Question

According to the penetration theory of mass transfer the mass transfer coefficient varies with diffusion coefficient (D) of the diffusing species as

• A. D
• B. D\(^0.5\)
• C. D\(^0.5\)
• D. D\(^1.5\)

Hint:

In mass transfer, the penetration theory suggests that the mass transfer coefficient is proportional to the square root of the diffusion coefficient.

Answer 1

The Correct Option is B

Step 1: Understanding the Penetration Theory. 
The penetration theory is used to describe the mass transfer in a system where a species diffuses into a stagnant fluid. According to this theory, the mass transfer coefficient (\(k\)) is related to the diffusion coefficient (\(D\)) by the following relationship: \[ k \propto D^{0.5} \] This means that the mass transfer coefficient varies as the square root of the diffusion coefficient. 
Step 2: Conclusion. 
The correct answer is (2), as per the penetration theory, the mass transfer coefficient varies with \(D^{0.5}\). 
 

Answer 2

Penetration theory (Higbie) models mass transfer at a fluid interface as unsteady-state diffusion into a fluid element that is exposed to the interface for a short contact time before being replaced by fresh fluid. The mass transfer coefficient predicted by this theory is \( k = 2\sqrt{\dfrac{D}{\pi t_c}} \), where \( t_c \) is the contact time. Let's check the dependence on \( D \) implied by each option.

1. D: A first-power dependence on \( D \) is what film theory predicts (\( k = D/\delta \), with a fixed film thickness), not penetration theory.
2. D\(^{0.5}\): Since \( k = 2\sqrt{D/(\pi t_c)} \), the mass transfer coefficient scales with the square root of the diffusion coefficient, exactly matching this option.
3. D\(^{0.5}\): This is the same square-root dependence as above, and it is likewise consistent with the penetration theory result.
4. D\(^{1.5}\): A three-halves power dependence is far stronger than what unsteady-state diffusion into a semi-infinite fluid element predicts.

Penetration theory gives a square-root dependence of the mass transfer coefficient on the diffusion coefficient.

Therefore, the correct answer is \( k \propto D^{0.5} \).