Distillation (McCabe–Thiele): the slope of the feed line (q-line) in terms of the liquid fraction q of the feed is equal to which expression?

Introduction / Context:
In binary distillation design using the McCabe–Thiele method, the feed condition is incorporated via the q-line. The q-line slope depends on the fraction of the feed that is liquid (q). Correctly using this slope is essential to intersect the operating lines at the proper location.

Given Data / Assumptions:

• q is the liquid fraction of feed (0 ≤ q ≤ 1 for saturated two-phase feeds; can be < 0 or > 1 for subcooled or superheated feeds).
• Vapor–liquid equilibrium representation on x–y diagram.
• Constant molal overflow approximation.

Concept / Approach:
The q-line relation is y = (q/(q−1)) x − z_F/(q−1), where z_F is the feed composition. Thus, the slope equals q/(q−1). Algebraically, q/(q−1) is identical to −q/(1−q). Hence the negative-form expression is often used in option sets.

Step-by-Step Solution:
Start from component balances around the feed stage.Derive q-line: y = (q/(q−1)) x − z_F/(q−1).Identify slope m_q = q/(q−1) = −q/(1−q).Match to options: −q/(1−q) is present.

Verification / Alternative check:
Check special cases: q = 1 (saturated liquid) → slope → ∞ (vertical line). q = 0 (saturated vapor) → slope 0 (horizontal). The expression −q/(1−q) reproduces these limits.

Why Other Options Are Wrong:
−q alone is not dimensionally or physically correct.−q/(q−1) equals q/(1−q), not the standard slope; option (b) is the correct equivalent form.'None of these' is incorrect because a valid expression is provided.

Common Pitfalls:
Sign mistakes; misinterpreting q for subcooled/superheated feeds; mixing compositions z_F with x and y.

Final Answer:
-q/(1-q)