A solid compound contains X,Y and Z atoms in a cubic lattice with X atoms occupying the corners ,Yatoms in the body centred position and Z atoms at the centre of the faces of the unit cell .What is the empirical formula of the compound?

Options

A. `xy_2z_3`
B. `x_2y_2z_3`
C. `xyz_3`
D. `xyz`

Answer

To find the empirical formula of the compound containing X, Y, and Z atoms in a cubic lattice, we will analyze the contributions of each type of atom based on their positions in the unit cell.

### Step-by-Step Solution:

1. **Identify the Positions of Atoms:**
- X atoms are located at the corners of the cubic unit cell.
- Y atoms are located at the body center of the cubic unit cell.
- Z atoms are located at the centers of the faces of the cubic unit cell.

2. **Calculate the Contribution of X Atoms:**
- There are 8 corners in a cubic unit cell.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared among 8 unit cells.
- Therefore, the total contribution of X atoms is:
\[ \text{Total contribution of X} = 8 \times \frac{1}{8} = 1 \]

3. **Calculate the Contribution of Y Atoms:**
- There is 1 body-centered atom in the unit cell.
- Since it is entirely contained within the unit cell, its contribution is:
\[ \text{Total contribution of Y} = 1 \]

4. **Calculate the Contribution of Z Atoms:**
- There are 6 faces in a cubic unit cell.
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell because it is shared between 2 unit cells.
- Therefore, the total contribution of Z atoms is:
\[ \text{Total contribution of Z} = 6 \times \frac{1}{2} = 3 \]

5. **Determine the Ratio of Atoms:**
- From the contributions calculated:
- X: 1
- Y: 1
- Z: 3
- The ratio of X:Y:Z is 1:1:3.

6. **Write the Empirical Formula:**
- Based on the ratio, the empirical formula of the compound can be written as:
\[ \text{Empirical Formula} = XYZ_3 \]

### Conclusion:
The empirical formula of the compound is \( XYZ_3 \).