Rectifying section operating line: When does its slope become unity in a binary distillation column?

Introduction / Context:
The operating line in the rectifying section relates vapour and liquid compositions via the external reflux. Its slope is a direct function of the reflux ratio and is a key feature of McCabe–Thiele construction.

Given Data / Assumptions:

• Binary distillation, constant molar overflow approximation.
• Rectifying operating line: y = (R/(R+1)) x + x_D/(R+1).

Concept / Approach:
The slope of the rectifying operating line equals R/(R+1). As R → ∞ (total reflux), slope → 1 and the intercept tends to 0, collapsing the operating line onto the 45° diagonal. This condition yields the minimum number of theoretical stages (but infinite utilities).

Step-by-Step Solution:

Verification / Alternative check:
Graphing m versus R confirms the asymptote at 1 as R grows large, consistent with textbook derivations.

Why Other Options Are Wrong:

• R = 0 gives slope 0, not 1.
• R = 1 gives slope 1/2.
• Minimum reflux gives a line tangent to the equilibrium curve, not slope 1.

Common Pitfalls:
Confusing minimum reflux (minimum stages infinite) with total reflux (minimum stages finite); they are opposite extremes.

Final Answer:
At total (infinite) reflux.