What is the simplest formula of a solid whose unit cell has the atom A at each corner, the atom B at each face centre and a atom C at the body centre.

Options

A. `A_(2)BC`
B. `AB_(2)C`
C. `AB_(3)C`
D. `ABC_(2)`

Answer

To find the simplest formula of the solid whose unit cell contains atoms A, B, and C in specific positions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Positions of Atoms**:
- Atoms A are located at each corner of the cube.
- Atoms B are located at the center of each face of the cube.
- Atom C is located at the center of the cube (body center).

2. **Calculate Contribution of Corner Atoms (A)**:
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because each corner atom is shared by 8 adjacent unit cells.
- There are 8 corners in a cube.
- Therefore, the total contribution of A per unit cell is:
\[ \text{Contribution of A} = 8 \times \frac{1}{8} = 1 \]

3. **Calculate Contribution of Face Center Atoms (B)**:
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell because each face atom is shared by 2 adjacent unit cells.
- There are 6 faces in a cube.
- Therefore, the total contribution of B per unit cell is:
\[ \text{Contribution of B} = 6 \times \frac{1}{2} = 3 \]

4. **Calculate Contribution of Body Center Atom (C)**:
- The body-centered atom contributes fully to the unit cell, so:
\[ \text{Contribution of C} = 1 \]

5. **Summarize Contributions**:
- From the calculations:
- Contribution of A = 1
- Contribution of B = 3
- Contribution of C = 1

6. **Write the Simplest Formula**:
- The simplest formula can be expressed as:
\[ \text{Formula} = A^1B^3C^1 \Rightarrow AB_3C \]

### Final Answer:
The simplest formula of the solid is \( AB_3C \).

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