A rule for finding the mass fraction of each phase in a two-phase region of a phase diagram from ratios of distances along the tie line. The fraction of a phase is proportional to the distance from the alloy composition to the opposite phase.
Japanese name: てこの規則
Lever rule appears in 5 practice questions across 1 topic, including Phase diagrams and the lever rule.
Consider an alloy in a two-phase region of a binary phase diagram. If you draw a horizontal line (the tie line) at that temperature, its two ends meet the boundaries of the two phases, and the compositions at those intersections are the compositions of the two phases. The overall alloy composition C₀ lies somewhere along the tie line; because the compositions of the two phases are fixed by temperature alone, changing the alloy composition does not change the phase compositions, only the proportions of the phases. The lever rule gives these proportions and is derived from the overall alloy composition and the conservation of mass of each component.
With Cα the composition of the α phase and C_L that of the other phase (the liquid, say), the mass fraction of liquid is W_L = (Cα − C₀)/(Cα − C_L) and the mass fraction of α is Wα = (C₀ − C_L)/(Cα − C_L). The name comes from the resemblance to the balance of a lever (seesaw): the fraction of a phase is proportional to the distance from the alloy composition to the composition of the opposite phase. A common mistake is to take the distance on the same side. You can check the result by confirming that the two fractions sum to 1. With compositions in wt% you get mass fractions; in at%, fractions by number of atoms.
In eutectic and eutectoid systems it is important to distinguish the fractions of phases from those of microconstituents. For example, a hypoeutectoid steel just below 727 °C has two phases, ferrite and cementite, but two microconstituents, proeutectoid ferrite and pearlite. The fraction of proeutectoid ferrite is found with the lever rule in the α + γ region just above 727 °C, while the fraction of all ferrite as a phase is found in the α + Fe₃C region. Being conscious of which tie line at which temperature you are using prevents mix-ups.
The lever rule assumes equilibrium. With rapid cooling, as in real casting, diffusion in the solid cannot keep up and segregation remains, so fractions and compositions do not match the calculation. Even so, measuring the area fraction of pearlite in the micrograph of a slowly cooled steel lets you back-calculate its approximate carbon content. Also, when the two phases differ greatly in density, mass fractions do not match the area (volume) fractions seen in a microscope, so convert between mass and volume fractions using the density of each phase as needed.
Answer: B. About 0.68
With the lever rule, the fraction of a phase is the distance from the alloy composition to the composition of the opposite phase, divided by the total length of the tie line. The liquid fraction is W_L = (Cα − C₀)/(Cα − C_L) = (42.5 − 35)/(42.5 − 31.5) = 7.5/11.0 ≈ 0.68. About 0.32 is the mistake of taking the distance on the same side, (35 − 31.5)/11.0, which is the fraction of α. About 0.82 is the ratio of the alloy composition to the solid composition, 35/42.5, which is not the lever rule. The fractions of the two phases always sum to 1, which you can use as a check once you have found one of them.
Answer: A. About 0.27
The eutectic structure is what the liquid (61.9 wt% Sn) remaining just above the eutectic temperature turns into through the eutectic reaction. So apply the lever rule in the α + L region just above the eutectic temperature: W_eutectic = W_L = (30 − 18.3)/(61.9 − 18.3) = 11.7/43.6 ≈ 0.27. About 0.73 is the fraction of the remaining primary α, the mistake of taking the opposite distance. About 0.15 is the fraction of all β phase just below the eutectic temperature, (30 − 18.3)/(97.8 − 18.3). All the β is inside the eutectic structure, but the eutectic structure also contains α, so the two do not match. The key to this kind of problem is distinguishing microconstituents (primary and eutectic) from phases (α and β).
In a binary system three phases coexist only at invariant points such as the eutectic temperature, where the fractions are not uniquely determined. Find them with the lever rule in the two-phase regions just above or just below the eutectic temperature.
If you calculate in the same units as the phase diagram, wt% gives mass fractions and at% gives fractions by number of atoms. The important thing is not to mix units partway through.
Because the closer the alloy composition is to the composition of one phase, the more of that phase there should be. Solving the conservation of total solute, C₀ = Wα·Cα + W_L·C_L, confirms that the result is the ratio of distances to the opposite side.
With the lever rule, the fraction of a phase is the distance from the alloy composition to the composition of the opposite phase, divided by the total length of the tie line.
A Cu-35 wt% Ni alloy is held at a temperature where two pha…So apply the lever rule in the α + L region just above the eutectic temperature: W_eutectic = W_L = (30 − 18.3)/(61.9 − 18.3) = 11.7/43.6 ≈ 0.27.
In the Pb-Sn system, take the eutectic temperature as 183 °…From the lever rule, W_proeutectoid α = (0.76 − 0.40)/(0.76 − 0.022) = 0.36/0.738 ≈ 0.49.
A 0.40 wt% C hypoeutectoid steel is cooled under equilibriu…The overall alloy composition does not change, which is consistent with the lever rule showing the solid fraction increasing.
When a Cu-Ni alloy is cooled under equilibrium through the …From the lever rule, W_primary Si = (18 − 12.6)/(100 − 12.6) = 5.4/87.4 ≈ 0.062, or about 6.2%.
The Al-Si system has a eutectic point at 577 °C and 12.6 wt…Eutectic reaction Pearlite Eutectoid reaction Gibbs phase rule