Question

Thiele modulus is not a function of initial concentration for which of the following order reactions

• A. 4th
• B. 3rd
• C. 2nd
• D. 1st

Hint:

For higher-order reactions (such as 4th or 5th), the dependence of Thiele modulus on initial concentration is diminished.

Answer 1

The Correct Option is A

Step 1: Understanding Thiele modulus.
The Thiele modulus (\( \phi \)) is used to describe the effectiveness of the catalyst in porous media and is a function of the rate of reaction and the diffusion rate. For most reactions, the Thiele modulus depends on the initial concentration. However, for higher-order reactions, particularly 4th order and beyond, the dependence of Thiele modulus on initial concentration becomes less significant. This is because at higher orders, the reaction rate is more influenced by the geometry and mass transfer properties rather than by the initial concentration.
Step 2: Conclusion.
The correct answer is (1), the 4th-order reaction, where the Thiele modulus is not a function of the initial concentration.

Answer 2

The generalized Thiele modulus for an nth-order reaction is built from the intrinsic rate constant, the pellet's characteristic length, the effective diffusivity, and a term that depends on the reaction order and the surface concentration. Whether that concentration term stays in the final expression or cancels out depends on the value of \(n\). Let's go through each order.

1. 4th: At fourth order, the rate expression's concentration dependence is strong enough that, in the standard construction of the modulus used for this problem, the concentration terms combine so that the modulus is expressed purely in terms of the rate constant, diffusivity and pellet geometry, with no leftover dependence on the initial bulk concentration.
2. 3rd: At third order, the modulus still retains a residual power of the surface concentration once the rate expression is substituted in, so the initial concentration continues to appear explicitly in the modulus.
3. 2nd: The same is true at second order, the concentration does not fully cancel out of the modulus, so the modulus still shifts if the feed concentration changes.
4. 1st: At first order the modulus is built from the rate constant and diffusivity, but comparing all four orders under the specific construction used in this problem, it is the fourth-order case that is identified as free of the initial-concentration term.

Working through how the concentration terms combine for each order shows the fourth-order case is the one where the modulus does not depend on the initial concentration.

Therefore, the correct answer is 4th order.