In the sphalerite (ZnS) structure `S^(2-)` ions form a face-centred cubic lattice. Then `Zn^(2+)` ions are present on the body diagonals at

Options

A. `(1)/(3)` rd of the distance
B. `(1)/(4) th` of the distance
C. `(1)/(6)th` of the distance
D. `(1)/(8)th` of the distance

Answer

To solve the problem regarding the position of `Zn^(2+)` ions in the sphalerite (ZnS) structure, we will follow these steps:

### Step 1: Understand the Structure
The sphalerite structure consists of `S^(2-)` ions forming a face-centered cubic (FCC) lattice. In an FCC lattice, the ions are located at the corners and the centers of each face of the cube.

**Hint:** Recall the arrangement of ions in an FCC lattice and how many ions are present in one unit cell.

### Step 2: Calculate the Number of `S^(2-)` Ions
In an FCC lattice, the number of `S^(2-)` ions can be calculated as follows:
- There are 8 corner atoms, each contributing 1/8 to the unit cell: \( 8 \times \frac{1}{8} = 1 \)
- There are 6 face-centered atoms, each contributing 1/2 to the unit cell: \( 6 \times \frac{1}{2} = 3 \)

Thus, the total number of `S^(2-)` ions in the unit cell is:
\[ 1 + 3 = 4 \]

**Hint:** Remember the contributions of corner and face-centered atoms in an FCC lattice.

### Step 3: Determine the Ratio of `Zn^(2+)` to `S^(2-)`
In the sphalerite structure, the ratio of `Zn^(2+)` to `S^(2-)` ions is 1:1. Therefore, there will also be 4 `Zn^(2+)` ions in the unit cell.

**Hint:** Use the stoichiometry of the compound to find the number of `Zn^(2+)` ions.

### Step 4: Identify the Position of `Zn^(2+)` Ions
The `Zn^(2+)` ions are located along the body diagonals of the cube. A cube has 4 body diagonals.

**Hint:** Recall how many body diagonals a cube has and how ions are positioned along these diagonals.

### Step 5: Calculate the Position Along the Body Diagonal
Each body diagonal connects two opposite corners of the cube. The `Zn^(2+)` ions are positioned at 1/4 of the distance along the body diagonal from one corner.

**Hint:** Consider the geometry of the cube and how distances are measured along the body diagonal.

### Final Conclusion
The `Zn^(2+)` ions in the sphalerite structure are located at a distance of 1/4 along the body diagonal from the corner of the cube.

**Final Answer:** The `Zn^(2+)` ions are present at 1/4 of the distance along the body diagonal.