P$4-22$ Alkylated cyclohexanols are important intermediates in the fragrance and perfume industry $[$Ind$.$ Eng. Chem. Res., $28,693(1989)].$ Recent work has focused on gas$-$phase catalyzed hydrogenation of $o-$cresol to $2-$methylcyclohexanone, which is then hydrogenated to $2-$methylcyclohexanol. In this problem we focus on only the first step in the reaction $($Figure P$4-22).$ The reaction on a nickel$-$silica catalyst was found to be zero$-$order in o$-$cresol and first$-$order in hydrogen with a specific reaction rate at $170C$ of $1\mathrm{.}74$mol of $o-$ cresol $\frac{?}{(kgcat*min*atm)}.$ The reaction mixture enters the packed$-$bed reactor at a total pressure of $5$atm. The molar feed consists of $67\%H_2$ and $33\%o-$cresol at a total molar rate of $40mo\frac{l}{m}in.$

$($a$)$ Neglecting pressure drop, plot the rate of reaction of $o-$cresol and the concentrations of each species as a function of catalyst weight. What is the ratio of catalyst weight needed to achieve the last $5\%$ conversion to the weight necessary to achieve the first $5\%$ conversion $(0$ to $5\%)$ in the plug$-$flow reactor?

$($b$)$ Accounting for the pressure drop in the packed bed using a value of $=0\mathrm{.}034k{g}^{-1},$ redo part $($a$)$ along with a plot of pressure versus catalyst weight.