The second order decomposition reaction

$2A(g)B(g)+C(g)$

is carried out in a $20cm$ ID tubular reactor packed with catalyst pellets of $1cm$ diameter. The reaction is internal$-$diffusion limited$.$ Pure A enters the reactor at a superficial velocity of $1m{s}^{-1},$ a temperature of $250C,$ and a pressure of $500$kPa. Experiments carried out on smaller pellets where the surface reaction is limiting yielded a reaction rate constant of $0\mathrm{.}5m^{6}mo{l}^{-1}me{n}^{-2}{s}^{-1}($see the additional information at the end of the question$)$

i$.$ Assuming the reactor is isothermal and pressure drop is negligible, derive an equation to calculate the length $($L$)$ of this packed bed reactor $($PBR$)$ required to achieve a conversion $($X$).$

ii$.$ Calculate the length of bed necessary to achieve $95\%$ conversion.

iii. Calculate the pressure drop in the bed, if the reactor is isothermal. If you are not sure about your answer in part ii$,$ assume $L=1m.$

iv$.$ Discuss how a large pressure drop affects the length of bed necessary to achieve $95\%$ conversion and whether the assumption of constant pressure is reasonable for a gas flowing through a PBR$.$

Additional Data:

Effective diffusivity: $1{10}^{-10}{m}^2{s}^{-1}$

Bed Porosity: $0\mathrm{.}5$

Density of $A$ at inlet conditions:

${}_A=0\mathrm{.}7kg{m}^{-3}$

Gas constant $R=8\mathrm{.}314m^{2}Pa{K}^{-1}mo{l}^{-1}$

Bulk density of bed: $8{10}^{2}g{m}^{-3}$ Internal surface area: $400m_{\text{c}\text{a}\text{t}}^2{g}^{-1}$ Viscosity of $A$ at inlet conditions:

$H_A=1\mathrm{.}3{10}^{-5}kg{m}^{-1}{s}^{-1}$