Gas–liquid mass transfer theory:\nUnder the two-film theory, which relationship correctly links the liquid-side mass transfer coefficient KL with diffusivity DAB and effective film thickness Zf?

Introduction / Context:
The two-film (double-film) theory models interphase mass transfer by assuming thin stagnant films adjacent to the interface through which transport occurs by molecular diffusion. The bulk phases are well mixed outside these films. Recognizing the proportionality between the mass transfer coefficient and diffusivity divided by film thickness is foundational in reactor and separation design.

Given Data / Assumptions:

• Steady-state diffusion across a liquid film of thickness Zf.
• Fickian diffusion with diffusivity DAB.
• Concentration gradient is approximately linear across the film.

Concept / Approach:
For a species A diffusing through a stagnant film, the molar flux NA is given by NA = -DAB * dCA/dz. With a linear concentration drop ΔC across thickness Zf, NA ≈ DAB * (ΔC / Zf). By definition, NA = KL * (C* - Cb). Therefore, KL ≈ DAB / Zf when the driving force is represented by the concentration difference across the film. This captures the intuitive result: faster diffusion (higher DAB) and thinner films (smaller Zf) both increase KL.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis: DAB has units of length^2/time; dividing by length gives length/time, consistent with KL units. Empirical correlations refine this ideal with turbulence-based film thickness estimates.

Why Other Options Are Wrong:

• KL = DAB: missing film thickness; dimensionally incorrect.
• Zf / DAB or DAB * Zf: inverted or multiplied; wrong scaling and units.
• 1 / (DAB * Zf): unit mismatch; does not produce length/time.

Common Pitfalls:
Confusing overall Kla with liquid-side KL; a is interfacial area per volume and must not be conflated with film thickness.

Final Answer:
KL = DAB / Zf