Question

A first-order heterogeneous reaction \( \text{A} \rightarrow \text{B} \) is carried out using a porous spherical catalyst. Assume isothermal conditions, and that intraphase diffusion controls the reaction rate. At a bulk \( \text{A} \) concentration of \( 0.3 \, \text{mol/L} \), the observed reaction rate in a \( 3 \, \text{mm} \) diameter catalyst particle is \( 0.2 \, \text{mol/s} \cdot \text{L}^{-1} \, \text{catalyst volume} \). At a bulk \( \text{A} \) concentration of \( 0.1 \, \text{mol/L} \), the observed reaction rate, in \( \text{mol/s} \cdot \text{L}^{-1} \, \text{catalyst volume} \), in a \( 6 \, \text{mm} \) diameter catalyst particle, is:

• A. 0.011
• B. 0.033
• C. 0.022
• D. 0.005

Hint:

For diffusion-controlled reactions, the observed rate is inversely proportional to the particle diameter and directly proportional to the concentration.

Answer 1

The Correct Option is B

Step 1: Intraphase diffusion-controlled reaction. For intraphase diffusion-controlled reactions, the observed reaction rate is proportional to the concentration and inversely proportional to the particle diameter. The relationship is: \[ r_{\text{obs}} \propto \frac{C}{D_p}. \] Step 2: Relate the observed rates. Let the observed rates for the \( 3 \, \text{mm} \) and \( 6 \, \text{mm} \) particles be \( r_1 \) and \( r_2 \), respectively. The relationship is: \[ \frac{r_2}{r_1} = \frac{C_2}{C_1} \cdot \frac{D_{p1}}{D_{p2}}. \] Substitute \( C_1 = 0.3 \, \text{mol/L}, C_2 = 0.1 \, \text{mol/L}, D_{p1} = 3 \, \text{mm}, D_{p2} = 6 \, \text{mm}, r_1 = 0.2 \): \[ r_2 = r_1 \cdot \frac{C_2}{C_1} \cdot \frac{D_{p1}}{D_{p2}} = 0.2 \cdot \frac{0.1}{0.3} \cdot \frac{3}{6}. \] Step 3: Simplify the calculation. \[ r_2 = 0.2 \cdot \frac{1}{3} \cdot \frac{1}{2} = 0.033 \, \text{mol/s} \cdot \text{L}^{-1}. \] Step 4: Conclusion. The observed reaction rate is \( 0.033 \, \text{mol/s} \cdot \text{L}^{-1} \, \text{catalyst volume} \).