P$5-13$ Balter White, Jesse Pinkman, and Mike Ehrmantraut stole $1000$ gallons of methylamine during an episode of the TV series Breaking Bad. Shortly thereafter, Jesse and Mike decided they would get out of the cooking business and sell their shares of the methylamine. Walter wanted to keep all of the methylamine for future meth cooks and not let Jesse and Mike have their shares. Suppose Jesse and Mike decided to sabotage Walter's cooking operation by ruining the methylamine using the following gas phase reaction:

$2C{H}_{3}N{H}_2$harr$(C{H}_3{)}_{2}NH+N{H}_3$

or$,$ in short:

$2AB+C$

This reaction converts the methylamine to dimethylamine, using a silica$-$alumina catalyst. The molar flow rate into a packed$-$bed reactor is $23\mathrm{.}6mo\frac{l}{s}$ and the entering pressure is $18$atm. Assume there is no pressure drop or temperature change in the reactor. The reaction rate follows an elementary rate law in terms of partial pressure.

$k_1=4\mathrm{.}25{10}^{-6}\frac{\text{m}\text{o}\text{l}}{at{m}^2*\text{g}\text{c}\text{a}\text{t}\text{a}\text{l}\text{y}\text{s}\text{t}*s}$ and $K_e=2\mathrm{.}5$

$($a$)$ Write the mole balance.

$($b$)$ Write the rate law in terms of partial pressures.

$($c$)$ Set up a stoichiometric table for this reaction.

$($d$)$ Write the partial pressures in terms of conversion.

$($e$)$ Write the rate law solely in terms of conversion.

$($f$)$ What is the equilibrium conversion, $x_e?$

$($g$)$ Write out your algorithm in terms of conversion.

$($h$)$ How many kilograms of catalyst would Jesse need to load in a PBR to obtain a conversion of $0\mathrm{.}9**x_e?$ For $x=0\mathrm{.}75x_e?$

$($i$)$ How many kilograms of catalyst would be needed to obtain $90\%$ of the equilibrium conversion in a fluidized$-$bed reactor? If this weight is very, very large, what might you suggest to reduce the weight? $($Ans$.$: $W=207\mathrm{.}2kg$ catalyst$)$

$($j$)$ What conversion would be achieved in a $100kg$ PBR with pressure drop and $=0\mathrm{.}0098k{g}^{-1}?$ At what catalyst weight does the exit pressure fall below $1\mathrm{.}0$atm $?$

$($k$)$ Repeat $($j$)$ when pressure drop is accounted for with $=6{10}^{-4}k{g}^{-1}.$ Ph$.$D$.$ student Julia Faeth created this problem using modified data from J$.$ W$.$ Mitchell, et al$.,$ Ind. Eng. Chem. Res. $33,1994,$ pp$.181-184.$