A particular porous catalyst pellet has a void fraction of $0\mathrm{.}40,$ a specific internal surface area of $180\frac{m^2}{g},$ and a pellet density of $1\mathrm{.}40\frac{g}{c}{m}^3.$ The tortuosity of the catalyst pores is equal to $3.$

$($a$)$ Compute the effective pore diameter of this catalyst. $(10$p$.)$

$($b$)$ Estimate the effective diffusion coefficient for dilute thiophene in hydrogen at a pressure of $1$ atm and temperature of $600K$ and identify the predominant type of diffusion in the pores for the given conditions. $(10$p$)$

Note: The bulk $($Fickian$)$ diffusion coefficient of thiophene in hydrogen in the gas phase can be estimated using the Fuller$-$Schettler$-$Giddings equation $($correlation$)$:

$D_{AB}=\frac{{10}^{-1}{T}^{1\mathrm{.}7v}[\frac{1}{M_A}+\frac{1}{M_n}{]}^{\frac{1}{2}}}{P[({\displaystyle \underset{?}{\overset{?}{}}}v{)}_A^{13}+({\displaystyle \underset{?}{\overset{?}{}}}v{)}_n^{13}{]}^2}$

$D_A$: Binary gas phase diffusivity of $A$ in $B$ in $c\frac{m^2}{s}.$

$T$ : Absolute temperature in Kelvin

$P$ : Absolute pressure in atmospheres

$M_A,M_B$ : Molecular weights of A and $B,$ respectively

$({\displaystyle \underset{?}{\overset{?}{}}}v{)}_A,({\displaystyle \underset{?}{\overset{?}{}}}v{)}_g$ : molecular volumes in $c\frac{m^3}{g}-$mol

The molecular diffusion volumes can be estimated using additive rules and tables. Thiophene: $v_1=70\mathrm{.}72c\frac{m^3}{g}-$mol. Hydrogen: $v_2=7\mathrm{.}07c\frac{m^3}{g}-$mol.

$($c$)$ If this catalyst is formed into spherical pellets with a diameter of $5mm,$ estimate the value of the surface reaction rate constant for which internal diffusion limitations will reduce the reaction rate by a factor of $2.(10$p$.)$