A mineral having the formula `AB_(2)` crytallises in the cubic close-packed lattice, with the A atoms occupying the lattice points. What is the coordination number of the A atoms and B atoms ? What percentage fraction of the tetrahedral sites is occupied by B atoms ?

Options

A. 0.25
B. 0.5
C. 0.75
D. 1

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the structure and coordination numbers
The mineral has the formula `AB2` and crystallizes in a cubic close-packed (CCP) lattice. In this structure, the A atoms occupy the lattice points.

**Coordination Number of A:**
In a cubic close-packed structure, each A atom (which is the cation) is surrounded by 8 B atoms (which are the anions). Therefore, the coordination number of A is 8.

**Coordination Number of B:**
Each B atom is surrounded by 4 A atoms. Thus, the coordination number of B is 4.

### Step 2: Determine the number of tetrahedral sites
In a cubic close-packed lattice, there are tetrahedral sites available for the smaller ions (in this case, B atoms). The number of tetrahedral sites in a CCP structure is given by the formula:
- Number of tetrahedral sites = 2 × (number of atoms in the unit cell)

Since there are 4 A atoms in the unit cell (because A occupies the lattice points in a CCP structure), the total number of tetrahedral sites is:
- Total tetrahedral sites = 2 × 4 = 8

### Step 3: Calculate the percentage of tetrahedral sites occupied by B atoms
In the AB2 structure, there are 2 B atoms for every unit cell. Since each B atom occupies one tetrahedral site, the number of tetrahedral sites occupied by B atoms is 4 (because there are 2 B atoms, and each occupies 2 tetrahedral sites).

To find the percentage of tetrahedral sites occupied by B atoms:
- Percentage fraction = (Number of tetrahedral sites occupied by B / Total number of tetrahedral sites) × 100
- Percentage fraction = (4 / 8) × 100 = 50%

### Final Answers:
- Coordination number of A atoms = 8
- Coordination number of B atoms = 4
- Percentage fraction of tetrahedral sites occupied by B atoms = 50%

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