MCEN90014 · Materials Engineering
Phase Diagrams & the Lever Rule
Phase Diagrams & the Lever Rule is the Week-3 core of University of Melbourne MCEN90014 Materials Engineering, the map that turns a chosen composition and temperature into the phases a metal or ceramic actually forms. You read it with three tools: the Gibbs phase rule F = C − P + 2 for degrees of freedom, the liquidus/solidus boundaries and tie-lines for phase compositions, and the lever rule for phase amounts. It is the standard quantitative exam question, running from Cu–Ni and Pb–Sn to the Fe–C eutectoid and oxide-ceramic diagrams.
What this chapter covers
- 01Distinguish state, phase and component precisely
- 02Apply the Gibbs phase rule F = C - P + 2 (constant pressure: F = C - P + 1)
- 03Read liquidus and solidus lines and identify one- vs two-phase fields
- 04Use a tie-line to read the composition of each coexisting phase
- 05Apply the lever rule with the opposite-arm convention to get phase fractions
- 06Tell a eutectic (L -> alpha + beta) from a eutectoid (gamma -> alpha + beta) reaction
- 07Interpret the Cu-Ni isomorphous and Pb-Sn eutectic diagrams
- 08Locate the Fe-C eutectoid (0.76 wt% C, 727 degC -> pearlite) and hypo/hyper-eutectoid alloys
- 09Read isomorphous (Al2O3-Cr2O3) and limited-solubility (SiO2-Al2O3) ceramic diagrams
Lever rule: phase fractions and masses in a Cu-Ni alloy
- +1Identify the phases and the three compositions. Two phases: liquid L and solid alpha. Overall C_0 = 32, liquid C_L = 25, solid C_alpha = 38 (all wt% Ni).
- +1Total tie-line length = C_alpha - C_L = 38 - 25 = 13 wt%.
- +1Liquid fraction uses the OPPOSITE (solid-side) arm: W_L = (C_alpha - C_0)/(C_alpha - C_L) = (38 - 32)/13 = 6/13 = 0.462.
- +1Solid fraction uses the liquid-side arm: W_alpha = (C_0 - C_L)/(C_alpha - C_L) = (32 - 25)/13 = 7/13 = 0.538.
- +1Check the fractions add to one: W_L + W_alpha = 0.462 + 0.538 = 1.000.
- +1Convert to mass for the 500 g charge: liquid = 0.462 x 500 = 231 g; solid = 0.538 x 500 = 269 g.
Key terms
- Phase
- A homogeneous region of a material with a uniform crystal structure, composition and density throughout, e.g. the liquid or the solid alpha solution in an alloy.
- Component
- A distinct chemical species (element or compound) from which the phases are built, e.g. Cu and Ni in a Cu-Ni alloy, or MgO and NiO in a ceramic.
- Gibbs phase rule
- F = C - P + 2 relates degrees of freedom F to components C and phases P; the +2 counts temperature and pressure. At fixed pressure (a T-composition diagram) it becomes F = C - P + 1.
- Liquidus / solidus
- The liquidus is the boundary above which the alloy is fully liquid; the solidus is the boundary below which it is fully solid. The two-phase field lies between them.
- Tie-line
- A horizontal (constant-temperature) line drawn across a two-phase field; its two ends give the compositions of the coexisting phases.
- Lever rule
- W_L = (C_alpha - C_0)/(C_alpha - C_L) and W_alpha = (C_0 - C_L)/(C_alpha - C_L), with W_L + W_alpha = 1. The fraction of a phase equals the tie-line arm on the OPPOSITE side of the overall composition C_0, divided by the whole tie-line.
- Eutectic reaction
- An invariant reaction where a liquid transforms into two solids on cooling, L -> alpha + beta. In Pb-Sn this occurs at 61.9 wt% Sn and 183 degC.
- Eutectoid reaction
- An invariant reaction where one solid transforms into two other solids, gamma -> alpha + beta. In Fe-C, austenite decomposes at 0.76 wt% C and 727 degC into pearlite (alpha-ferrite + Fe3C lamellae).
Phase Diagrams & the Lever Rule FAQ
What is the difference between the liquidus and the solidus?
The liquidus is the line above which the alloy is completely liquid; the solidus is the line below which it is completely solid. Between them the liquid and solid coexist, and a horizontal tie-line drawn at any temperature in that field gives the composition of each phase at its two ends.
How do I tell a eutectic reaction from a eutectoid reaction?
Look at the starting phase. A eutectic starts from a LIQUID: L -> alpha + beta on cooling (the classic Pb-Sn solder point at 61.9 wt% Sn, 183 degC). A eutectoid starts from a SOLID: gamma -> alpha + beta (the Fe-C point at 0.76 wt% C, 727 degC, giving pearlite). Both are invariant points where F = 0.
Can AI help me with phase diagrams and the lever rule in MCEN90014?
Yes, as a study aid. Sia can walk you step by step through applying the Gibbs phase rule, reading a tie-line, and setting up the lever rule with the opposite-arm convention, and it can check your reasoning on a practice problem. Use it to understand the method and build intuition, not to produce answers for assessed work, since the University of Melbourne generative-AI policy for this subject is strict and the final exam is invigilated with a printed formula sheet. Always confirm the exam format on Canvas.
Studying with AI? Sia — free AI mechanical engineering tutor works through MCEN90014 step by step.
Exam move
Phase-diagram questions are among the most reliable marks in the paper because the method is fixed. Drill the lever-rule routine until it is automatic: name the phases from the field label, read the two tie-line ends for the phase compositions, apply W = (opposite arm)/(whole tie-line), and finish with the fractions-sum-to-one check. Keep composition (a tie-line end) and amount (a lever arm) strictly separate, since confusing them is the most common error. Memorise the two invariant reactions by their starting phase (eutectic from liquid, eutectoid from solid) and the Fe-C and Pb-Sn landmark points, and be ready to pair a short definition with a full lever-rule calculation.
