Calculate phase fractions, phase masses, or overall composition with the lever rule from tie-line endpoint compositions and total mass.

Use one component and one composition basis for the same equilibrium tie line. Masses require mass-based compositions.

Lever Rule Formula

The lever rule calculates phase fractions on the same mass or molar basis as the endpoint and overall compositions. The endpoints must be distinct and the overall composition must lie between them. Mass calculations require mass-based compositions.

Xa = (B – T) / (B – A)
Xb = (T – A) / (B – A)
Variables:

  • Xa is the phase A fraction on the selected basis
  • Xb is the phase B fraction on the selected basis
  • A is the composition of phase A (in mole fraction or mass fraction)
  • B is the composition of phase B (in mole fraction or mass fraction)
  • T is the overall system composition (in mole fraction or mass fraction)

To calculate each phase fraction, subtract the overall system composition from the composition of the other phase and divide by the difference in composition between the two phases. This is done for both phases. The two phase fractions sum to 1 by conservation on the selected basis. Mass-based compositions give mass fractions; molar or atomic compositions do not directly give masses.

What is a Lever Rule?

The Lever Rule is a tool used in thermodynamics to determine the mole fraction (or mass fraction) of each phase of a binary equilibrium phase diagram. It is based on the principle of conservation of mass and is called the “Lever Rule” because it can be visually represented as a lever on a fulcrum, where the lengths of the lever arms are inversely proportional to the amounts of the phases present.

How to Calculate Lever Rule?

The following steps outline how to calculate the Lever Rule:

  1. First, determine the composition of phase A (A) and phase B (B) in mole fraction or mass fraction.
  2. Next, determine the overall system composition (T) in mole fraction or mass fraction.
  3. Next, use the formula Xa = (B – T) / (B – A) to calculate the phase A fraction on the selected basis.
  4. Finally, use the formula Xb = (T – A) / (B – A) to calculate the phase B fraction on the selected basis.

Example Problem:

Use the following variables as an example problem to test your knowledge:

Composition of phase A (A) = 0.4

Composition of phase B (B) = 0.6

Overall system composition (T) = 0.5