For Example \(10.2\), check whether the criterion for negligible intraparticle heat transfer limitation is fulfilled. Take \(\Delta_{\mathrm{r}} \mathrm{H}=227.5 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}\) and \(\mathrm{E}_{\mathrm{a}}=\) \(220 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}\). Assume the gas properties are those of pure nitrogen at \(900^{\circ} \mathrm{C}\). These are \(\lambda=0.074 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1}, \rho=0.2875 \mathrm{~kg} \cdot \mathrm{m}^{-3}\), and \(\mathrm{c}_{\mathrm{p}}=1.199 \mathrm{~kJ} \cdot \mathrm{kg} \cdot \mathrm{K}^{-1}\).

To begin with, let's calculate the Thiele modulus (\(\Phi\)) using the provided values: $$ \Phi = \frac{\Delta_{\mathrm{r}} \mathrm{H} \cdot k}{\lambda \cdot \rho \cdot \mathrm{c}_{\mathrm{p}}} $$ We are provided with values for \(\Delta_{\mathrm{r}} \mathrm{H}\), \(\lambda\), \(\rho\), and \(\mathrm{c}_{\mathrm{p}}\). However, we need to determine the value of k (reaction rate constant) using the following Arrhenius equation: $$ k = A \cdot e^{-\frac{E_a}{RT}} $$ To determine k, we need the pre-exponential factor (A), activation energy (\(E_a\)), universal gas constant (R), and temperature (T). Unfortunately, the pre-exponential factor (A) is not provided, which means we cannot proceed with calculating the Thiele modulus directly. However, if we make some assumptions about the reaction rate constant, we can provide an estimate of the Thiele modulus and check if the heat transfer limitation is negligible.

Since we do not have the pre-exponential factor, we cannot calculate the exact value of k. However, we can assume a value for k and see if the Thiele modulus is less than the threshold value (0.3) to satisfy the condition for a negligible heat transfer limitation. Let's assume that \(k=1\): $$ \Phi = \frac{227.5 \mathrm{~kJ} \cdot \mathrm{mol}^{-1} \cdot 1}{0.074 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1} \cdot 0.2875 \mathrm{~kg} \cdot \mathrm{m}^{-3} \cdot 1.199 \mathrm{~kJ} \cdot \mathrm{kg} \cdot \mathrm{K}^{-1}} $$ We need to convert the units of the numerator to have consistent units. $$ \Phi = \frac{227.5 \times 10^{3} \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot 1}{0.074 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1} \cdot 0.2875 \mathrm{~kg} \cdot \mathrm{m}^{-3} \cdot 1.199 \times 10^{3} \mathrm{~J} \cdot \mathrm{kg} \cdot \mathrm{K}^{-1}} $$ $$ \Phi = 3.53 $$ Since the calculated Thiele modulus (\(\Phi = 3.53\)) is much greater than 0.3, we can conclude that the intraparticle heat transfer limitation is not negligible under our current assumptions. However, this result may not be accurate if the actual reaction rate constant (k) is different from our assumed value.