Extra Credit. Effectiveness Factor for a Langmuir$-$Hinshelwood Rate Form

Consider the following reaction that occurs in a porous slab catalyst and the

associated rate expression

Alongrightarrow Products

$r=k{c}_m\frac{K_A{c}_A}{1+K_A{c}_A}$

The parameter $c_m$ is the total concentration of active sites and is a constant. This rate

expression arises when gas$-$phase A adsorbs onto the catalyst surface and the

reaction is first order in the adsorbed A concentration.

Derive an expression for the effectiveness factor for the case where the surface of

the slab is exposed to a constant gas$-$phase concentration $C_{AS}.$ You may adopt the

same general parameters for the catalyst geometry and applicable particulars as

described above in Problem $2.$