Dilute solutions — A very dilute solution is prepared by dissolving x1 moles of solute in x2 moles of solvent. The mole fraction of solute is approximately equal to:

Introduction / Context:
Mole fraction is a fundamental composition variable in thermodynamics and phase equilibria. For very dilute solutions, convenient approximations simplify calculations while maintaining acceptable accuracy. This question tests your ability to derive a first-order approximation for the mole fraction of solute when the solute amount is much smaller than the solvent amount.

Given Data / Assumptions:

• Moles of solute = x1, moles of solvent = x2.
• Very dilute condition: x1 ≪ x2.
• Ideal mixing and no volume change on mixing (approximation irrelevant to mole fraction but common context).

Concept / Approach:

By definition, mole fraction of solute X_solute = x1 / (x1 + x2). Under the dilute condition x1 ≪ x2, the denominator x1 + x2 ≈ x2. Therefore X_solute ≈ x1/x2. This approximation is widely used to linearize expressions in mass transfer and colligative property estimations for trace solutes.

Step-by-Step Solution:

Verification / Alternative check:

Numerical example: x1 = 0.01, x2 = 100 → exact X = 0.01 / 100.01 ≈ 9.999e-5; approximate X ≈ 0.01 / 100 = 1.0e-4. Relative error is <0.001%.

Why Other Options Are Wrong:

B, D invert or distort the relationship. C approaches 1 and is nonsensical for a dilute solute. E is exact but the question asks for an approximate expression capturing the dilute simplification.

Common Pitfalls:

Applying the approximation outside the very dilute regime; forgetting that exact mole fraction is bounded between 0 and 1.

Final Answer:

x1/x2