Film theory – dependence of mass transfer coefficient on diffusivity According to the classical film theory for gas–liquid or liquid–liquid mass transfer, how does the mass transfer coefficient K depend on molecular diffusivity D?

Introduction / Context:
Rate theories for mass transfer (film, penetration, surface renewal) differ in their predicted dependence of the mass transfer coefficient on diffusivity. Choosing the appropriate scaling helps compare systems, estimate rates, and understand when enhancement mechanisms dominate.

Given Data / Assumptions:

• Film theory assumes a stagnant film of thickness δ next to the interface.
• Within the film, transport is by steady-state molecular diffusion.
• Bulk phases are well mixed beyond the film.

Concept / Approach:
In film theory, the mass transfer coefficient is K ≈ D/δ. If δ is treated as independent of D for a given hydrodynamic condition, the proportionality reduces to K ∝ D. In contrast, the penetration and surface renewal theories predict K ∝ D^0.5. Recognizing these different exponents allows selection of the correct model for unsteady interfacial contact or turbulent renewal conditions.

Step-by-Step Solution:

Verification / Alternative check:
Measured Sherwood–Reynolds–Schmidt correlations often show D^n with n between 0.33 and 0.67, but the idealized film model specifically yields n = 1 when δ is fixed by hydrodynamics.

Why Other Options Are Wrong:

• D^0.5 corresponds to penetration/surface-renewal models, not classical steady film.
• D^1.5 or D^2 drastically overstate the role of diffusivity and lack theoretical basis here.

Common Pitfalls:
Assuming one theory fits all regimes; in highly turbulent interfaces with short contact times, D^0.5 behavior is more appropriate than the linear dependence.

Final Answer:
K ∝ D