Consider a chemical reaction, (a ightarrow b), taking place in a bed of spherical catalyst pellets.

Consider a chemical reaction, \(a \rightarrow b\), taking place in a bed of spherical catalyst pellets. The true reaction rate is \(\mathrm{n}^{\text {th }}\)-order in the concentration of \(a\) and it has an activation energy of \(E_{a}\). You decide to measure this in your reactor and find that the reaction order is \(n^{\prime}\) with activation energy \(E_{a}^{\prime}\). You know that the Thiele modulus for your pellets is large so that you have severe diffusion limitations. Speculate on whether that has any effect on the kinetics.

a. What is the Thiele modulus for a spherical catalyst pellet assuming an \(\mathrm{n}^{\text {th }}\)-order reaction?

b. For large values of the Thiele modulus, the effectiveness factor for a spherical catalyst pellet is:

\[\eta=\frac{3}{\phi_{n}} \sqrt{\frac{2}{n+1}}\]

What is the relationship between the observed reaction rate, \(-r_{a, o b s}\) and the actual reaction rate, \(-r_{a}\) using this approximation? Does it affect \(n\) ?