Membrane rejection: If a membrane completely rejects the solute (no solute passes), what is the value of the rejection coefficient R (sometimes denoted s or σ in simplified forms)?

Introduction:Rejection quantifies how effectively a membrane excludes a solute. It is widely used in UF/NF/RO specifications and process control. Understanding its limiting values helps interpret analyzer readings and calculate product purity.

Given Data / Assumptions:

• Definition: R = 1 − (C_p/C_f), where C_p is permeate concentration and C_f is feed (or bulk) concentration.
• Complete rejection implies C_p ≈ 0 relative to C_f.
• Well-mixed bulk; concentration polarization effects are ignored for the definition.

Concept / Approach:By definition, if no solute appears in the permeate, C_p → 0 and R → 1. If solute freely passes, C_p ≈ C_f and R ≈ 0. Values greater than 1 or negative are unphysical for the simple definition and indicate measurement or definition issues (unless using specialized “sieving” forms with different baselines).

Step-by-Step Solution:Write R = 1 − C_p/C_f.Set C_p = 0 for complete rejection ⇒ R = 1 − 0 = 1.Interpretation: 100% of solute is retained by the membrane.

Verification / Alternative check:Manufacturer specs listing “rejection ≥ 95%” correspond to R ≥ 0.95, consistent with the definition. A perfect, idealized membrane would report R = 1 for that solute.

Why Other Options Are Wrong:R = 0: implies no rejection.

R > 1 or R < 1 (other than 1): inconsistent with the standard definition for simple systems.

Common Pitfalls:

• Mixing up rejection with the reflection coefficient σ in coupled-flow equations; many texts align σ with R under certain limits, but definitions differ by context.
• Using retentate concentration instead of feed in the R formula without noting the basis.

Final Answer:1