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 \( A^{+}B^{-} \), we can follow these steps:

### Step 1: Understand the Structure
We know that \( B^{-} \) ions constitute the lattice, and \( A^{+} \) ions occupy 25% of the tetrahedral holes.

### Step 2: Define Variables
Let the number of \( B^{-} \) ions in the lattice be \( n \). In a crystal structure, the number of tetrahedral voids is always double the number of atoms in the lattice. Therefore, the number of tetrahedral voids is \( 2n \).

### Step 3: Calculate the Number of \( A^{+} \) Ions
Since \( A^{+} \) ions occupy 25% of the tetrahedral holes, we can calculate the number of \( A^{+} \) ions as follows:
\[ \text{Number of } A^{+} \text{ ions} = 25\% \text{ of } 2n = \frac{25}{100} \times 2n = \frac{2n}{4} = \frac{n}{2} \]

### Step 4: Determine the Ratio of \( A^{+} \) to \( B^{-} \)
Now we have:
- Number of \( A^{+} \) ions = \( \frac{n}{2} \)
- Number of \( B^{-} \) ions = \( n \)

The ratio of \( A^{+} \) to \( B^{-} \) can be expressed as:
\[ \text{Ratio of } A^{+} : B^{-} = \frac{n/2}{n} = \frac{1}{2} \]

### Step 5: Simplify the Ratio
To express this ratio in whole numbers, we can multiply both parts of the ratio by 2:
\[ 1 : 2 \]

### Step 6: Write the Formula
From the ratio \( 1 : 2 \), we can deduce that the formula of the solid is:
\[ AB_{2} \]

### Final Answer
The formula of the binary solid is \( AB_{2} \).

---