Problem $1.$ Effect of Catalyst Particle Shape on the Observed Rate. The production rate of a heterogeneously catalyzed, first$-$order reaction in a $0\mathrm{.}75-$cm diameter spherical pellet is RA $=-3\mathrm{.}25$ x$10-5$ mol$/$cm$3-$s when the catalyst is exposed to pure, gaseous A at a total pressure of $1$ atm and a temperature of $525$ K$.$ The activation energy EA $=18,600$ cal$/$mol$.$ Furthermore, the effective diffusivity of A in the pellet is DKA,e $=0\mathrm{.}009$ cm$2/$s at $525$ K and the mean pore size is small enough that the gaseous diffusion is primarily controlled by the Knudsen regime. The bulk fluid and the external surface concentrations can be assumed the same. Find the production rate if the catalyst is changed to a cylindrical pellet that is $0\mathrm{.}5$ cm in diameter and has an overall length of $1\mathrm{.}0$ cm$,$ and if the temperature is increased from $525$ K to $600$ K$.$