For an immobilized enzyme particle, how does the internal effectiveness factor (η) change with solute diffusivity and particle size, assuming reaction is at least partly diffusion-limited?

Introduction:
The internal effectiveness factor η compares the actual rate inside a porous catalyst or gel particle to the rate if the entire interior were at the bulk substrate concentration. It captures diffusion-reaction interplay and guides selection of particle size and support porosity for immobilized enzymes.

Given Data / Assumptions:

• Reaction exhibits some degree of internal diffusion limitation.
• Particles are porous beads or gels containing the enzyme.
• Diffusivity refers to effective diffusivity inside pores, not in bulk liquid.

Concept / Approach:
Higher effective diffusivity reduces concentration gradients inside particles, raising η toward 1. Larger particles increase diffusion path length and the Thiele modulus, steepening internal gradients and lowering η. Thus η increases with diffusivity and typically decreases as particle size increases, all else equal.

Step-by-Step Solution:

Verification / Alternative check:
Modeling via Thiele modulus (phi) shows η decreases as phi increases; phi grows with particle size and decreases with higher diffusivity, matching the qualitative trends.

Why Other Options Are Wrong:

• Increasing size increasing η contradicts diffusion limitations.
• Decreasing diffusivity decreasing η is true, but option wording combines trends incorrectly.
• Option d mixes a correct effect of diffusivity with an incorrect size trend.

Common Pitfalls:
Ignoring external film resistance can misattribute low rates to internal diffusion. Ensure proper agitation and avoid bead swelling that reduces pore connectivity.

Final Answer:
increase of diffusivity and decreased with the increase of particle size