In a fcc arrangement of A and B atoms, where A atoms are at the Corners of the unit cell, B atoms at the face centers, two atoms are missing from two corners in each unit cell, then the simplest formula of the compound is

Options

A. `A_(7)B_(6)`.
B. `A_(6)B_(7)`
C. `A_(7)B_(24)`
D. `AB_(4)`

Answer

To find the simplest formula of the compound formed by A and B atoms in a face-centered cubic (FCC) arrangement with the given conditions, we will follow these steps:

### Step 1: Determine the contribution of A atoms
In a face-centered cubic (FCC) unit cell:
- A atoms are located at the corners.
- There are 8 corners in a unit cell.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.

**Calculation:**
\[ \text{Total contribution from A atoms} = 8 \times \frac{1}{8} = 1 \]

### Step 2: Account for missing A atoms
According to the problem, 2 A atoms are missing from the corners. Since each missing atom contributes \( \frac{1}{8} \):
- Contribution of 2 missing A atoms:
\[ \text{Contribution from missing A atoms} = 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \]

Now, we subtract this from the total contribution:
\[ \text{Remaining contribution from A} = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 3: Determine the contribution of B atoms
B atoms are located at the face centers:
- There are 6 faces in a unit cell.
- Each face atom contributes \( \frac{1}{2} \).

**Calculation:**
\[ \text{Total contribution from B atoms} = 6 \times \frac{1}{2} = 3 \]

### Step 4: Write the formula based on contributions
Now, we have:
- Contribution from A = \( \frac{3}{4} \)
- Contribution from B = 3

To write the formula, we express the ratio of A to B. We can multiply both contributions by 4 to eliminate the fraction:
\[ \text{A} = \frac{3}{4} \times 4 = 3 \]
\[ \text{B} = 3 \times 4 = 12 \]

Thus, the ratio of A to B is:
\[ A : B = 3 : 12 \]

### Step 5: Simplifying the ratio
To simplify the ratio:
\[ \frac{3}{3} : \frac{12}{3} = 1 : 4 \]

### Final Answer
The simplest formula of the compound is:
\[ \text{AB}_4 \]

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