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 $0$

$x<mi\; id="EditorElement\_675844"></mi><mi\; id="EditorElement\_675846">L</mi>$ is 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$.,c=c_o$

at $x=L).$

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 $c_o,$ the concentration outside the

pellet.

c$.$ Define an ideal rate in the absence of diffusional gradients and derive an expression

for the effectiveness factor.