In a crystallizes solid, anions `B` are arranged in cubic close packing, cations `A` are equally distributed between octahedral and tetrahedral voids, what is the formula of the solid?

Explanation

To determine the formula of the crystalline solid where anions \( B \) are arranged in cubic close packing and cations \( A \) are equally distributed between octahedral and tetrahedral voids, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Cubic Close Packing (CCP):**
- In cubic close packing (CCP), which is equivalent to face-centered cubic (FCC), the number of atoms (or anions in this case) per unit cell is 4.
- This is calculated as follows:
- There are 8 corners in a cube, each contributing \( \frac{1}{8} \) of an atom: \( 8 \times \frac{1}{8} = 1 \).
- There are 6 face centers, each contributing \( \frac{1}{2} \) of an atom: \( 6 \times \frac{1}{2} = 3 \).
- Total: \( 1 + 3 = 4 \).

2. **Identifying the Number of Voids:**
- In a cubic close-packed structure, the number of octahedral voids is equal to the number of atoms in the lattice, which is 4.
- The number of tetrahedral voids is double the number of atoms in the lattice: \( 2 \times Z = 2 \times 4 = 8 \).

3. **Distribution of Cations \( A \):**
- The problem states that cations \( A \) are equally distributed between octahedral and tetrahedral voids.
- Therefore, if there are 4 octahedral voids and 8 tetrahedral voids, cations \( A \) will occupy:
- 4 octahedral voids (1 cation per void) = 4 cations.
- 4 tetrahedral voids (1 cation per void) = 4 cations.
- Total number of cations \( A \) = \( 4 + 4 = 8 \).

4. **Calculating the Formula of the Solid:**
- Now we have:
- Number of anions \( B = 4 \)
- Number of cations \( A = 8 \)
- To write the formula, we express the ratio of cations to anions:
- The simplest ratio of \( A \) to \( B \) is \( 8:4 \), which simplifies to \( 2:1 \).
- Therefore, the formula of the solid is \( A_2B \).

### Final Answer:
The formula of the solid is \( A_2B \).

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