A cubic solid is made up of two elements X and Y. Atoms of X are present on every alternate cornerand one at the center of cube. Y is at `1/3`rd of the total faces. The empirical formula of the compound is

Options

A. ` X_2Y_1.5`
B. `X_2.5 Y`
C. `XY_2.5`
D. `X_1.5 Y_2`

Answer

To determine the empirical formula of the compound made up of elements X and Y, we will analyze the contributions of each element based on the information given in the question.

### Step-by-Step Solution:

1. **Identify the Contribution of Element X:**
- Atoms of X are located at every alternate corner of the cube. A cube has 8 corners, so if X is present at every alternate corner, it means X is present at 4 corners.
- The contribution of each corner atom of X is \( \frac{1}{8} \) (since each corner atom is shared by 8 adjacent cubes).
- Therefore, the total contribution of X from the corners is:
\[ \text{Contribution from corners} = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]
- Additionally, there is 1 atom of X at the center of the cube, which contributes fully (1).
- Thus, the total contribution of X is:
\[ \text{Total contribution of X} = \frac{1}{2} + 1 = \frac{3}{2} \]

2. **Identify the Contribution of Element Y:**
- Y is present at \( \frac{1}{3} \) of the total faces. A cube has 6 faces, so \( \frac{1}{3} \) of 6 faces means Y is present on 2 faces.
- The contribution of each face atom of Y is \( \frac{1}{2} \) (since each face atom is shared by 2 adjacent cubes).
- Therefore, the total contribution of Y from the faces is:
\[ \text{Contribution from faces} = 2 \times \frac{1}{2} = 1 \]

3. **Combine the Contributions:**
- Now we have the contributions:
- Contribution of X = \( \frac{3}{2} \)
- Contribution of Y = \( 1 \)
- The empirical formula can be expressed in the ratio of X to Y:
\[ \text{Ratio of X to Y} = \frac{3/2}{1} = \frac{3}{2} \]
- To express this in whole numbers, we can multiply both parts of the ratio by 2:
\[ \text{Empirical formula} = X^{3}Y^{2} \]

### Final Answer:
The empirical formula of the compound is \( X_3Y_2 \).

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