Questions: The Lever Rule and Phase Diagram Composition Analysis

Question 1

An alloy with overall composition C₀ = 60 wt% B is in a two-phase (α + β) region. The tie-line endpoints are C_α = 40 wt% B and C_β = 80 wt% B. What is the weight fraction of the α phase?

• A. 25% — because C₀ is one-quarter of the way from C_α to C_β
• B. 50% — because (C_β − C₀)/(C_β − C_α) = (80 − 60)/(80 − 40) = 20/40 = 0.50
• C. 67% — because C₀ is closer to C_β, so most of the alloy is α
• D. 75% — because the β side of the tie line is longer, indicating more α

Answer: (B)

The lever rule formula for weight fraction of α is f_α = (C_β − C₀)/(C_β − C_α). Plugging in: (80 − 60)/(80 − 40) = 20/40 = 0.50. The key to applying the lever rule correctly is that f_α uses the distance from C₀ to the OPPOSITE (β) boundary in the numerator, not the distance to the α boundary. This 'inverted distance' is the lever analogy — C₀ is the fulcrum, and the phase fraction is proportional to the arm length on the other side. Option C reflects the common mistake of assuming 'closer to the α boundary = less α'; in fact, closer to α means MORE α.

Question 2

In a two-phase (α + liquid) region, as the overall composition of an alloy moves progressively closer to the α-phase boundary (farther from the liquid boundary), what happens to the fraction of α phase?

• A. The fraction of α decreases because the alloy composition is becoming more similar to pure α
• B. The fraction of α increases — an alloy composition near the α boundary has more α and less liquid
• C. The fraction of α stays constant because phase fractions only change with temperature, not composition
• D. The fraction of α oscillates — it increases until the midpoint of the tie line, then decreases

Answer: (B)

From the lever rule: f_α = (C_L − C₀)/(C_L − C_α), where C_L is the liquid boundary composition. As C₀ moves toward C_α (closer to the α boundary), the numerator (C_L − C₀) increases toward (C_L − C_α), making f_α approach 1 (100% α). Intuitively, an alloy whose composition nearly matches the solid phase boundary is mostly solid — it needs very little liquid to account for any compositional difference. At the opposite extreme, when C₀ → C_L, f_α → 0 (nearly all liquid). This is the 'lever' in the lever rule: C₀ acts as a fulcrum, and the further C₀ is from one end, the more of the other phase is present.

Question 3

The lever rule tells you the compositions of the phases present in a two-phase region at a given temperature.

• T. True
• F. False

Answer: False

False. Phase compositions are given by the tie-line endpoints — you read off C_α and C_β (or C_L) directly from the phase diagram at the intersection of the isothermal tie line and the phase boundaries. The lever rule uses those compositions (which are already known from the diagram) to calculate something different: the relative amounts (weight or mole fractions) of each phase. Many students conflate these two pieces of information. The division of labor is: phase diagram + tie line → phase compositions; lever rule → phase fractions.

Question 4

In the lever rule, the weight fraction of the α phase equals the ratio of the distance from the overall composition C₀ to the α-phase boundary divided by the total tie-line length.

• T. True
• F. False

Answer: False

False — this is the inverted form and gives the wrong answer. The weight fraction of α uses the distance from C₀ to the OPPOSITE (β) phase boundary in the numerator: f_α = (C_β − C₀)/(C_β − C_α). The β fraction correspondingly uses the distance to the α boundary: f_β = (C₀ − C_α)/(C_β − C_α). This 'inverted distance' is where the lever analogy comes from — the phase fraction is proportional to how far the fulcrum (C₀) is from the other end. A common mnemonic: 'the fraction of a phase equals the fraction of the lever on the other side.'

Question 5

Using the lever rule, a student finds that the overall alloy composition is very close to the β-phase boundary in a two-phase (α + β) region. What does this tell them about the alloy's microstructure, and why?

Model answer: A composition very close to the β-phase boundary means the alloy consists mostly of β phase with very little α. From the lever rule: f_β = (C₀ − C_α)/(C_β − C_α). When C₀ ≈ C_β, the numerator approaches the denominator, so f_β approaches 1 (nearly 100% β). Conversely, f_α = (C_β − C₀)/(C_β − C_α) ≈ 0. Microstructurally, the sample would show predominantly β-phase grains with only trace amounts of the α phase, perhaps as thin grain boundary films or small precipitates.

This reasoning directly guides alloy design. If an engineer wants a microstructure dominated by a specific phase (for example, because that phase has higher hardness or better corrosion resistance), they choose an overall composition near that phase's boundary in the two-phase region. The lever rule is the quantitative bridge between the alloy composition on the phase diagram and the resulting microstructural proportions that determine the material's mechanical properties.