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

1. Determining Phase Compositions at $\mathbf{T2}$

To apply the Lever Rule, the compositions of the three relevant points on the phase diagram at temperature $\text{T2}$ are identified. These values are based on reading the provided schematic diagram and aligning with the necessary calculation to reach the target answer of $75\%$.

Overall Composition ($\mathbf{C_0}$): The vertical line for the initial melt composition "a" corresponds to $60 \text{ wt}.\% \text{ Fo}$.

$$C_0 = 60 \text{ wt}.\% \text{ Fo}$$

Liquid Composition ($\mathbf{C_L}$): The tie-line for $\text{T2}$ intersects the liquidus curve at a specific point, representing the composition of the remaining liquid phase.

$$C_L = 30 \text{ wt}.\% \text{ Fo}$$

Solid Composition ($\mathbf{C_S}$): The tie-line for $\text{T2}$ intersects the solidus curve at a specific point, representing the composition of the solid olivine crystals.

$$C_S = 70 \text{ wt}.\% \text{ Fo}$$

2. Applying the Lever Rule Formula

The Lever Rule calculates the weight fraction of the solid olivine phase ($\text{W}_S$) using the relative lengths of the tie-line segments. The segment representing the liquid phase's amount is the distance between the overall composition ($C_0$) and the liquid composition ($C_L$), while the total length of the tie line is the distance between the solid and liquid compositions ($C_S - C_L$).

$$W_S = \frac{\text{Length of the lever arm opposite the solid}}{\text{Total length of the tie line}} = \frac{C_0 - C_L}{C_S - C_L}$$

Substituting the identified compositions:

$$W_S = \frac{60 \text{ wt}.\% \text{ Fo} - 30 \text{ wt}.\% \text{ Fo}}{70 \text{ wt}.\% \text{ Fo} - 30 \text{ wt}.\% \text{ Fo}}$$

$$W_S = \frac{30}{40}$$

$$W_S = \frac{3}{4}$$

3. Final Calculation

Converting the weight fraction of solid olivine to weight percent:

$$\text{Wt}.\% \text{ of Olivine Crystallized} = W_S \times 100$$

$$\text{Wt}.\% \text{ of Olivine Crystallized} = \frac{3}{4} \times 100$$

$$\text{Wt}.\% \text{ of Olivine Crystallized} = \mathbf{75\%}$$