A solid with formula `ABC_(3)` would probably have

Options

A. A at body centre, B at face centres and C at corners of the cube
B. A at corners of cube, B at body centre C at face centre
C. A at corners of hexagon, B at centres of the hexagon and C inside the hexagonal unit cell
D. A at corner, B at face centre, C at body centre

Answer

To determine the probable arrangement of atoms in a solid with the formula \( ABC_3 \), we need to analyze the positions of the atoms A, B, and C based on the given options. Here’s a step-by-step breakdown of the solution:

### Step 1: Analyze the given formula
The formula \( ABC_3 \) indicates that:
- There is 1 atom of A
- There is 1 atom of B
- There are 3 atoms of C

### Step 2: Evaluate the first option
The first option states:
- A is at the body center
- B is at the face centers

**Calculating contributions:**
- A (body center): There is 1 body center atom contributing 1 atom of A.
- B (face centers): There are 6 face centers, and each contributes \( \frac{1}{2} \). Thus, total contribution from B = \( 6 \times \frac{1}{2} = 3 \).

**Total atoms from this option:**
- A = 1
- B = 3
- C = 0 (not mentioned)

This does not match the formula \( ABC_3 \). Hence, this option is incorrect.

### Step 3: Evaluate the second option
The second option states:
- A is at the corners
- B is at the body center
- C is at the face centers

**Calculating contributions:**
- A (corners): There are 8 corner atoms, each contributing \( \frac{1}{8} \). Thus, total contribution from A = \( 8 \times \frac{1}{8} = 1 \).
- B (body center): There is 1 body center atom contributing 1 atom of B.
- C (face centers): There are 6 face centers, and each contributes \( \frac{1}{2} \). Thus, total contribution from C = \( 6 \times \frac{1}{2} = 3 \).

**Total atoms from this option:**
- A = 1
- B = 1
- C = 3

This matches the formula \( ABC_3 \). Hence, this option is correct.

### Conclusion
The correct arrangement for the solid with the formula \( ABC_3 \) is:
- A at corners
- B at body center
- C at face centers