A binary solid `A^(+)B^(-)` has a structure with `B^(-)` ions constituting the lattice and `A^(+)` ions occupying 25% tetrahedral holes. Formula of the solid is

Options

A. `A_(2)B`
B. AB
C. `AB_(2)`
D. `AB_(4)`

Answer

To determine the formula of the binary solid \( A^{+}B^{-} \) with the given conditions, we will follow these steps:

### Step 1: Identify the structure of the lattice
The problem states that \( B^{-} \) ions constitute the lattice. We will assume that \( B^{-} \) forms a face-centered cubic (FCC) lattice. In an FCC lattice, there are 4 atoms per unit cell.

**Hint:** Remember that in an FCC lattice, the contribution of atoms from corners and face centers adds up to 4 atoms per unit cell.

### Step 2: Calculate the number of tetrahedral holes
In an FCC lattice, there are 8 tetrahedral holes per unit cell. Since \( A^{+} \) ions occupy 25% of the tetrahedral holes, we can calculate the number of \( A^{+} \) ions.

**Hint:** Tetrahedral holes are formed in FCC and HCP structures; knowing the number of holes is crucial for determining how many ions can occupy them.

### Step 3: Calculate the number of \( A^{+} \) ions
Since there are 8 tetrahedral holes in the FCC lattice and \( A^{+} \) occupies 25% of these, the number of \( A^{+} \) ions is:
\[ \text{Number of } A^{+} = 0.25 \times 8 = 2 \]

**Hint:** Calculating the percentage of occupied holes helps in determining the number of ions present in the structure.

### Step 4: Determine the total number of ions in the unit cell
From the FCC structure, we have:
- Number of \( B^{-} \) ions = 4 (from the FCC lattice)
- Number of \( A^{+} \) ions = 2 (from the tetrahedral holes)

**Hint:** Always ensure to account for both types of ions in the unit cell when determining the formula.

### Step 5: Write the empirical formula
The empirical formula can be derived from the ratio of \( A^{+} \) to \( B^{-} \):
\[ \text{Formula} = A^{+}_2B^{-}_4 \]
This can be simplified to:
\[ \text{Formula} = AB_2 \]

**Hint:** When writing the empirical formula, ensure that the subscripts are in the simplest whole number ratio.

### Final Answer
The formula of the solid is \( AB_2 \).