$(55$ pts$)1.$ You have been assigned a new fluidized bed reactor that cracks a feed stock into small molecules. The overall stoichiometry of this reaction is: $.$ This fluidized bed reactor can be modeled as a CSTR$.$ This reactor is jacketed and you have available both cooling and heating fluids. The heat transfer fluid can be cooled or heated within the temperature range of $200K{T}_{a}500K.$ The mass of catalyst in the reactor is $5kg.$ The feed to this reactor is gas phase at $300K$ and consists of the reactant A and an inert I. The pressure inside the reactor vessel can be assumed to be uniform at a value of $P=1\mathrm{.}1{10}^{5}Pa.$ Assume that the reaction is reversible and elementary. Physical and chemical properties are given in the table below.

$W=5$kgcat

a$)$ Give the elementary reaction rate expression for the removal of $A.$

b$)$ To get you started in this reactor design, your boss asked the chemist intern to make a plot of the equilibrium conversion as a function of temperature which is given on the following page. This is plotted as the dashed line. Since your job is on the line you are required to derive an analytical expression for equilibrium conversion as a function of temperature. Calculate the equilibrium conversion at $490K$ and compare it to the value on the dashed line on the plot on the next page. $($e$.$g$.)$ Show all of your work including plotting the point.

c$)$ Also as an extra added calculation the chemist intern calculated the conversion as a function of temperature from the mole balance following company manuals. He never took heat transfer and didn't know what equations to use for this problem. So he used the mole balance and assumed a temperature to create the solid curve shown on the next page. Derive the analytical expression for conversion from the mole balance as a function of temperature and calculate one of the points on the curve. $($e$.$g$.)$ Show all of your work.

d$)$ Derive an analytical expression for the conversion in the fluidized bed $($CSTR$)$ as a function of temperature using the energy balance. $($e$.$g$.)$ Calculate the conversion from the mole balance at a temperature of $520K$ and verify that this is one of the points on the solid line on the graph on the next page. Show all of your work

e$)$ Using both the energy and mole balance determine the maximum conversion and subsequent reactor temperature that can be achieved in this reactor with the above operating conditions. You must choose whether or not you are using the heating$/$cooling jacket and if using the jacket then the jacket temperature, $T_a$ Remember that this reactor has the option to run as an adiabatic reactor or you could use the jacketed section for cooling or heating. The heat transfer fluid can be cooled or heated within the temperature range of $200K$ to $500K.$ Show all of your work including work on the graph.

g$)$ For your chosen reactor operating conditions in part e$)$ give all of the possible steady$-$state operating conditions. State whether or not the maximum steady$-$state conversion that you found is a stable or unstable operating condition.