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Question #44720

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The Question

In an industrial reverse osmosis (RO) plant, an aqueous solution of a reactive pollutant \ce{A} (bulk concentration $C_b = 10 \text{ mol m}^{-3}$ ) is processed at $298 \text{ K}$ . The RO membrane perfectly rejects \ce{A} ( $R = 100\%$ ). Due to concentration polarization, a boundary layer exists at the membrane surface with a mass transfer coefficient $k_m = 1.0 \times 10^{-5} \text{ m s}^{-1}$ . The steady-state water flux through the membrane is $J_w = 2.0 \times 10^{-5} \text{ m s}^{-1}$ .

At the membrane surface, \ce{A} undergoes a first-order degradation reaction $\ce{A -> B}$ with a surface rate constant $k_s = 1.0 \times 10^{-5} \text{ m s}^{-1}$ . This reaction occurs exclusively at the membrane surface.

Assuming steady-state film theory, what is the rate of degradation of \ce{A} per unit membrane area ( $r''$ )?

(Note: $e$ is the base of the natural logarithm)

Options

A.
$\frac{2 \times 10^{-4} e^2}{e^2 + 1} \text{ mol m}^{-2} \text{s}^{-1}$
B.
$1 \times 10^{-4} e^2 \text{ mol m}^{-2} \text{s}^{-1}$
C.
$1 \times 10^{-4} \text{ mol m}^{-2} \text{s}^{-1}$
D.
$\frac{2 \times 10^{-4} e^2}{e^2 - 1} \text{ mol m}^{-2} \text{s}^{-1}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Numerical_Precision

Improve calculation stamina and avoid rounding errors in high-stakes competitive examinations.

Concepts Behind This Question

Reverse Osmosis Dynamics|Stoichiometric Coefficient Division|Rate Law Derivation from Mechanism

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of Reverse Osmosis Dynamics and Chemical Kinetics.

The 'Hidden Variable' is the modification of the standard concentration polarization boundary condition.

Students typically memorize the standard formula $C_m = C_b \exp(J_w/k_m)$ , which assumes zero solute flux at the membrane surface ( $J_A = 0$ ).

However, the presence of a surface degradation reaction acts as a continuous sink, changing the boundary condition to $J_A = k_s C_m$ .

Failing to account for this coupled transport-reaction mechanism leads to the incorrect application of the standard formula (Option B) or ignoring polarization entirely (Option C).

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Solutions - Class 12
Concept	Reverse Osmosis Dynamics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free