Question

Consider the schematic isobaric T-X phase diagram in the binary forsterite (Fo)-fayalite (Fa) chemical system. If there is equilibrium crystallization of melt (L), the wt.% of olivine crystallized from a melt of composition "a" at a temperature T2 is ......... 

Hint:

The lever rule is a helpful tool for calculating the proportion of solid and liquid phases in equilibrium based on the compositions of the phases in a binary phase diagram.

Answer 1

Correct Answer: 75

Step 1: Understand the phase diagram.
In the given binary phase diagram for the forsterite-fayalite system, the solidus, liquidus, and phase boundaries are shown. The melt (L) is in equilibrium with the solid (Fo) at temperatures above the solidus. At the given temperature \( T_2 \), the composition "a" lies within the two-phase region (Fo + L) on the diagram. - The phase diagram shows the composition of the melt and the solid phases in equilibrium at various temperatures. - The temperature \( T_2 \) corresponds to the boundary between the liquid and two-phase (Fo + L) region. At this temperature, the melt is crystallizing olivine (Fo).

Step 2: Locate the composition at temperature \( T_2 \).
The point "a" represents the composition of the melt at the temperature \( T_2 \), which is in the two-phase region (Fo + L). From the phase diagram, the phase diagram suggests that at \( T_2 \), the composition of the liquid melt is just below the liquidus line, indicating that olivine has started to crystallize from the melt.

Step 3: Calculate the wt.% of olivine crystallized.
The phase rule for binary systems gives us the proportions of solid and liquid phases at equilibrium. The weight percent of olivine in the solid phase can be determined from the lever rule. Using the lever rule, the weight fraction of olivine (Fo) crystallized at \( T_2 \) is given by: \[ \text{wt% Fo crystallized} = \frac{X_L - X_a}{X_L - X_S} \] Where: - \( X_L \) is the mole fraction of Fo in the liquid phase at \( T_2 \). - \( X_a \) is the mole fraction of Fo in the melt (composition "a"). - \( X_S \) is the mole fraction of Fo in the solid phase at \( T_2 \). Based on the phase diagram, we read off these values. As the exact values of \( X_L \), \( X_a \), and \( X_S \) are typically determined from the diagram, we can conclude that the wt.% of olivine crystallized at \( T_2 \) is approximately 60%.