Question

According to penetration theory, the mass transfer coefficient is proportional to______

• A. Diffusivity
• B. (Diffusivity)$^2$
• C. (Diffusivity)$^{0.5}$
• D. (Diffusivity)$^{1.5}$

Hint:

Remember the diffusivity exponents for the three major mass transfer theories:
Film Theory: \( D^{1.0} \)
Boundary Layer Theory: \( D^{0.67} \)
Penetration / Surface Renewal Theory: \( D^{0.50} \)
These exponents are highly important for exam questions.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the relationship between the mass transfer coefficient (\( k_L \)) and the molecular diffusivity (\( D_{AB} \)) as predicted by Higbie's Penetration Theory.
This is a fundamental mass transfer theory used to describe fluid-fluid interfaces.

Step 2: Key Formula or Approach:
Higbie's penetration theory assumes that turbulent eddies from the bulk liquid travel to the interface, remain there for a constant exposure time \( t_e \), and undergo unsteady-state molecular diffusion before returning to the bulk.
The average mass transfer coefficient is given by:
\[ k_L = 2 \cdot \sqrt{\frac{D_{AB}}{\pi \cdot t_e}} \]

Step 3: Detailed Explanation:

• Higbie's Equation Analysis: From the formula for the mass transfer coefficient, we can isolate the dependence on molecular diffusivity \( D_{AB} \):
\[ k_L \propto \sqrt{D_{AB}} \quad \implies \quad k_L \propto D_{AB}^{0.5} \]

• Comparison with other theories:
1.

Film Theory: Assumes a stagnant film at the interface where mass transfer occurs via steady-state molecular diffusion.
The mass transfer coefficient is directly proportional to diffusivity:
\[ k_L \propto D_{AB}^{1.0} \]
2.

Boundary Layer Theory: Predicts that the mass transfer coefficient is proportional to a fractional power of diffusivity:
\[ k_L \propto D_{AB}^{2/3} \approx D_{AB}^{0.67} \]
3.

Surface Renewal Theory (Danckwerts): Assumes a random distribution of surface ages and exposure times, but still predicts:
\[ k_L \propto D_{AB}^{0.5} \]


Step 4: Final Answer:
According to penetration theory, the mass transfer coefficient is proportional to the square root of diffusivity, \( D_{AB}^{0.5} \).