A solid between `A` and `B` has the following arrangement of atoms `:`
`(i)` Atoms `A` are arranged in `c.c.p.` array.
`(ii)` Atoms `B` occupy all the all the octahedral voids and half the tetrahedral voids. What is the formula of the compound ?

Options

A. `A_(2)B`
B. AB
C. `AB_(2)`
D. `AB_(4)`

Answer

To find the formula of the compound formed by atoms A and B, we will follow these steps:

### Step 1: Determine the number of A atoms in the c.c.p. (cubic close-packed) arrangement
- In a c.c.p. or FCC (Face-Centered Cubic) lattice, there are:
- 8 atoms at the corners, each contributing \( \frac{1}{8} \) (total contribution from corners = \( 8 \times \frac{1}{8} = 1 \))
- 6 face-centered atoms, each contributing \( \frac{1}{2} \) (total contribution from faces = \( 6 \times \frac{1}{2} = 3 \))
- Therefore, the total number of A atoms in one unit cell is:
\[ 1 + 3 = 4 \]

### Step 2: Determine the number of B atoms occupying the voids
- In a c.c.p. lattice, the number of octahedral voids is equal to the number of atoms, which is 4.
- The number of tetrahedral voids is twice the number of octahedral voids, so:
\[ \text{Number of tetrahedral voids} = 2 \times 4 = 8 \]
- According to the question, B occupies all the octahedral voids and half of the tetrahedral voids:
- All octahedral voids = 4
- Half of tetrahedral voids = \( \frac{8}{2} = 4 \)
- Therefore, the total number of B atoms is:
\[ 4 + 4 = 8 \]

### Step 3: Write the empirical formula
- Now that we have the number of A and B atoms:
- Number of A atoms = 4
- Number of B atoms = 8
- The formula can be written as:
\[ A_4B_8 \]
- To simplify this formula, we divide both subscripts by the greatest common divisor, which is 4:
\[ A_{4/4}B_{8/4} = AB_2 \]

### Final Answer
The formula of the compound is:
\[ AB_2 \]