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 find the formula of the solid formed by the three types of atoms X, Y, and Z, we will follow these steps:

### Step 1: Determine the number of X atoms in the FCC lattice
- In a face-centered cubic (FCC) lattice, there are 4 atoms of X per unit cell.
- Therefore, the number of X atoms = 4.

### Step 2: Calculate the number of Y atoms occupying tetrahedral voids
- In an FCC lattice, there are 8 tetrahedral voids.
- Given that Y occupies one-fourth of these tetrahedral voids, we calculate:
\[ \text{Number of Y atoms} = \frac{1}{4} \times 8 = 2 \]

### Step 3: Calculate the number of Z atoms occupying octahedral voids
- In an FCC lattice, there are 4 octahedral voids.
- Given that Z occupies half of these octahedral voids, we calculate:
\[ \text{Number of Z atoms} = \frac{1}{2} \times 4 = 2 \]

### Step 4: Write the empirical formula of the solid
- Now we have:
- X = 4
- Y = 2
- Z = 2
- The empirical formula can be written as:
\[ \text{Formula} = X_4Y_2Z_2 \]

### Step 5: Simplify the formula
- To simplify the formula, we can divide all the subscripts by 2:
\[ \text{Simplified formula} = X_2Y_1Z_1 \text{ or simply } X_2YZ \]

### Final Answer
The formula of the solid is \( X_2YZ \).

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