Surface renewal (Danckwerts) theory predicts how the liquid-side mass transfer coefficient K_L depends on molecular diffusivity. Which statement best describes this dependence?

Introduction:
Surface renewal theory extends penetration ideas by introducing a renewal rate, describing the stochastic replacement of surface elements. It is widely used to interpret k_La trends in agitated tanks and gas–liquid contactors.

Given Data / Assumptions:

• Random renewal of surface elements with frequency s (s^-1).
• Unsteady diffusion dominates during each exposure period.
• Diffusivity D determines how quickly concentration boundary layers develop.

Concept / Approach:
Danckwerts derived K_L proportional to (D * s)^0.5, so holding renewal rate s constant, K_L ∝ D^0.5. This square-root dependence reflects diffusion length scaling with time to the power 0.5 in unsteady processes, linking molecular properties with hydrodynamic renewal dynamics.

Step-by-Step Solution:

Verification / Alternative check:
Experimental correlations of K_L at different solutes (varying D) often follow a half-power law, consistent with surface renewal predictions.

Why Other Options Are Wrong:

• Inverse or cube-root dependences are not supported by the Danckwerts derivation.

Common Pitfalls:
Interpreting square-root dependence as universal; hydrodynamics (s) also changes with agitation and aeration, affecting K_L beyond molecular properties alone.

Final Answer:
directly proportional to the square root of the molecular diffusivity