In the fcc arrangement of `A` and `B` atoms whose `A` atoms are at corners of the unit cell and `B` are at the face centres one of the `A` atom is missing from one corner in each unit cell. What is the simplest formula of the compound?

Explanation

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

### Step 1: Identify the positions of the atoms in the unit cell
In a face-centered cubic (FCC) arrangement:
- `A` atoms are located at the corners of the unit cell.
- `B` atoms are located at the face centers of the unit cell.

### Step 2: Calculate the contribution of `A` atoms
In a standard FCC unit cell, there are 8 corners. Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell. However, since one `A` atom is missing from one corner, we only have 7 corner atoms contributing.

- Contribution from `A` atoms:
\[ \text{Total contribution of A} = 7 \times \frac{1}{8} = \frac{7}{8} \]

### Step 3: Calculate the contribution of `B` atoms
In the FCC structure, there are 6 face centers, and each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell.

- Contribution from `B` atoms:
\[ \text{Total contribution of B} = 6 \times \frac{1}{2} = 3 \]

### Step 4: Write the empirical formula
Now, we have:
- Total `A` atoms = \( \frac{7}{8} \)
- Total `B` atoms = \( 3 \)

The simplest formula can be expressed as:
\[ A_{\frac{7}{8}}B_{3} \]

### Step 5: Clear the fraction for the formula
To eliminate the fraction, we can multiply the entire formula by 8 to get whole numbers:
\[ A_{7}B_{24} \]

### Final Answer
The simplest formula of the compound is:
\[ \text{A}_7\text{B}_{24} \]

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