A compound of formula `A_(2)B_(3)` has the hcp lattice. Which atom forms the hcp lattice and what fraction of tetrahedral voids is occupied by the other atoms

Options

A. hcp lattice-B, `(2)/(3)` Tetrachedral voids-A
B. hcp lattice-A, `(2)/(3)` Tetrachedral voids-B
C. hcp lattice-A, `(1)/(3)` Tetrachedral voids-B
D. hcp lattice-B, `(1)/(3)` Tetrachedral voids-A

Explanation

To solve the problem step by step, we need to analyze the given compound with the formula \( A_2B_3 \) and determine which atom forms the hcp lattice and what fraction of tetrahedral voids is occupied by the other atom.

### Step 1: Understanding the hcp lattice
In a hexagonal close-packed (hcp) lattice, there are two types of voids: octahedral and tetrahedral. Each unit cell of hcp contains:
- 6 atoms (2 atoms per unit cell).
- 12 tetrahedral voids.

### Step 2: Assigning atoms to the lattice
We need to determine whether atom A or atom B forms the hcp lattice. Let's assume that atom B forms the hcp lattice.

### Step 3: Counting the number of tetrahedral voids
If B forms the hcp lattice, the total number of tetrahedral voids in the lattice will be:
- Total tetrahedral voids = \( 2n \) (where n is the number of B atoms in the lattice).

### Step 4: Determining the occupancy of tetrahedral voids
According to the problem, we need to find out how many of these tetrahedral voids are occupied by atom A.

### Step 5: Checking the occupancy fraction
If we assume that a certain fraction of the tetrahedral voids is occupied by atom A, we can express this as:
- Let \( x \) be the fraction of tetrahedral voids occupied by A.

### Step 6: Formulating the compound
If \( x \) of the tetrahedral voids is occupied by A, then:
- The total number of A atoms = \( x \times 2n \).
- The total number of B atoms = \( n \).

From the formula \( A_2B_3 \), we can set up the following equations:
- \( 2 = x \times 2n \) (for A)
- \( 3 = n \) (for B)

### Step 7: Solving the equations
From the equation for B:
- \( n = 3 \).

Substituting \( n \) into the equation for A:
- \( 2 = x \times 2 \times 3 \)
- \( 2 = 6x \)
- \( x = \frac{1}{3} \).

### Step 8: Conclusion
Thus, if B forms the hcp lattice, then \( \frac{1}{3} \) of the tetrahedral voids are occupied by A.

### Final Answer
- Atom B forms the hcp lattice.
- The fraction of tetrahedral voids occupied by atom A is \( \frac{1}{3} \).

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