In the unit cell of the crystal formed by the ionic compound of X and Y, the corners are occupied by X and the centers of the faces by Y. The empirical forumal of the compound is

Options

A. XY
B. `X_(2)Y`
C. `XY_(3)`
D. `X_(8)Y_(6)`

Answer

To determine the empirical formula of the ionic compound formed by X and Y, we need to analyze the arrangement of the ions in the unit cell.

### Step-by-Step Solution:

1. **Identify the Positions of Ions**:
- The problem states that the corners of the unit cell are occupied by X ions and the centers of the faces are occupied by Y ions.

2. **Calculate the Contribution of X Ions**:
- In a cube (which is the shape of the unit cell), there are 8 corners.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared among 8 adjacent unit cells.
- Therefore, the total contribution of X ions from the corners is:
\[ \text{Total contribution of X} = 8 \times \frac{1}{8} = 1 \]
- Thus, there is 1 X ion in the unit cell.

3. **Calculate the Contribution of Y Ions**:
- There are 6 faces in a cube, and each face has 1 Y ion at its center.
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell because it is shared between 2 adjacent unit cells.
- Therefore, the total contribution of Y ions from the face centers is:
\[ \text{Total contribution of Y} = 6 \times \frac{1}{2} = 3 \]
- Thus, there are 3 Y ions in the unit cell.

4. **Write the Empirical Formula**:
- Now that we have the number of ions in the unit cell, we can write the empirical formula.
- The empirical formula is based on the ratio of the number of X ions to the number of Y ions:
\[ \text{Empirical formula} = X_1Y_3 \text{ or simply } XY_3 \]

### Final Answer:
The empirical formula of the compound is \( XY_3 \).

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