According to the penetration theory of mass transfer, which was proposed by Higbie, the mass transfer process occurs through transient diffusion. This theory is particularly used when a gas comes into contact with a liquid for a short period.
Under these transient conditions, the mass transfer coefficient \( k \) is derived from the unsteady-state diffusion equation and is found to be proportional to the square root of the diffusivity \( D \). Mathematically, the relationship is expressed as:
\[
k \propto \sqrt{D} \text{or} k \propto D^{0.5}
\]
This implies that as the diffusivity increases, the mass transfer coefficient increases, but at a decreasing rate (due to the square root dependency).
The other options represent incorrect dependencies:
- \( k \propto D^{1.5} \): Overestimates the influence of diffusivity.
- \( k \propto D^2 \): Not supported by penetration theory.
- \( k \propto D \): Linear dependence is not applicable in transient diffusion.
Hence, the correct relationship as per penetration theory is \( k \propto D^{0.5} \).