A solid is formed and it has three types of atoms X, Y, Z. X forms an FCC lattice with Y atoms occupying one-fourth of tetrahedral voids and Z atoms occupying half of the octahedral voids. The formula of the solid is

Options

A. `X_4 Y_4 Z_2`
B. `X_2 Y Z_2`
C. `X_4 Y Z`
D. `X_2 Y Z`

Answer

To determine the formula of the solid formed by the atoms X, Y, and Z, we need to analyze the contributions of each type of atom based on the information provided.

### Step-by-Step Solution:

1. **Identify the structure of atom X**:
- Atom X forms a Face-Centered Cubic (FCC) lattice.
- In an FCC lattice, there are 4 atoms per unit cell.

2. **Determine the number of tetrahedral voids**:
- In an FCC lattice, the number of tetrahedral voids (THV) is double the number of atoms in the unit cell.
- Therefore, the number of tetrahedral voids = 2 × 4 = 8.

3. **Calculate the number of Y atoms**:
- It is given that Y atoms occupy one-fourth of the tetrahedral voids.
- Number of Y atoms = (1/4) × 8 = 2.

4. **Determine the number of octahedral voids**:
- In an FCC lattice, the number of octahedral voids (OHV) is equal to the number of atoms in the unit cell.
- Therefore, the number of octahedral voids = 4.

5. **Calculate the number of Z atoms**:
- It is given that Z atoms occupy half of the octahedral voids.
- Number of Z atoms = (1/2) × 4 = 2.

6. **Combine the contributions to find the formula**:
- Now we have:
- X: 4 atoms
- Y: 2 atoms
- Z: 2 atoms
- The empirical formula can be expressed as X4Y2Z2.

7. **Simplify the formula**:
- To simplify, we can divide all the subscripts by 2:
- X: 4/2 = 2
- Y: 2/2 = 1
- Z: 2/2 = 1
- Thus, the simplified formula of the solid is X2Y1Z1.

### Final Answer:
The formula of the solid is **X2Y1Z1**.

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