Math 2.0

Classical thermodynamics, heat transfer Equations of state and phases Proposition

Lever rule

Builds on Phase diagramThermodynamic equilibriumGibbs phase rule

Statement

Consider a mixture of two components AA and BB at fixed temperature and pressure. It has nn moles in total, of which a fraction x0x_0 is BB. Assume that the mixture is in thermodynamic equilibrium as two coexisting phases α\alpha and β\beta. Let xαx_\alpha and xβx_\beta be their mole fractions of BB, where xα<x0<xβx_\alpha < x_0 < x_\beta. Let nαn_\alpha and nβn_\beta be the numbers of moles in the two phases. Then nα(x0−xα)=nβ(xβ−x0),nβn=x0−xαxβ−xα.n_\alpha (x_0 - x_\alpha) = n_\beta (x_\beta - x_0), \qquad \frac{n_\beta}{n} = \frac{x_0 - x_\alpha}{x_\beta - x_\alpha}.

Remarks

The proof is a mass balance: the total amount of BB gives nx0=nαxα+nβxβn x_0 = n_\alpha x_\alpha + n_\beta x_\beta, and n=nα+nβn = n_\alpha + n_\beta. Substituting n=nα+nβn = n_\alpha + n_\beta and rearranging gives both formulas. For example, if xα=0.2x_\alpha = 0.2, xβ=0.7x_\beta = 0.7 and x0=0.4x_0 = 0.4, then nβ/n=0.2/0.5=0.4n_\beta / n = 0.2/0.5 = 0.4, so 4040 percent of the material is in phase β\beta. The same rule holds with mass fractions and masses in place of mole fractions and moles. The name comes from a picture in the phase diagram. On the tie line from xαx_\alpha to xβx_\beta, the amounts balance like weights on a lever with its fulcrum at x0x_0.

Going deeper

For a student

In a binary phase diagram at fixed pressure, the Gibbs phase rule gives one degree of freedom in a two-phase region. So fixing the temperature fixes xαx_\alpha and xβx_\beta: they are the ends of the horizontal tie line. The lever rule then fixes the amounts, which the phase rule does not. It is used daily in metallurgy, for example to find the fractions of solid and liquid in a cooling alloy, and in distillation.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Chemical potential, concept, no evidence levelChemical potentialconceptIntensive and extensive properties, concept, no evidence levelIntensive and extensi…conceptPhase diagram, concept, no evidence levelPhase diagramconceptPhase transition, concept, no evidence levelPhase transitionconceptThermodynamic equilibrium, concept, no evidence levelThermodynamic equilib…conceptTriple point, concept, no evidence levelTriple pointconceptGibbs phase rule, statement, no evidence levelGibbs phase rulestatementLever rule, statement, no evidence levelLever rulestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Chemical potential concept
  2. Intensive and extensive properties concept
  3. Phase diagram concept
  4. Phase transition concept
  5. Thermodynamic equilibrium concept
  6. Triple point concept
  7. Gibbs phase rule statement
  8. Lever rule statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lever_rule
About this page
TagBXY69X7T75
URIm2://BXY69X7T75
Kindstatement, proposition
WritingW0 stub
MSC80A22, 80A50
StatusThis copy of the site has no build data. Run m2 status BXY69X7T75.
AboutPhase diagram
UsesPhase diagram, Thermodynamic equilibrium, Gibbs phase rule
Same as
Changed3 November 2026

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