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The lever rule calculates the amounts of two coexisting phases from the overall composition and their equilibrium compositions.
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Formula
β, xA, yA and zA are dimensionless mole fractions
| Symbol | Meaning | Unit |
|---|---|---|
| Molar fraction of the vapor or second phase | dimensionless | |
| Mole fraction of component A in the liquid phase | mol/mol | |
| Mole fraction of component A in the vapor phase | mol/mol | |
| Overall mole fraction of component A in the two-phase mixture | mol/mol |
\beta=\frac{z_A-x_A}{y_A-x_A},\quad 1-\beta=\frac{y_A-z_A}{y_A-x_A}The lever rule follows from the material balance for a binary mixture in the two-phase region of an equilibrium diagram. In vapor-liquid equilibrium, β is the molar vapor fraction and 1−β is the liquid fraction. Compositions xA and yA must be taken from the same tie line at the stated temperature or pressure. The rule gives phase amounts; the Gibbs phase rule counts degrees of freedom and Raoult law models equilibrium vapor pressures.
A binary mixture lies in a two-phase region and xA and yA come from the same tie line.
The overall material balance determines molar fractions of coexisting phases.
Overall composition
Material balance between two phases.
Given xA=0.20, yA=0.80 and zA=0.50, find vapor fraction β.
β=(0.50−0.20)/(0.80−0.20)=0.50.
Combining mole fractions from different temperatures.
Only endpoints from the same equilibrium tie line belong in the lever rule.
Formula cluster
If zA lies midway between xA and yA, the molar phase fractions are equal.
On one tie line let xA=0.20 in the liquid, yA=0.80 in the vapor and zA=0.50 overall. Then β=(0.50−0.20)/(0.80−0.20)=0.50, and the liquid fraction is also 0.50. Check: 0.5·0.8+0.5·0.2=0.5, which recovers overall composition zA.
Calculate phase amounts from binary phase diagrams and overall composition
Here is how to work through a typical Lever Rule for Phase Fractions (β = (zA−xA)/(yA−xA)) task step by step:
Task
Given xA=0.20, yA=0.80 and zA=0.50, find vapor fraction β.
Solution path
β=(0.50−0.20)/(0.80−0.20)=0.50.
The relation describes phase fractions of a binary mixture from its position in a two-phase region. Its output is β as the fraction of the second phase and 1−β as the first-phase fraction.. If zA lies midway between xA and yA, the molar phase fractions are equal. It answers a bounded question and places Lever Rule for Phase Fractions in a defined measurement or model context. It does not replace a complete process or system description. State the reference quantity and conditions before interpreting the number; only then can you tell whether it represents an amount, correction or efficiency.
The quantities are xA and yA are phase compositions; zA is overall composition. Their units are All quantities are mole fractions and dimensionless.. Use values on the same reference basis; a ratio across inconsistent phases, time intervals or standards would not be meaningful. Check the dimension of every term before calculating and keep conversions visible. This shows whether the result should be dimensionless or carry a unit. Dimensional analysis cannot repair a bad measurement, but it catches many calculation errors early.
The stated validity is: The equation assumes a binary mixture and a matching tie line. The calculation requires Both phases coexist at equilibrium at the same state point.. Outside this range, additional interactions, changed reference states or different geometries can alter the relation. If an assumption fails, do not transfer the result without qualification; name the limitation and use a more suitable model or measurement. This matters especially when comparing metrics collected under different experimental conditions. This assumption determines which samples and states the calculated value actually represents, so report it alongside the numerical result.
A representative case is: xA=0.20, yA=0.80 and zA=0.50 give β=0.50. The check gives: 0.5·0.8+0.5·0.2 returns zA=0.5.. Write substituted values with units, then put the result back into the original relation or balance. A successful check still depends on the model assumptions. Also test whether limiting cases behave plausibly, for example whether an ideal limit is recovered or a balance returns the overall composition. These checks are more informative than merely rounding to several decimal places. For an additional plausibility check, change one input at a time and confirm that the result moves in the expected direction.
A nearby concept is the Gibbs phase rule.. The key difference is that The lever rule gives phase amounts; the phase rule gives equilibrium degrees of freedom. Both relations may appear in related problems, but they answer different questions and do not automatically use the same data. Choose the equation by the requested output: If zA lies midway between xA and yA, the molar phase fractions are equal. If a problem asks for the Gibbs phase rule. instead, use that definition. This distinction prevents similarly named metrics or models from being treated as interchangeable. This distinction also determines which measurements, reference states and intermediate steps must be documented for a reproducible analysis.
Quanta exam set
Curated exam set for "Lever Rule for Phase Fractions":
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Which formula describes Lever Rule for Phase Fractions?
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How do you rearrange Lever Rule for Phase Fractions for Overall composition?
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Which common mistake happens with Lever Rule for Phase Fractions?
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