At a eutectic point in a binary alloy system, how many phases coexist in equilibrium?

Introduction / Context:
The eutectic point is a classic feature of binary phase diagrams. It is the unique composition and temperature at which a liquid transforms into two solid phases simultaneously upon cooling (or the reverse upon heating). Understanding the number of phases present is vital for materials design, solder selection, and heat-treatment scheduling.

Given Data / Assumptions:

• Eutectic refers to a binary system (two components).
• At the eutectic temperature and composition, the system is at invariant equilibrium under constant pressure.
• Phases are defined as physically distinct, homogeneous regions.

Concept / Approach:
Gibbs’ phase rule for condensed systems at constant pressure is F = C − P + 1. At the eutectic, degrees of freedom F = 0 (invariant), and components C = 2; solving for P gives P = C + 1 = 3. These three phases are the liquid plus two distinct solids (often denoted α and β). Thus, at the eutectic point, exactly three phases coexist in equilibrium.

Step-by-Step Solution:
Write phase rule at constant pressure: F = C − P + 1.For eutectic, F = 0 and C = 2.Solve: 0 = 2 − P + 1 → P = 3.Interpretation: liquid + two solids coexist.

Verification / Alternative check:
Inspection of typical Pb–Sn or NaCl–H2O phase diagrams shows a “V” shaped liquidus meeting at a eutectic point where solid α and solid β precipitate together from the liquid—three phases at once.

Why Other Options Are Wrong:
1 or 2 phases do not satisfy the invariant condition at the eutectic.“Unpredictable” is incorrect; the phase rule precisely predicts three phases.

Common Pitfalls:
Confusing eutectic isotherm with peritectic reactions; assuming only two solids without the liquid present at the point; misapplying the full (non-condensed) phase rule.

Final Answer:
3