A body centre cubic lattice is made up of two different types of atoms A and B. Atom A occupies the body centre and B occupying the corner positions. One of the corners is left unoccupied per unit cell. Empirical formula of such a solid is

Options

A. AB
B. `A_(2) B_(2)`
C. `A_(5)B_(7)`
D. `A_(8)B_(7)`

Answer

To find the empirical formula of the solid made up of two different types of atoms A and B in a body-centered cubic lattice, we can follow these steps:

### Step 1: Understand the Structure
In a body-centered cubic (BCC) lattice:
- There is 1 atom at the body center (Atom A).
- There are 8 corner positions, each potentially occupied by Atom B.

### Step 2: Determine the Number of Atoms
- **Atom A**: Since there is 1 atom at the body center, the contribution of Atom A per unit cell is:
\[ \text{Number of A atoms} = 1 \]

- **Atom B**: There are 8 corners in the unit cell, but one corner is left unoccupied. Therefore, there are 7 occupied corners. Each corner atom is shared by 8 unit cells, so the contribution of Atom B per unit cell is:
\[ \text{Number of B atoms} = 7 \times \frac{1}{8} = \frac{7}{8} \]

### Step 3: Calculate the Ratio of A to B
Now, we can find the ratio of the number of atoms of A to B:
\[ \text{Ratio of A to B} = \frac{1}{\frac{7}{8}} = \frac{1 \times 8}{7} = \frac{8}{7} \]

### Step 4: Write the Empirical Formula
The empirical formula can be expressed based on the ratio of A to B:
\[ \text{Empirical formula} = A_8B_7 \]

### Final Answer
Thus, the empirical formula of the solid is:
\[ \text{Empirical formula} = A_8B_7 \]

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