A solid crystal is composed of X, Y and Z atoms. Y atoms are occupying `50%` of octahedral voids, whereas X atoms are occupying the `100%` tetrahedral voids while Z atoms occupy CCP arrangement. The formula of the compound is:

Options

A. `X_(8) Y_(2)Z_(4)`
B. `X_(5)Y_(10)Z_(8)`
C. `X_(4)YZ_(2)`
D. `X_(10)Y_(4)Z_(2)`

Explanation

To determine the formula of the compound composed of X, Y, and Z atoms, we will follow these steps:

### Step 1: Identify the arrangement of Z atoms
- Z atoms occupy a close-packed (CCP) arrangement. In a CCP arrangement, there are 4 atoms per unit cell.

**Hint:** Remember that in a CCP structure, the number of atoms per unit cell is always 4.

### Step 2: Determine the number of octahedral and tetrahedral voids
- In a CCP arrangement, the number of octahedral voids is equal to the number of atoms, which is 4.
- The number of tetrahedral voids in a CCP structure is double the number of atoms, which means there are 8 tetrahedral voids.

**Hint:** For every atom in a CCP structure, there are specific ratios of octahedral and tetrahedral voids.

### Step 3: Calculate the number of Y atoms
- According to the problem, Y atoms occupy 50% of the octahedral voids. Since there are 4 octahedral voids, Y atoms will occupy:
\[ \text{Number of Y atoms} = 50\% \times 4 = 2 \]

**Hint:** To find the number of Y atoms, calculate half of the total octahedral voids.

### Step 4: Calculate the number of X atoms
- The problem states that X atoms occupy 100% of the tetrahedral voids. Since there are 8 tetrahedral voids, X atoms will occupy:
\[ \text{Number of X atoms} = 100\% \times 8 = 8 \]

**Hint:** When an atom occupies 100% of a void type, simply take the total number of those voids.

### Step 5: Write the empirical formula
- Now we have:
- X = 8
- Y = 2
- Z = 4
- The empirical formula can be written as:
\[ \text{Formula} = X_8Y_2Z_4 \]
- To simplify, we can divide all the subscripts by 2:
\[ \text{Simplified Formula} = X_4YZ_2 \]

**Hint:** Always simplify the empirical formula by dividing the subscripts by their greatest common divisor.

### Final Answer
The formula of the compound is:
\[ \text{Answer: } X_4Y_2Z \]

### Summary of Steps
1. Identify the number of Z atoms in CCP (4).
2. Determine the number of octahedral (4) and tetrahedral voids (8).
3. Calculate Y atoms (2 from 50% of 4 octahedral voids).
4. Calculate X atoms (8 from 100% of 8 tetrahedral voids).
5. Write and simplify the empirical formula (X4Y2Z).