Question 10.8: Application of the Lever Rule Calculate the amounts of α and......
Application of the Lever Rule
Calculate the amounts of α and L at 1250 °C in the Cu-40% Ni alloy shown in Figure 10-12.
Let’s say that x = mass fraction of the alloy that is solid α. Since we have only two phases, the balance of the alloy must be in the liquid phase (L). Thus, the mass fraction of liquid will be 1 – x. Consider 100 grams of the alloy. This alloy will consist of 40 grams of nickel at all temperatures. At 1250 °C, let us write an equation that will represent the mass balance for nickel. At 1250 °C, we have 100x grams of the α phase.
We have 100(1 – x) grams of liquid.
Total mass of nickel in 100 grams of the alloy = mass of nickel in liquid + mass of nickel in α
∴ (100)(% Ni in alloy) = [(100)(1 – x)](% Ni in L) + (100)(x)(% Ni in α)
∴ (% Ni in alloy) = (% Ni in L)(1 – x) + (% Ni in α)(x)
By multiplying and rearranging,
From the phase diagram at 1250 °C:
x = \frac{40 \ – \ 32 }{45 \ – \ 32 } = \frac{8}{13} = 0.62If we convert from mass fraction to mass percent, the alloy at 1250 °C contains 62% α and 38% L. Note that the concentration of nickel in the α phase (at 1250 °C) is 45%, and the concentration of nickel in the liquid phase (at 1250 °C) is 32%.