Hydrodealkylation is a process in which side chain alkyl groups $($like methyl$)$ are removed from aromatics by reaction with

hydrogen. It is an important process in refining of crude oil to higher value fuels or into other valuable chemicals. The following

reactions take place in the reactor:

$C_6{H}_{5}C{H}_3{?}_{\text{t}\text{o}\text{l}\text{u}\text{e}\text{n}\text{e}}+H_2{C}_6{H}_6{?}_{\text{b}\text{e}\text{n}\text{z}\text{e}\text{n}\text{e}}+C{H}_4$

$C_6{H}_4(C{H}_3{)}_2+H_2{C}_6{H}_{5}C{H}_3+C{H}_4$

xylene

toluene

$C_6{H}_3(C{H}_3{)}_3+H_2{C}_6{H}_4(C{H}_3{)}_2+C{H}_4$

pseudocumene

xylene

In addition, an unwanted side reaction occurs, in which two benzenes react to form diphenyl:

$2C_6{H}_6{C}_6{H}_5{C}_6{H}_5+H_2$

benzene diphenyl

A process stream containing $10$mol$\%$ benzene $(C_6{H}_6),20$mol$\%$ toluene $(C_6{H}_{5}C{H}_3),30$mol$\%$ xylene $C_6{H}_4(C{H}_3{)}_2,$ and $40$

mol$\%$ pseudocumene $C_6{H}_3(C{H}_3{)}_3$ is fed at a rate of $100$gmol to a hydrodealkylation plant. This process stream is mixed with

$H_2$ at $5-$molar excess $H_2$ before being fed to the reactor. Hydrogen and methane are separated from the remaining compounds

$($lumped together as aromatics$).$ The aromatics are analyzed and found to contain $28$mol$\%$ pseudocumene, $1$mol$\%$ diphenyl,

$19\%$ benzene, and $1\%$ toluene. Calculate the extents of reaction and fractional conversions of all reactants, the methane

production rate, and the mol$\%$ methane in the gas stream.