$(30$ points$)$ The water$-$gas shift reaction is carried out isothermally in a flow reactor with no pressure

drop.

$CO(g)+H_{2}O(g)C{O}_2(g)+H_2(g)$

Initial feed is composed of carbon monoxide, water, hydrogen, and argon $($inert$).$ The partial pressure

of initial feed is shown below.

$P_{CO}^0=0\mathrm{.}12$atm,$P_{H_{2}O}^0=0\mathrm{.}38$atm,$P_{H_2}^0=0\mathrm{.}38$atm,$P_{N_2}^0=0\mathrm{.}54$atm

At the reaction temperature of $503K,$ determine the following.

$(1)(7pts)$ Construct a stoichiometric table of water$-$gas shift reaction with the given information.

$(2)(8$ pts$)$ The rate law of water$-$gas shift reaction can be expressed by the following Langmuir$-$

Hinshelwood kinetics.

$-r_{CO}=\frac{k(P_{CO}{P}_{H_{2}O}-\frac{P_{C{O}_2}{P}_{H_2}}{K_{eq}})}{(1+K_{CO}{P}_{CO}+K_{H_{2}O}{P}_{H_{2}O}+K_{C{O}_2}{P}_{C{O}_2}+K_{H_2}{P}_{H_2}{)}^2}$

$K_{CO}=2\mathrm{.}21at{m}^{-1},K_{H_{2}O}=0\mathrm{.}0047at{m}^{-1},K_{C{O}_2}=0\mathrm{.}4at{m}^{-1},K_{H_2}=0\mathrm{.}052at{m}^{-1},K_{eq}=38\mathrm{.}8$

Pre$-$exponential factor $=0\mathrm{.}92$mol$*g^{-1}*s^{-1}*at{m}^{-2},$ Activation energy $=4\mathrm{.}08k\frac{J}{m}ol$

Calculate all parameters and express the rate law as a function of the conversion X$.$

$(3)(8$ pts$)$ Calculate the equilibrium conversion $x_e$ and sketch the Levenspiel plot from $x=0$ to $x=x_e.$

$(4)(7$ pts$)$ Calculate the volume of an ideal CSTR operating at $503K,$ which achieves a conversion of

$0\mathrm{.}9x_e.$