Q.54 An enzyme immobilised in a porous spherical pellet, catalyses a strongly
mass-transfer limited first-order reaction. The effectiveness factor for the
immobilised enzyme reaction increases with the
(A) decrease in the size of the pellet
(B) increase in the pore diffusivity within the pellet
(C) decrease in the enzyme turnover number
(D) increase in the enzyme concentration within the pellet
The effectiveness factor (η) measures how efficiently an immobilized enzyme utilizes its catalytic potential, particularly under mass-transfer limitations in porous supports like spherical pellets.
Correct Answer
For a strongly mass-transfer limited first-order reaction in a porous spherical pellet, the effectiveness factor increases with (A) decrease in the size of the pellet. Smaller pellets reduce the Thiele modulus (φ), which quantifies diffusion-reaction competition, leading to higher η closer to 1.
Key Concept: Thiele Modulus
The Thiele modulus φ for first-order kinetics is given by φ = (R/3) * sqrt(k / D_eff), where R is pellet radius, k is the reaction rate constant (proportional to enzyme concentration and turnover), and D_eff is effective pore diffusivity.
Effectiveness factor η = (3/φ) * (1/tanh(φ) – 1/φ), which increases as φ decreases.
In mass-transfer limited regimes (high φ), internal diffusion dominates, making η sensitive to pellet geometry and transport properties.
Option Analysis
| Option | Explanation | Impact on η |
|---|---|---|
| (A) Decrease in pellet size | Reduces R, lowering φ and diffusion path length, minimizing concentration gradients inside the pellet . | Increases η |
| (B) Increase in pore diffusivity | Raises D_eff, reducing φ and enhancing substrate access to enzyme sites within pores . | Increases η |
| (C) Decrease in enzyme turnover number | Lowers k (turnover number k_cat reflects intrinsic speed), reducing φ and mass-transfer limitation . | Increases η |
| (D) Increase in enzyme concentration | Raises k (via higher V_max), increasing φ and strengthening reaction-diffusion competition . | Decreases η |
Options A, B, and C all increase η by lowering φ, but the question implies a single best answer in standard GATE context, with A directly addressing geometric control in pellets.
Practical Implications
In bioprocess design, smaller pellets or higher diffusivity optimize immobilized enzyme reactors for first-order kinetics, though pressure drop and mechanical stability must balance.
This principle applies to biotechnological applications like biosensors and packed-bed reactors.
