Effectiveness factor η for immobilized enzymes: how do we quantify the extent to which diffusion slows the reaction rate relative to the intrinsic (no-diffusion-limit) rate?

Introduction:
The effectiveness factor η captures how internal and external mass transfer resistances depress the observable reaction rate below the intrinsic kinetic potential. It is widely used with porous catalysts and immobilized enzymes to indicate whether pore diffusion or film limitations are significant.

Given Data / Assumptions:

• Intrinsic rate refers to the rate at uniform surface/interior concentration equal to bulk (no diffusion limitation).
• Actual rate is the measured volumetric rate under operating conditions, including transport resistances.
• η is dimensionless and bounded 0 < η ≤ 1 in classical definitions.

Concept / Approach:
By definition, η = (actual rate) / (rate if not slowed by diffusion). Values much less than 1 indicate significant diffusion limitation; values close to 1 indicate kinetic control. Engineers use η to decide whether to decrease particle size, increase temperature (raising diffusivity), or change support porosity.

Step-by-Step Solution:
Step 1: Determine the intrinsic kinetic expression (e.g., Michaelis–Menten or power-law) at the relevant C_s.Step 2: Measure or compute the observed rate under the same bulk conditions.Step 3: Take the ratio η = r_actual / r_intrinsic.Step 4: Interpret η to guide process changes to mitigate diffusion effects.

Verification / Alternative check:
Thiele modulus and effectiveness factor relationships for porous catalysts provide analytical connections between η and internal diffusion severity, supporting the above definition.

Why Other Options Are Wrong:

• Inverted ratios correspond to 1/η, not η itself.
• Mass transfer rate over reaction rate is a different diagnostic, not the standard η definition.

Common Pitfalls:
Comparing rates at different bulk concentrations or temperatures; η must compare under identical intrinsic driving forces to be meaningful.

Final Answer:
actual reaction rate/ rate if not slowed by diffusion