McCabe–Thiele analysis: what is the slope of the operating line in the rectifying section (assume constant molal overflow and total condenser)?

Introduction / Context:Binary distillation under the constant-molal-overflow assumption yields straight operating lines on the y–x diagram. The rectifying section line characterizes mass balances above the feed tray and is central to stage-count estimation.

Given Data / Assumptions:

• Total condenser; no side draws.
• Constant molal overflow; negligible heat of mixing.
• Reflux ratio R = L/V.

Concept / Approach:The rectifying line can be written as y = (R/(R+1)) x + xD/(R+1). The slope is L/V = R/(R+1), which is always less than 1 for finite reflux (R > 0). Only in the limiting case of infinite reflux does the slope approach 1, not exceed it.

Step-by-Step Solution:Write the rectifying operating line: y = (L/V) x + xD (1 − L/V).Recognize slope = L/V = R/(R+1).Since R > 0, R/(R+1) < 1 → choose “< 1.”

Verification / Alternative check:At minimum reflux, the slope is relatively small; as R increases, the slope approaches 1 from below, consistent with textbook diagrams.

Why Other Options Are Wrong:0: would imply no internal liquid flow (not rectifying).∞: vertical line—nonsensical for operating lines.> 1: impossible under constant molal overflow without side draws.

Common Pitfalls:Mixing rectifying and stripping lines; forgetting the intercept is xD/(R+1).

Final Answer:<1