A compound is formed by two elements X and Y. The element Y forms cubic close packed arrangement and those of element X occupy one third of the tetrahedral voids. What is the formula of the compound?

Options

A. `X_2Y_3`
B. `X_3Y_2`
C. `X_3Y`
D. `XY_3`

Answer

To solve the problem, we need to determine the formula of the compound formed by elements X and Y based on the information provided about their arrangements in the crystal lattice.

### Step-by-Step Solution:

1. **Identify the arrangement of element Y**:
- The problem states that element Y forms a cubic close-packed (CCP) arrangement. In crystallography, the cubic close-packed structure is also known as face-centered cubic (FCC).

2. **Determine the number of atoms of Y in the unit cell**:
- In an FCC unit cell, there are 4 atoms of the element present. This is calculated as follows:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) to the unit cell (total contribution from corners = \( 8 \times \frac{1}{8} = 1 \)).
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) to the unit cell (total contribution from faces = \( 6 \times \frac{1}{2} = 3 \)).
- Therefore, total number of Y atoms in the unit cell = \( 1 + 3 = 4 \).

3. **Identify the arrangement of element X**:
- The problem states that element X occupies one third of the tetrahedral voids in the CCP structure.

4. **Calculate the number of tetrahedral voids in the FCC unit cell**:
- In an FCC unit cell, there are 8 tetrahedral voids.

5. **Determine how many tetrahedral voids are occupied by X**:
- Since X occupies one third of the tetrahedral voids:
\[ \text{Number of tetrahedral voids occupied by X} = \frac{1}{3} \times 8 = \frac{8}{3} \]

6. **Establish the ratio of X to Y**:
- We have:
- Number of Y atoms = 4
- Number of X atoms = \( \frac{8}{3} \)
- To find the simplest ratio, we can express both in whole numbers:
- Multiply both by 3 to eliminate the fraction:
- Number of X atoms = \( 8 \)
- Number of Y atoms = \( 4 \times 3 = 12 \)

7. **Write the empirical formula**:
- The ratio of X to Y is \( 8:12 \), which simplifies to \( 2:3 \).
- Therefore, the formula of the compound can be expressed as \( X_2Y_3 \).

### Final Answer:
The formula of the compound is \( X_2Y_3 \).

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