Membrane separations — If a solute passes through a membrane freely, what is the value of the rejection coefficient s?

Introduction:
The rejection coefficient s (also called sigma) quantifies solute retention by a membrane. It is defined such that s = 1 indicates complete rejection and s = 0 indicates no rejection (free passage). Understanding s is essential for designing ultrafiltration, nanofiltration, and reverse osmosis steps.

Given Data / Assumptions:

• Definition: s = 1 − (C_p / C_f), where C_p is permeate concentration and C_f is feed or bulk concentration at the membrane surface.
• Steady operating conditions; negligible concentration polarization for the qualitative conclusion.
• Solute passes freely.

Concept / Approach:
If the solute passes freely, then C_p ≈ C_f, giving s ≈ 0. As retention increases, C_p decreases relative to C_f and s increases toward 1. Values outside 0 to 1 are nonphysical for ideal definitions and data without artifacts.

Step-by-Step Solution:
Start from s = 1 − (C_p / C_f).For free passage: C_p / C_f ≈ 1.Compute s ≈ 1 − 1 = 0.

Verification / Alternative check:
Membrane characterization tables list s near zero for fully permeating tracers and near unity for fully retained macromolecules, matching the calculation.

Why Other Options Are Wrong:

• 1 implies total rejection, not free passage.
• > 1 or negative contradict the standard definition.
• < 1 but greater than 0 implies partial rejection, not free passage.

Common Pitfalls:
Using bulk feed concentration instead of membrane-surface concentration when polarization is significant; misinterpreting observed s due to analytical noise at very low C_p.

Final Answer:
0