Calculate Weight Fraction Phase Diagram | Lever Rule Calculator
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Weight Fraction of Phase α
57.14%
Weight Fraction of Phase β
42.86%
Mass of Phase α
571.43 units
Mass of Phase β
428.57 units
Phase Distribution
Detailed Phase Breakdown
| Phase |
Composition Limit |
Weight Fraction |
Calculated Mass |
| Phase α |
10% |
57.14% |
571.43 |
| Phase β |
80% |
42.86% |
428.57 |
Formula Applied: The Lever Rule
W_alpha = (C_beta – C₀) / (C_beta – C_alpha)
W_beta = (C₀ – C_alpha) / (C_beta – C_alpha)
The fraction of a phase is proportional to the length of the tie-line "arm" opposite to it.
Error: Overall composition (C₀) must be between Phase α and Phase β compositions.
Comprehensive Guide to Calculate Weight Fraction Phase Diagram
What is the Weight Fraction Phase Diagram Calculation?
To calculate weight fraction phase diagram data means to determine the relative amounts of two coexisting phases within a binary alloy system at a specific temperature. This calculation is a fundamental cornerstone of Materials Science and Engineering, specifically within the study of metallurgy and thermodynamics.
When an alloy cools and enters a two-phase region (for example, Liquid + Solid, or Alpha + Beta), the material splits into two distinct phases with different compositions. Engineers and metallurgists must quantify exactly how much of the material is solid and how much is liquid (or how much is Alpha vs. Beta) to predict mechanical properties like tensile strength, ductility, and solidification behavior.
Common misconceptions include assuming that the composition of the phases is the same as the overall alloy composition, or that the amounts are split 50/50. In reality, the proportions are strictly governed by the distance of the overall composition from the phase boundaries, a concept known as the Lever Rule.
Practical Examples (Real-World Use Cases)
Example 1: Copper-Nickel (Cu-Ni) Cooling
Consider a Cu-Ni alloy with 40 wt% Nickel (C₀ = 40) at a temperature where the liquidus is at 45% Ni (C_liquid) and the solidus is at 32% Ni (C_solid).
- Input C₀: 40%
- Input C_alpha (Solid): 45%
- Input C_beta (Liquid): 32%
- Result:
- Weight Fraction Solid = (40 – 32) / (45 – 32) = 8 / 13 = 0.615 (61.5%)
- Weight Fraction Liquid = (45 – 40) / (45 – 32) = 5 / 13 = 0.385 (38.5%)
Interpretation: The alloy is mostly solid, approaching complete solidification.
Example 2: Lead-Tin (Pb-Sn) Solder
A classic solder alloy (60% Sn) is cooled into the α + β region. At equilibrium, the α phase contains 19% Sn, and the β phase contains 97% Sn.
- Input C₀: 60%
- Input C_alpha: 19%
- Input C_beta: 97%
- Result:
- Weight Fraction α = (97 – 60) / (97 – 19) = 37 / 78 = 0.474 (47.4%)
- Weight Fraction β = (60 – 19) / (97 – 19) = 41 / 78 = 0.526 (52.6%)
How to Use This Calculator
- Identify Compositions: Look at your phase diagram at the specific temperature of interest. Read the x-axis values for the phase boundaries (the intersection of the tie-line with the phase fields).
- Enter Overall Composition (C₀): Input the nominal composition of your alloy in the first field.
- Enter Phase Boundaries: Input the composition of the left phase (usually α) and the right phase (usually β or Liquid).
- Add Total Mass (Optional): If you want to know the specific mass in grams or kilograms, enter the total system mass.
- Review Results: The calculator will instantly generate the weight fractions and update the visual chart.
Key Factors That Affect Results
When you calculate weight fraction phase diagram results, several physical and thermodynamic factors influence the outcome:
1. Temperature
Temperature is the primary driver. As temperature changes, the phase boundaries ($C_\alpha$ and $C_\beta$) shift along the solvus, solidus, or liquidus lines. A small change in temperature can drastically alter the weight fractions.
2. Equilibrium vs. Non-Equilibrium Cooling
The Lever Rule assumes thermodynamic equilibrium (infinitely slow cooling). In real-world casting (Scheil cooling), diffusion is limited, leading to coring and segregation, which alters the actual phase fractions compared to the theoretical calculation.
3. Pressure
While often ignored in condensed metal systems, pressure changes can shift phase boundaries, particularly in systems involving gases or high-pressure allotropes.
4. Impurities (Ternary Elements)
The presence of a third element turns a binary line into a ternary surface. Even small impurities can change the solubility limits ($C_\alpha$ and $C_\beta$), rendering simple binary calculations inaccurate.
5. Solid Solubility Limits
The maximum amount of solute a phase can hold determines the endpoints of your tie-line. Systems with low solubility will have wider tie-lines, resulting in very different fraction ratios.
6. Density Differences
This calculator provides Weight/Mass fractions. If the two phases have significantly different densities (e.g., a heavy intermetallic vs. a light matrix), the Volume fraction (what you see under a microscope) will differ from the calculated weight fraction.
Frequently Asked Questions (FAQ)
Why must the overall composition be between the phase compositions?
If C₀ is not between C_alpha and C_beta, the alloy is not in a two-phase region. It exists as a single phase (100% α or 100% β), and the Lever Rule does not apply.
Can I use atomic percent instead of weight percent?
Yes, the math works identically for atomic percent to calculate atomic fractions. However, to convert atomic fractions to mass fractions, you must account for the atomic weights of the elements.
What is the difference between weight fraction and volume fraction?
Weight fraction is based on mass. Volume fraction is based on space occupied. They are only equal if the densities of both phases are identical.
Does this work for ternary diagrams?
No. For ternary (3-component) systems, the Lever Rule becomes a "plane" calculation usually solved using a triangular grid or center-of-gravity method, not a simple linear tie-line.
What if my result is negative?
A negative result indicates an input error, usually meaning the overall composition (C₀) lies outside the range defined by the two phase compositions.
Is the Lever Rule applicable during rapid quenching?
Generally, no. Rapid quenching freezes the microstructure in a non-equilibrium state. The Lever Rule predicts the state only if the material has had time to diffuse and settle.
What units should I use for mass?
You can use any unit (grams, kg, lbs, tons) as long as you are consistent. The fraction is a dimensionless ratio.
How accurate is this for steels?
It is highly accurate for the Iron-Carbon equilibrium diagram (slow cooling). For heat-treated steels (martensite formation), kinetics dominate over thermodynamics, so this calculator would not apply.
Related Tools and Internal Resources
Explore more engineering and metallurgical calculators to assist with your materials design:
// Initialize calculator
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// Initial Calculation
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calculatePhaseFraction();
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function calculatePhaseFraction() {
// 1. Get Inputs
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var cAlpha = parseFloat(document.getElementById('compAlpha').value);
var cBeta = parseFloat(document.getElementById('compBeta').value);
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if (cAlpha === cBeta) {
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// Logic Check: Is C0 between CAlpha and CBeta?
// Normalize range so min is alpha, max is beta for math simplicity
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var maxC = Math.max(cAlpha, cBeta);
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drawChart(0, 0);
return;
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// 2. Perform Calculation (Lever Rule)
// W_alpha = (C_beta – C0) / (C_beta – C_alpha)
// W_beta = (C0 – C_alpha) / (C_beta – C_alpha)
// Ensure we calculate based on the specific inputs regardless of which is larger
// Formula technically uses absolute differences for lengths
var denominator = Math.abs(cBeta – cAlpha);
// Fraction of Alpha is proportional to distance from Beta to C0
var wAlpha = Math.abs(cBeta – c0) / denominator;
// Fraction of Beta is proportional to distance from C0 to Alpha
var wBeta = Math.abs(c0 – cAlpha) / denominator;
// 3. Update Results
var massAlpha = wAlpha * totalMass;
var massBeta = wBeta * totalMass;
document.getElementById('resFractionAlpha').innerText = (wAlpha * 100).toFixed(2) + "%";
document.getElementById('resFractionBeta').innerText = (wBeta * 100).toFixed(2) + "%";
document.getElementById('resMassAlpha').innerText = massAlpha.toFixed(2);
document.getElementById('resMassBeta').innerText = massBeta.toFixed(2);
// Update Table
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"
" +
"| Phase α | " +
"" + cAlpha + "% | " +
"" + (wAlpha * 100).toFixed(2) + "% | " +
"" + massAlpha.toFixed(2) + " | " +
"
" +
"
" +
"| Phase β | " +
"" + cBeta + "% | " +
"" + (wBeta * 100).toFixed(2) + "% | " +
"" + massBeta.toFixed(2) + " | " +
"
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var total = wAlpha + wBeta;
var alphaAngle = (wAlpha / total) * 2 * Math.PI;
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ctx.fillText("Phase β", 30, 47);
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function resetCalculator() {
document.getElementById('compOverall').value = 40;
document.getElementById('compAlpha').value = 10;
document.getElementById('compBeta').value = 80;
document.getElementById('totalMass').value = 1000;
calculatePhaseFraction();
}
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var beta = document.getElementById('resFractionBeta').innerText;
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var text = "Phase Diagram Weight Fractions:\n" +
"Input Composition (C0): " + c0 + "%\n" +
"Phase Alpha Fraction: " + alpha + "\n" +
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