Engineering & Science
Binary Phase Diagram Lever Rule: Liquid & Solid Equilibrium Mass Fractions
Calculate equilibrium liquid mass fraction w_L = (C0 - C_S) / (C_L - C_S) and solid mass fraction w_S = (C_L - C0) / (C_L - C_S) along a horizontal isothermal tie-line in binary alloy phase diagrams.
Apply the opposite-arm tie-line balance to solder and alloy mushy zones!
Inside a two-phase solid + liquid ("mushy") region of a binary phase diagram (such as Pb-Sn solder or Cu-Ni alloys), drawing a horizontal isothermal tie-line at temperature T fixes solidus composition C_S on one boundary and liquidus composition C_L on the other! Mass conservation of solute (C0 = w_S * C_S + w_L * C_L with w_S + w_L = 1) yields the famous metallurgical Lever Rule—where each phase's mass fraction equals the OPPOSITE arm of the tie-line divided by total tie-line length!
Specifically, for C_S <= C0 <= C_L (or generally whenever C0 lies between C_S and C_L), liquid mass fraction is w_L = (C0 - C_S) / (C_L - C_S) and solid mass fraction is w_S = (C_L - C0) / (C_L - C_S)! Notice that signed subtraction (C0 - C_S) / (C_L - C_S) automatically works whether C_L > C_S (eutectic right-sloping liquidus) OR C_L < C_S! For a 35 wt% Sn solder alloy at 200 °C where alpha-solidus is C_S = 18 wt% Sn and liquidus is C_L = 62 wt% Sn, w_L = (35 - 18) / (62 - 18) = 17 / 44 = 0.3864 (38.64% liquid) and w_S = 27 / 44 = 0.6136 (61.36% alpha-solid)!
| Overall Alloy C0 (Tie-Line: CS=18%, CL=62%) | Liquid Mass Fraction w_L | Solid Mass Fraction w_S | Check Sum (w_L + w_S) |
|---|---|---|---|
| C0 = 35 wt% Sn (Plumbing Solder) | 17 / 44 = 0.3864 (38.64%) | 27 / 44 = 0.6136 (61.36%) | 1.0000 |
| C0 = 50 wt% Sn (50/50 Solder) | 32 / 44 = 0.7273 (72.73%) | 12 / 44 = 0.2727 (27.27%) | 1.0000 |
| Outside Two-Phase Zone (C0 = 10% < CS) | Guarded Single-Phase Alert | 100% Solid Single Phase | "Outside Tie-Line Two-Phase Region" |
Equilibrium Liquid Mass Fraction w_L = (C0 - C_S) / (C_L - C_S)
Verifies C0 lies inclusively between C_S and C_L (with C_L <> C_S) before computing the opposite-arm ratio.
=IF(OR(B3 = B4, B2 < MIN(B3, B4), B2 > MAX(B3, B4)), "Outside Tie-Line Two-Phase Region", ROUND((B2 - B3) / (B4 - B3), 4))Equilibrium Solid Mass Fraction w_S = (C_L - C0) / (C_L - C_S)
Computes the solid mass fraction using the liquidus-to-overall tie-line arm.
=IF(OR(B3 = B4, B2 < MIN(B3, B4), B2 > MAX(B3, B4)), "Outside Tie-Line Two-Phase Region", ROUND((B4 - B2) / (B4 - B3), 4))Try the example
Use the original values or change them to test your understanding.