Question

For a first order isothermal catalytic reaction A to B, occurring in an infinitely long cylindrical pore, the relationship between effectiveness factor \(\epsilon\) and thiele modulus \(\phi\), is

• A. \(\epsilon = \frac{1}{\phi^2}\)
• B. \(\epsilon = \phi\)
• C. \(\epsilon = 1\)
• D. \(\epsilon = \frac{1}{\phi}\)

Hint:

For reactions in porous media, understanding the relationship between the effectiveness factor and thiele modulus is crucial for determining catalyst efficiency.

Answer 1

The Correct Option is A

Step 1: Understanding the relation.
The effectiveness factor \(\epsilon\) and thiele modulus \(\phi\) are related in catalytic reactions occurring in porous media. The relationship for a first-order isothermal catalytic reaction in an infinitely long cylindrical pore is given by: \[ \epsilon = \frac{1}{\phi^2} \] This relationship reflects the behavior of the catalyst effectiveness depending on the diffusion and reaction rates.
Step 2: Analyzing the options.
(1) \(\epsilon = \frac{1}{\phi^2}\): This is the correct relationship for a first-order reaction occurring in an infinitely long cylindrical pore.
(2) \(\epsilon = \phi\): This is incorrect because it does not match the expected relationship.
(3) \(\epsilon = 1\): This is incorrect because it does not account for the influence of the thiele modulus.
(4) \(\epsilon = \frac{1}{\phi}\): This is incorrect as it does not reflect the correct relationship for the given scenario.
Step 3: Conclusion.
The correct answer is \(\epsilon = \frac{1}{\phi^2}\), as it matches the established relationship for this reaction type.

Answer 2

The effectiveness factor \( \epsilon \) measures how much internal diffusion resistance suppresses the observed reaction rate compared to the intrinsic rate, and it must always lie between 0 and 1, approaching a small fraction as the Thiele modulus \( \phi \) grows large (strong diffusion limitation). Let's test each proposed relationship against this requirement.

1. \( \epsilon = \frac{1}{\phi^2} \): As \( \phi \) grows large, this expression falls off steeply toward zero, consistent with an effectiveness factor that shrinks sharply once diffusion resistance dominates strongly in a long, narrow cylindrical pore geometry.
2. \( \epsilon = \phi \): This grows without bound as \( \phi \) increases, which is not physically sensible for an effectiveness factor, since \( \epsilon \) must stay bounded between 0 and 1 by definition; this relationship must be rejected on that basis alone.
3. \( \epsilon = 1 \): A constant effectiveness factor of exactly one would mean the observed rate always equals the intrinsic kinetic rate regardless of \( \phi \), which ignores diffusion resistance altogether.
4. \( \epsilon = \frac{1}{\phi} \): This decays more gradually than the correct relationship for this pore geometry and reaction order, understating how strongly diffusion resistance suppresses the effective rate in an infinitely long cylindrical pore under first-order, isothermal conditions.

Ruling out the relationships that either grow without bound, stay fixed at unity, or decay too gradually, the steep inverse-square decay is the one consistent with strong diffusional control in this geometry.

Therefore, the correct answer is \( \epsilon = \dfrac{1}{\phi^2} \).