The Lever Rule In Binary Systems

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The Lever RuleIn Binary Systems

• The relative amounts of two coexisting phases
  are INVERSELY proportional to their distances
  from a point

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Translating This Statement

• two coexisting phases
• Means you are in a 2 phase region.

• their distances from a point
• Pick an arbitrary point C,

L
The distances are AC and CB
A L
B L
C
A B
A
B
3
Tie Lines

• The line from A to B through C is a tie line.
• A tie line is,
• An isothermal line
• That connects two equilibrium phases

4
Translating This Statement

• The amounts . . . Are inversely proportional
• Means AC / AB is the fraction of B

L
B
And, CB / AB is the fraction of A
A L
B L
A
C
A B
A
B
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A Sample Calculation

• Draw a horizontal tie line through the point.

• Identify the phases.

• Measure its length

L
L (liquid)

• Measure the length of
• each side

C
D
AD 1 cm
A L
B L
AC .75 cm
CD .25 cm

• Calculate the amounts A L

A B
A
B
A (solid)
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A more complex system . . .

• Draw a tie line.

• Identify the phases.

• Measure the line lengths.

• Calculate the amounts of
• each phase present.

A Li2O-B2O3
CB .562 cm
B Li2O-2B2O3
AC .188 cm
C
AB .75 cm
A
B
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One Last Note.

• If a point is in a single phase region
  (including a solid solution), NO tie line is
  used. There is 100 of that phase.

C
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Summary

• The lever rule is used to calculate the relative
  percents of each phase when 2 or more phases are
  present.
• The first step in lever rule calculation is to
  draw a tie line through the composition.
• Next one measures the lengths of the tie line,
  and the distance from the composition to each
  phase.
• The relative concentration of a phase is
  proportional to the distance from the other phase
  to the composition, divided by the length of the
  tie line. (Opposite length / total length)