversion $(x_e)$ in two extreme temperatures, $T0$ and $T.$

$($c$)$ Sketch the equilibrium conversion $(x_e)$ as a function of temperature $($T$).$

$($d$)$ In the conversion $(x)-$temperature $(T)$ space, sketch at least three contour lines with constant reaction rates $(-r_A).$

$($e$)$ Find and sketch the reaction rate $(-r_A)$ as a function of temperature $(T).$

$($f$)$ In the $x-T$ space, sketch the energy balance line in an adiabatic reactor.

$($g$)$ What is the adiabatic equilibrium temperature?

$($h$)$ Sketch the Levenspiel plot vs$.($:$x\}$ in an adiabatic reactor.

$($i$)$ Find the volume of one CSTR operated adiabatically to get $x=0\mathrm{.}9x_e.$

$($j$)$ What is the operating temperature range if a PFR $)=(10(L)$ with a heat exchanger and $T_a=300K$ constant$)$ is used?

$($k$)$ What is the optimal $T$ progression to get use $x=0\mathrm{.}7,$ if you have unlimited options of reactors and heat exchangers?