Questions: Binary Phase Diagrams and Equilibrium

Question 1

An alloy has an overall composition of 30 wt% B. At a given temperature it falls in the α + liquid two-phase region. The tie line shows the α phase has composition 20 wt% B and the liquid phase has composition 50 wt% B. What is the weight fraction of liquid?

• A. 50% — the overall composition is midway between the two endpoints
• B. 33% — calculated as (30 − 20) / (50 − 20)
• C. 67% — the overall composition is closer to the liquid endpoint
• D. 30% — the weight fraction equals the overall composition

Answer: (B)

The lever rule gives the fraction of each phase by mass balance. Fraction of liquid = (x_overall − x_alpha) / (x_liquid − x_alpha) = (30 − 20) / (50 − 20) = 10/30 ≈ 33%. The key insight is that you measure the distance from the overall composition to the *opposite* phase's endpoint. Because the overall composition is only 10 units from the α endpoint but 20 units from the liquid endpoint, less of the liquid is needed to balance the mass — so the fraction of liquid is the smaller number. Option A incorrectly reads the midpoint as 50/50.

Question 2

At the eutectic point in a binary system at constant pressure, how many phases coexist simultaneously, and what does this imply about the system's degrees of freedom?

• A. One phase; one degree of freedom (temperature can vary)
• B. Two phases (liquid + one solid); one degree of freedom
• C. Three phases (liquid + two solids); zero degrees of freedom — temperature and compositions are all fixed
• D. Two phases; zero degrees of freedom because composition is fixed at the eutectic

Answer: (C)

At the eutectic point, liquid and both solid phases coexist simultaneously: three phases in a two-component system. Applying the Gibbs phase rule at constant pressure: F = C − P + 1 = 2 − 3 + 1 = 0. Zero degrees of freedom means the eutectic is an invariant point — both the temperature and the compositions of all three phases are fixed. You cannot change temperature or composition while keeping all three phases in equilibrium. This is why the eutectic appears as a single point on the diagram, not a region.

Question 3

In a two-phase region of a binary phase diagram, the composition of each phase depends on the overall alloy composition.

• T. True
• F. False

Answer: False

Phase compositions in a two-phase region are determined entirely by the tie line endpoints at that temperature — they do NOT change with overall composition. If you move horizontally across a two-phase region (changing overall composition while holding temperature constant), the tie line stays the same, so the compositions of both phases remain constant. What changes is only the *proportion* (fraction) of each phase, not what each phase is made of. This is one of the most important — and commonly missed — features of phase diagrams.

Question 4

The eutectic composition in a binary system melts at a lower temperature than either pure component.

• T. True
• F. False

Answer: True

This is the defining feature of a eutectic: it is the composition with the lowest melting point in the entire system, lower than either pure A or pure B. This happens because mixing lowers the free energy of the liquid phase more than that of the solids, stabilizing the liquid to lower temperatures. Practically, this is why lead-tin solder (near the eutectic) melts at ~183°C while pure lead melts at 327°C and pure tin at 232°C.

Question 5

Explain why the lever rule uses the distance to the *opposite* phase's endpoint to calculate the fraction of a given phase — what physical principle underlies this?

Model answer: The lever rule is a mass balance. For the overall composition to equal the weighted sum of the two phase compositions, a phase that is far from the overall composition must be present in a smaller amount, while a phase close to the overall composition must be present in a larger amount. Measuring the distance to the *opposite* endpoint captures how much of that phase must 'pull' the average toward the overall value. It works exactly like a lever or see-saw: the further a phase composition is from the fulcrum (overall composition), the less weight it needs on that side to balance.

If overall = 30%, α = 20%, liquid = 50%, then the liquid must be 'balanced' by enough α to bring the average to 30%. Since liquid (50%) is far above the target and α (20%) is close below, you need less liquid and more α — and the lever rule quantifies this. The formula fraction_liquid = (x_overall − x_α) / (x_liq − x_α) directly expresses this distance-weighting logic.