A sales representative of the company supplying the catalyst for unit $x$ came along yesterday. The person said that their company has a new, more active catalyst for the reaction.

$A$ Money

The catalytically active material in the catalyst has not changed and the intrinsic rate constant for this first order reaction is $k^{\text{'}\text{'}}=1\mathrm{.}0{\mathrm{.}10}^{-6}m{s}^{-1}$ and the improved catalyst activity is claimed by a more efficient diffusion of the reagent to the active sites. Table P$4$ shows some of the characteristics of the catalyst, which is currently used, and the newly improved catalyst.

Table P$4$: Characteristics of current and newly developed catalysts.

$\backslash$table$[[,,$Current catalyst,$\backslash$table$[[$Newly developed$],[$catalyst$]]],[d_{\text{p}\text{e}\text{i}\text{l}\text{e}\text{t}\text{,}}mm,$Diameter of pellet,$10($spherical$),10($spherical$)],[{}_{\text{p}\text{a}\text{r}\text{t}\text{i}\text{c}\text{l}\text{e}\text{,}},$Porosity of pellet,$0\mathrm{.}4,0\mathrm{.}5],[d_{\text{p}\text{a}\text{r}\text{e}\text{,}}nm,$Pore diameter,$40,75],[S,\frac{m^2}{g},$Specific surface area,$16\mathrm{.}7,13\mathrm{.}3],[{}_{\text{c}\text{a}\text{t}\text{a}\text{l}\text{y}\text{s}\text{t}\text{,}}\frac{g}{c}{m}^3,$Density of catalyst,$2\mathrm{.}4,2\mathrm{.}0]]$

a$)$ Determine the effectiveness factor for the currently used catalyst knowing that the effective diffusivity of $A$ in the pores of the currently catalyst is $8\mathrm{.}8{\mathrm{.}10}^{-7}\frac{m^2}{s},$ and comment on the influence of mass transfer on the reaction.

b$)$ Estimate the effective diffusivity of $A$ in the newly developed catalyst assuming that Knudsen diffusion is the dominant transport mechanism assuming that the tortuosity and the constriction factor remain the same.

c$)$ Estimate the effectiveness factor over the newly developed catalyst, and comment on the improved mass transfer.

d$)$ Currently, a conversion of A of $75\%$ using $1250kg$ of catalyst in an isothermal packed bed reactor is achieved. What conversion can be expected with the same feed and having $1000kg$ of the new catalysts?

$(7)$