$4\mathrm{.}51.$ Ethanol is produced commercially by the hydration of ethylene:

$C_2{H}_4+H_{2}Olongrightarrow{C}_2{H}_{5}OH$

Some of the product is converted to diethyl ether in the side reaction

$2C_2{H}_5$OHlongrightarrow$(C_2{H}_5{)}_{2}O+H_{2}O$

The feed to the reactor contains ethylene, steam, and an inert gas. A sample of the reactor effluent gas is analyzed and found to contain $43\mathrm{.}3$ mole $\%$ ethylene, $2\mathrm{.}5\%$ ethanol, $0\mathrm{.}14\%$ ether, $9\mathrm{.}3\%$ inerts, and the balance water.

$($a$)$ Take as a basis $100$mol of effluent gas, draw and label a flowchart, and do a degree$-$of$-$freedom analysis based on atomic species to prove that the system has zero degrees of freedom. ${?}^7$

$($b$)$ Calculate the molar composition of the reactor feed, the percentage conversion of ethylene, the fractional yield of ethanol, and the selectivity of ethanol production relative to ether production.

$($c$)$ The percentage conversion of ethylene you calculated should be very low. Why do you think the reactor would be designed to consume so little of the reactant? $($Hint: If the reaction mixture remained in the reactor long enough to use up most of the ethylene, what would the main product constituent probably be$?)$ What additional processing steps are likely to take place downstream from the reactor?