Question

In solid catalysed reactions, the diffusional effects are more likely to affect the overall rate of reaction for:

• A. Fast reactions in catalyst of small pore diameter
• B. Fast reactions in catalyst of large pore diameter
• C. Slow reactions in catalyst of small pore diameter
• D. Slow reactions in catalyst of large pore diameter

Hint:

To maximize internal pore diffusion limitations, look for conditions that maximize the Thiele Modulus (\(\phi = L\sqrt{k/\mathcal{D}_e}\)): - High intrinsic reaction rate (\(k \uparrow\)) \(\rightarrow\) Fast reaction. - Low internal diffusivity (\(\mathcal{D}_e \downarrow\)) \(\rightarrow\) Small pore diameter (restricts transport via Knudsen diffusion).

Answer 1

The Correct Option is A

Concept: In heterogeneous, solid-catalyzed reactions, the overall observed rate of reaction is determined by a combination of the intrinsic chemical reaction rate on the catalytic surface and the rates of mass transfer (diffusion) of the reactants through the catalyst pores. The importance of internal pore diffusion resistance relative to the intrinsic reaction rate is quantified using the dimensionless Thiele Modulus (denoted by $\phi$). For a first-order reaction occurring inside a porous catalyst pellet, the Thiele modulus is defined mathematically as: \[ \phi = L \cdot \sqrt{\frac{k}{\mathcal{D}_e}} \] Where:
• \( L \) represents the characteristic length scale of the catalyst pellet.
• \( k \) represents the intrinsic chemical reaction rate constant.
• \( \mathcal{D}_e \) represents the effective internal diffusion coefficient inside the pores.

Step 1: Evaluating the impact of reaction speed.
Let us analyze how the intrinsic reaction speed affects the Thiele modulus and diffusion limitations:
• For Fast Reactions, the rate constant $k$ is large ($k \uparrow\uparrow$). This causes the Thiele modulus ($\phi$) to become large: \[ k \uparrow \quad \Rightarrow \quad \phi \rightarrow \text{large} \] A large Thiele modulus means that the reactants are consumed rapidly as soon as they enter the pore mouths, before they can diffuse deep into the catalyst pellet. This creates a steep concentration gradient and makes the overall process highly limited by internal diffusion resistance.

Step 2: Evaluating the impact of pore diameter.
Let us analyze how the pore diameter affects the effective diffusion coefficient ($\mathcal{D}_e$):
• When a catalyst pellet has a small pore diameter, the pore dimensions are often smaller than the mean free path of the gas molecules. Under these conditions, the transport mechanism is dominated by Knudsen diffusion rather than bulk diffusion. The Knudsen diffusivity ($\mathcal{D}_K$) scales directly with the pore diameter ($d_p$): \[ \mathcal{D}_K \propto d_p \] As the pore diameter decreases ($d_p \downarrow$), the diffusion coefficient drops significantly ($\mathcal{D}_e \downarrow\downarrow$). A lower diffusion coefficient increases the value of the Thiele modulus: \[ \mathcal{D}_e \downarrow \quad \Rightarrow \quad \phi = L \cdot \sqrt{\frac{k}{\mathcal{D}_e}} \uparrow\uparrow \]

Step 3: Conclusion.
Combining both factors, a system with a large intrinsic reaction rate ($k$) and a small pore diameter (which lowers $\mathcal{D}_e$) maximizes the value of the Thiele modulus ($\phi \gg 1$). Under these conditions, the internal effectiveness factor ($\eta$) drops below 1 ($\eta \approx 1/\phi$), meaning that the overall reaction rate is heavily limited by internal pore diffusion resistance. Therefore, diffusional effects are most likely to limit the overall rate of reaction for fast reactions in catalysts of small pore diameter.