In a compound oxide ions are arranged in cubic close packing arrangement. Cations A occupy one-sixth of the tetrahedral voids and cations B occupy one-third of the octahdral voids. The formual of the compound is `A_(x), B_(y), O_(z)` then find the value of `x + y + z`

Answer

To solve the problem, we need to determine the values of x, y, and z in the formula \( A_x B_y O_z \) based on the arrangement of ions in the compound.

### Step-by-Step Solution:

1. **Identify the Arrangement of Oxide Ions**:
- The oxide ions (O²⁻) are arranged in a cubic close packing (CCP) arrangement, which is equivalent to a face-centered cubic (FCC) structure.

2. **Calculate the Number of Oxide Ions**:
- In a face-centered cubic (FCC) structure, the number of atoms (or ions) is given by:
\[ \text{Number of atoms} = \frac{1}{8} \times 8 + \frac{1}{2} \times 6 = 1 + 3 = 4 \]
- Therefore, there are 4 oxide ions (O²⁻) in the unit cell.

3. **Determine the Number of Tetrahedral and Octahedral Voids**:
- In a FCC structure:
- The number of tetrahedral voids = 8
- The number of octahedral voids = 4

4. **Calculate the Number of Cations A and B**:
- Cation A occupies one-sixth of the tetrahedral voids:
\[ \text{Cations A} = \frac{1}{6} \times 8 = \frac{8}{6} = \frac{4}{3} \]
- Cation B occupies one-third of the octahedral voids:
\[ \text{Cations B} = \frac{1}{3} \times 4 = \frac{4}{3} \]

5. **Write the Empirical Formula**:
- From the calculations, we have:
- \( A = \frac{4}{3} \)
- \( B = \frac{4}{3} \)
- \( O = 4 \)
- Thus, the formula can be expressed as:
\[ A_{\frac{4}{3}} B_{\frac{4}{3}} O_4 \]

6. **Normalize the Formula**:
- To express the formula in whole numbers, we can multiply each coefficient by 3:
\[ A_{4} B_{4} O_{12} \]
- This simplifies to:
\[ A_{1} B_{1} O_{3} \quad \text{(after dividing by 4)} \]

7. **Identify x, y, z**:
- From the normalized formula \( A_1 B_1 O_3 \), we find:
- \( x = 1 \)
- \( y = 1 \)
- \( z = 3 \)

8. **Calculate x + y + z**:
- Finally, we calculate:
\[ x + y + z = 1 + 1 + 3 = 5 \]

### Final Answer:
The value of \( x + y + z \) is **5**.