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How do you use lever law?

Published in Lever Rule Application 3 mins read

The lever rule, in the context of phase diagrams, is used to determine the weight fraction (or mole/atomic fraction, if the data is in those units instead of weight percent) of each phase in a two-phase region. Here's how it works:

Understanding the Lever Rule

The lever rule is a tool used in materials science and engineering to determine the relative amounts of each phase in a two-phase region of a phase diagram. It's analogous to a lever, where the fulcrum represents the overall composition, and the lengths of the lever arms represent the relative amounts of each phase.

Steps to Apply the Lever Rule

  1. Identify the Phase Diagram: Make sure you have the correct phase diagram for the system you are analyzing (e.g., binary eutectic, isomorphous).

  2. Determine the Temperature and Overall Composition: Identify the specific temperature and overall composition (e.g., weight percent of component B) for your alloy.

  3. Locate the Point on the Phase Diagram: Find the point on the phase diagram that corresponds to your temperature and overall composition.

  4. Determine the Phases Present: If the point falls within a single-phase region, only that phase is present. If it falls within a two-phase region, both phases bordering that region are present.

  5. Draw a Tie Line: At the temperature of interest, draw a horizontal line (the "tie line") across the two-phase region. This line connects the compositions of the two phases that are in equilibrium.

  6. Determine the Phase Compositions: The points where the tie line intersects the phase boundaries indicate the compositions of each phase. Read these composition values from the composition axis (usually weight percent).

  7. Calculate the Weight Fractions:

    • Let w1 be the weight fraction of phase 1, and w2 be the weight fraction of phase 2.

    • Let c0 be the overall composition of the alloy.

    • Let c1 be the composition of phase 1.

    • Let c2 be the composition of phase 2.

    • Then, the lever rule equations are:

      • w1 = (c2 - c0) / (c2 - c1)
      • w2 = (c0 - c1) / (c2 - c1)
      • Note that w1 + w2 = 1

Example

Imagine a binary alloy of A and B at a certain temperature where two phases, α and β, are present. Let's say:

  • The overall composition of the alloy (c0) is 40 wt% B.
  • The composition of the α phase (c1) is 10 wt% B.
  • The composition of the β phase (c2) is 70 wt% B.

Then, the weight fractions are:

  • wα = (70 - 40) / (70 - 10) = 30 / 60 = 0.5
  • wβ = (40 - 10) / (70 - 10) = 30 / 60 = 0.5

This means that the alloy contains 50% α phase and 50% β phase by weight.

Important Considerations

  • Temperature is Constant: The lever rule applies at a specific, constant temperature.
  • Equilibrium Conditions: The lever rule assumes that the system is at equilibrium.
  • Units: Make sure all compositions are in the same units (weight percent, atomic percent, or mole fraction). As stated in the provided video excerpt, if the composition is in weight percent, the lever rule calculates the weight fraction. If the composition is in mole or atomic fraction, the lever rule will determine the mole or atomic fraction.

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