To avoid catalyst poisoning, a particular catalyst is designed so that the

outside region is inert. Consider such a catalyst shown in the picture

below where only the region from $c=c_{0}x=L,L,k,D{c}_{0}0is$ active. $(A$ variation $of$

this was done $in$ some older automotive three$-$way catalysts $to$ avoid

poisoning $by$ lead added $to$ the gasoline.$)$ Assume the reaction $in$ the

active portion can $be$ treated $as$ first order and that mass transfer

external $to$ the particles $is$ fast $(i.e.,c=c_{0}atx=L).$

$a.$ Write the differential material balances for the two regions within

the particle. What are the boundary conditions?

$b.$ Determine the reaction rate $in$ terms $of,L,k,D,$ and $c_0,$ the

concentration outside the pellet.

$c.$ Define $an$ ideal rate $in$ the $absenceof$ diffusional gradients and

derive $an$ expression for the

effectiveness factor.