Find the molecular formula of a compound made by atoms A and B, where A atoms present at all corners of cubic lattice and B atoms at alternate face centre.

Options

A. AB
B. `AB_(2)`
C. `A_(2)B`
D. `A_(2)B_(3)`

Answer

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

### Step 1: Determine the contribution of A atoms in the cubic lattice.
- In a cubic lattice, A atoms are located at all corners.
- There are 8 corners in a cube.
- Each corner atom is shared by 8 adjacent cubes, so the contribution of one corner atom to the cube is \( \frac{1}{8} \).
- Therefore, the total contribution of A atoms in one cube is:
\[ \text{Total A atoms} = 8 \times \frac{1}{8} = 1 \]

### Step 2: Determine the contribution of B atoms in the cubic lattice.
- B atoms are located at alternate face centers.
- There are 6 faces on a cube, but since B atoms are present at alternate face centers, they will be present on 3 faces (for example, if B is on the front, back, and top faces).
- Each face-centered atom is shared by 2 adjacent cubes, so the contribution of one face-centered atom to the cube is \( \frac{1}{2} \).
- Therefore, the total contribution of B atoms in one cube is:
\[ \text{Total B atoms} = 3 \times \frac{1}{2} = \frac{3}{2} \]

### Step 3: Write the empirical formula.
- From the calculations:
- Number of A atoms = 1
- Number of B atoms = \( \frac{3}{2} \)
- To express this in whole numbers, we can multiply both the number of A and B atoms by 2 to eliminate the fraction:
- A: \( 1 \times 2 = 2 \)
- B: \( \frac{3}{2} \times 2 = 3 \)

### Step 4: Final molecular formula.
- Therefore, the molecular formula of the compound is:
\[ \text{Molecular Formula} = A_2B_3 \]