Why is it called the lever rule?

The overall composition is the fulcrum of a lever whose arms are the distances to the phase compositions on the tie line. The longer arm opposite a phase means that phase has the smaller fraction — just like a mechanical lever.

Do f_α and f_L always sum to one?

Yes for mass (or mole) fractions of the two coexisting phases in a closed binary system at equilibrium. Outside the two-phase field one fraction is 1 and the other is 0.

Binary Phase Diagram & Lever Rule

Isomorphous A–B T–x diagram: liquidus, solidus, tie line, and lever-rule phase fractions f_α and f_L for overall composition C₀.

About this model

A binary isomorphous phase diagram plots temperature against composition for an A–B alloy that forms a continuous solid solution. The liquidus is the upper boundary of the two-phase α+L field; the solidus is the lower boundary. At a fixed temperature inside that field a horizontal tie line connects the solidus composition x_α to the liquidus composition x_L. For an overall composition C₀ on that tie line the lever rule gives the phase mass fractions f_α = (x_L − C₀)/(x_L − x_α) and f_L = (C₀ − x_α)/(x_L − x_α). The simulator uses a schematic Cu–Ni-like lens diagram so students can move the pink probe and watch the tie line and fraction bar update.

Who it's for: Materials science, metallurgy, physical chemistry, and introductory solidification courses.

Key terms

Lever rule
Liquidus
Solidus
Tie line
Phase fraction
Binary phase diagram

How it works

Binary isomorphous phase diagram with liquidus and solidus curves. Set temperature and overall composition to see the tie line in the two-phase field and compute solid/liquid fractions with the lever rule.

Key equations

Tie line at T: solidus → x_α(T), liquidus → x_L(T)

f_α = (x_L − C₀)/(x_L − x_α) · f_L = (C₀ − x_α)/(x_L − x_α)

f_α + f_L = 1 (mass fractions of phases)

Frequently asked questions

Why is it called the lever rule?: The overall composition is the fulcrum of a lever whose arms are the distances to the phase compositions on the tie line. The longer arm opposite a phase means that phase has the smaller fraction — just like a mechanical lever.
Do f_α and f_L always sum to one?: Yes for mass (or mole) fractions of the two coexisting phases in a closed binary system at equilibrium. Outside the two-phase field one fraction is 1 and the other is 0.