SME
Materials Science & Material Properties · Problem type
Eutectic and primary α
Use this for “A 40 wt% Sn lead–tin alloy cools through its eutectic. With α at 18.3 and the eutectic at 61.9 wt% Sn, how much ends up eutectic?”
Diagram
Lever
rule:
Wₑ
=
C₀ − Cα
CE − Cα
=
21.7
43.6
=
0.498.
Wα′ = 1 − Wₑ = 0.502.
Each phase’s share is the arm on the far side of C₀.
This is an example. Type your own values to start a new problem.
Wₑ
=
C₀ (wt%)
−
Cα (wt%)
CE (wt%)
−
Cα (wt%)
Wα′
Primary α share
calculated
Cβ
β composition
example · wt%
Wα
Total α share
calculated
Wβ
Total β share
calculated
Formulas
Wₑ
=
C₀
− Cα
CE − Cα
0.4977
=
40 − 18.3
61.9 − 18.3
Wα′ = 1 − Wₑ
0.5023 = 1 − 0.4977
Wα
=
Cβ −
C₀
Cβ − Cα
0.727
=
97.8 − 40
97.8 − 18.3
Wβ = 1 − Wα
0.273 = 1 − 0.727
Assumptions
•
Just above the eutectic temperature: primary α at Cα and liquid at CE.
•
All of that liquid becomes the eutectic microconstituent as it cools through TE.
•
The alloy is hypoeutectic (between Cα and CE); compositions are wt% of tin.
•
Just below TE the whole alloy is α at Cα and β at Cβ (97.8 wt% Sn for lead–tin): the lever on that line gives the total α and β.
Step-by-step
WE KNOW
C₀ = 40 wt%, Cα = 18.3 wt%, CE = 61.9 wt%, Cβ = 97.8 wt%
FIND
eutectic share (Wₑ), primary α share (Wα′), total α share (Wα), total β share (Wβ)
Step 1 · Find eutectic share (Wₑ)
Use Wₑ = (C₀ − Cα) ÷ (CE − Cα)
The
lever
rule:
the
eutectic
share
is
the
arm
from
the
far
end
to
C₀
,
over
the
whole
tie
line.
Wₑ
=
40 − 18.3
61.9 − 18.3
Wₑ
=
21.7
43.6
Wₑ = 0.4977
Step 2 · Find primary α share (Wα′)
Use Wα′ = 1 − Wₑ
The two shares add to 1, so the primary α is the rest.
Wα′ = 1 − 0.4977
Wα′ = 0.5023
Step 3 · Find total α share (Wα)
Use Wα = (Cβ − C₀) ÷ (Cβ − Cα)
The
lever
rule:
the
total
α
share
is
the
arm
from
the
far
end
to
C₀
,
over
the
whole
tie
line.
Wα
=
97.8 − 40
97.8 − 18.3
Wα
=
57.8
79.5
Wα = 0.727
Step 4 · Find total β share (Wβ)
Use Wβ = 1 − Wα
The two shares add to 1, so the β is the rest.
Wβ = 1 − 0.727
Wβ = 0.273
Check
Put the numbers back in. Both sides should match.
0.4977
=
40 − 18.3
61.9 − 18.3
✓
0.5023 = 1 − 0.4977 ✓
0.727
=
97.8 − 40
97.8 − 18.3
✓
0.273 = 1 − 0.727 ✓
Topic check
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