The Lever Rule

Introduction

A phase diagram tells us what the liquid and vapour compositions are at equilibrium, but it does not by itself tell us how much of each phase a particular sample contains. The lever rule supplies those amounts from a component balance and the overall composition. Its simple-looking “opposite arm” formula is reliable only after the coexisting phases and their tie-line endpoints have been identified correctly.

Core explanation

Consider a binary sample with n L moles of liquid and n V moles of vapour. Let x A be A's mole fraction in liquid, y A its mole fraction in vapour, and z A its overall mole fraction. Conservation of A gives (n L+n V)z A=n L x A+n V y A. Set n=n L+n V and f V=n V/n. Then z A=(1−f V)x A+f V y A, so f V=(z A−x A)/(y A−x A) and f L=(y A−z A)/(y A−x A). Both fractions are between zero and one only if z A lies between the endpoint compositions. This is the algebra behind the lever analogy.

On a straight composition axis, the liquid amount is proportional to the distance from z to the vapour endpoint y, and the vapour amount is proportional to the distance from the liquid endpoint x to z. The opposite distances give the amounts: more material lies at the endpoint closer to the overall composition. An overall z close to x implies mostly liquid; z close to y implies mostly vapour. For mole fractions, the calculated fractions are mole fractions of phase amount. If a plot uses mass fractions, phase amounts must be expressed consistently on a mass basis.

At a specified temperature and pressure inside a binary two-phase region, equilibrium determines x and y. The total or overall composition z then sets the phase split. Changing z along the same tie line changes amounts but not endpoints. This is why the phase rule counts intensive equilibrium freedoms differently from the lever rule: the former determines allowable T, pressure and phase compositions, while material balance fixes quantities for a particular sample.

In a T–x–y diagram at fixed pressure, draw a horizontal line at the selected temperature. Read x from the bubble boundary and y from the dew boundary. An overall composition shown by a vertical line must fall between those endpoints to give both phases. If it lies outside, the sample at that temperature is one phase and a two-phase lever calculation is inappropriate. A positive numerical answer from mistakenly chosen endpoints is not evidence that the diagram was read correctly.

The same balance principle works for solid–liquid eutectic diagrams, liquid–liquid separations and other two-phase tie-line plots. The choice of composition variable and basis changes, but z=f α x α+f β x β remains. In a ternary diagram, tie-line geometry is not a single horizontal scalar axis; one uses vector balances and may use a graphical lever construction along a tie line. The essential requirement is still two known equilibrium endpoint compositions.

At an endpoint, one phase amount tends to zero. The arithmetic may produce f V=0 or 1, but at the exact boundary the incipient phase has infinitesimal amount in the ideal equilibrium limit. The lever rule does not determine the temperature or pressure of that boundary and cannot be used to infer kinetics, droplet sizes or nucleation rates. It is a conservation equation, not a mechanism of separation.

Step-by-step reasoning

First establish that two phases coexist and identify a tie line at a common T and pressure. Read or calculate its endpoint compositions x and y on one consistent mole or mass basis. Place overall z between them. Write total and component balances, solve for one phase fraction, and check that both fractions are nonnegative and sum to one.

Visual explanation

Picture a horizontal ruler with x on the left, z between and y on the right. The fraction of vapour is the left segment length (z−x) divided by the full length (y−x). The liquid fraction is the right segment (y−z) divided by the full length. This “opposite arm” relation comes directly from balancing component A.

Real-world analogy

Place a balance point between two weights on a beam. A larger weight must be closer to the pivot to balance a smaller weight farther away. Likewise, a sample composition close to the liquid endpoint implies much more liquid than vapour. The analogy illustrates ratios but does not replace the mole balance or determine equilibrium endpoints.

Real-world example

A flash drum partially vaporises a liquid feed at selected pressure and temperature. VLE determines the compositions of the exiting liquid and vapour. Engineers use feed composition and flow rate with the lever rule to estimate their relative molar flow rates, then size separation equipment using energy balance and real-mixture corrections.

Why?

Why are the arms opposite the phases? If the total composition moves toward the vapour endpoint, more of the sample must have vapour composition to pull the weighted average that way. Algebraically, z=x+f V(y−x), so the displacement from x is proportional to the vapour fraction, not the liquid fraction.

Common misconception

One cannot apply the lever rule to any two arbitrary points on a diagram; they must be endpoints of the same equilibrium tie line. Another mistake is using the distance from z to a phase's own endpoint for that phase's amount. The amount is proportional to the opposite segment, as the mass-balance derivation shows.

Worked example

At a specified T and pressure, liquid has x A=0.20 and vapour has y A=0.70. A sample has z A=0.40 and total 10.0 mol. The vapour fraction is (0.40−0.20)/(0.70−0.20)=0.40. Thus n V=4.0 mol and n L=6.0 mol. Check A balance: liquid contains 6.0(0.20)=1.2 mol A, vapour contains 4.0(0.70)=2.8 mol A, total 4.0 mol A, matching 10.0(0.40).

Quick check

1. If z is very close to liquid endpoint x, which phase dominates? Answer: Liquid dominates. The vapour fraction (z−x)/(y−x) is small, and the liquid fraction is correspondingly near one.

Exam focus

Derive or write z=f L x+f V y and f L+f V=1 before using the distance shortcut. Label units and basis. Confirm x≤z≤y when the axis is ordered that way; otherwise the assumed two-phase state or endpoint assignments need review.

Advanced insight

The lever rule is a convex-combination statement: an overall composition of a two-phase mixture lies on the line segment joining its phase compositions. This geometric fact generalises to multicomponent vectors. In a ternary diagram, the relative amounts along one tie line still follow distances, but locating the correct tie line requires independent phase-equilibrium information.

Summary

For two phases, overall composition is the amount-weighted average of equilibrium endpoint compositions. The vapour fraction is (z−x)/(y−x), and the liquid fraction is (y−z)/(y−x) on a consistent basis. The rule determines quantities only after a valid two-phase tie line is known.

Practice questions

1. A tie line has x A=0.10, y A=0.60 and z A=0.35. Find vapour fraction. Answer: (0.35−0.10)/(0.60−0.10)=0.50, so half the total moles are vapour. 2. If z A equals x A, what is the limiting vapour fraction? Answer: Zero; the overall composition coincides with the liquid endpoint, so the vapour amount tends to zero at the boundary. 3. Why is f V=−0.2 a warning rather than a valid negative amount? Answer: It indicates z is outside the chosen tie-line endpoints or the endpoints were misread; the assumed two-phase equilibrium is invalid. 4. Can a mole-basis lever fraction be used unchanged as a mass fraction? Answer: Not generally. Convert compositions and amounts consistently because phases can have different mean molar masses.