The binary phase diagram of metals P and Q is shown in the figure. An alloy X containing60% P and 40% Q (by weight) is cooled from liquid to solid state. The fractions of solid andliquid (in weight percent) at 1250 °C, respectively, will be

The correct option is: 22.2% and 77.8%.

1. Identifying Values from the Binary Phase Diagram:
The given phase diagram shows Temperature in ${°}^{\text{C}}$ on the vertical axis and the Concentration of Q (in weight %) on the horizontal axis.
The alloy X contains 60% P and 40% Q by weight. Since the horizontal axis represents the weight percentage of metal Q, the composition of the overall alloy, denoted as $C_0$, is:
$$C_0=40\text{\% Q}$$
At the specified temperature of 1250 °C, we draw a horizontal tie line across the two-phase (Liquid + Solid) region:
• The tie line intersects the Liquidus curve at a concentration of 32% Q. Therefore, the composition of the liquid phase ($C_L$) is 32% Q.
• The tie line intersects the Solidus curve at a concentration of 68% Q. Therefore, the composition of the solid phase ($C_S$) is 68% Q.

2. Applying the Lever Rule:
The Lever Rule is used to determine the phase fractions in a two-phase region of a binary phase diagram.
The weight fraction of the solid phase ($f_S$) is given by the formula:
$$f_S=\frac{C_0-C_L}{C_S-C_L}$$
Substituting the values obtained from the phase diagram:
$$f_S=\frac{40-32}{68-32}=\frac{8}{36}=\frac{2}{9}\approx 0.2222\text{ (or 22.2\%)}$$

Similarly, the weight fraction of the liquid phase ($f_L$) is calculated as:
$$f_L=\frac{C_S-C_0}{C_S-C_L}$$
Substituting the values:
$$f_L=\frac{68-40}{68-32}=\frac{28}{36}=\frac{7}{9}\approx 0.7778\text{ (or 77.8\%)}$$

3. Conclusion:
At 1250 °C, the weight fractions of solid and liquid phases in alloy X are 22.2% and 77.8% respectively.