It was found that in `30 unit` cells of ZnS, four atoms are of trivalent metal. How many vacant tetrahedral voids are present in `30 unit` cells of ZnS?

Options

A. `2`
B. `62`
C. `122`
D. `126`

Explanation

To solve the problem of how many vacant tetrahedral voids are present in 30 unit cells of ZnS, we can follow these steps:

### Step 1: Determine the number of atoms in one unit cell of ZnS
ZnS crystallizes in a face-centered cubic (FCC) lattice structure. In an FCC lattice, there are 4 atoms per unit cell.

### Step 2: Calculate the total number of tetrahedral voids in one unit cell
In an FCC lattice, the number of tetrahedral voids is given by the formula:
\[ \text{Number of tetrahedral voids} = 2 \times \text{Number of atoms in the unit cell} \]
Thus, for one unit cell of ZnS:
\[ \text{Tetrahedral voids} = 2 \times 4 = 8 \]

### Step 3: Calculate the total number of tetrahedral voids in 30 unit cells
For 30 unit cells, the total number of tetrahedral voids is:
\[ \text{Total tetrahedral voids} = 8 \times 30 = 240 \]

### Step 4: Determine how many tetrahedral voids are filled by Zn²⁺ ions
In the ZnS structure, it is given that 50% of the tetrahedral voids are filled with Zn²⁺ ions. Thus:
\[ \text{Filled tetrahedral voids} = 50\% \times 240 = 120 \]

### Step 5: Determine how many tetrahedral voids are occupied by trivalent metal ions
It is stated that there are 4 trivalent metal ions present. These ions can displace 6 Zn²⁺ ions to maintain electrical neutrality in the crystal structure.

### Step 6: Calculate the number of tetrahedral voids remaining vacant
From the total of 240 tetrahedral voids, if 6 Zn²⁺ ions are replaced by the 4 trivalent metal ions, we need to find out how many voids remain vacant:
- Initially, there are 240 tetrahedral voids.
- After replacing 6 Zn²⁺ ions, the number of tetrahedral voids occupied by Zn²⁺ becomes:
\[ 120 - 6 = 114 \]

### Step 7: Calculate the total number of vacant tetrahedral voids
The total number of vacant tetrahedral voids is:
\[ \text{Total vacant voids} = \text{Total tetrahedral voids} - \text{Occupied voids} \]
\[ \text{Total vacant voids} = 240 - 114 - 4 = 122 \]

### Final Answer
Thus, the total number of vacant tetrahedral voids in 30 unit cells of ZnS is **122**.

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