The volume of atoms present in body centred cubic unit cell of a metal of atomic radius r is,

Options

A. `(16)/(3)pi r^(3)`
B. `(8)/(3) pi r^(3)`
C. `(12)/(3)pi r^(3)`
D. `(24)/( 3) pi r^(3)`.

Answer

To find the volume of atoms present in a body-centered cubic (BCC) unit cell of a metal with atomic radius \( r \), we can follow these steps:

### Step 1: Understand the Structure of BCC
In a body-centered cubic (BCC) unit cell, there are 2 atoms per unit cell. One atom is located at each corner of the cube, and one atom is located at the center of the cube.

### Step 2: Calculate the Volume of One Atom
The volume \( V \) of a single atom, which we assume to be spherical, can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 3: Calculate the Total Volume of Atoms in BCC
Since there are 2 atoms in the BCC unit cell, the total volume of the atoms in the unit cell is:
\[ \text{Total Volume} = 2 \times V = 2 \times \left(\frac{4}{3} \pi r^3\right) \]
This simplifies to:
\[ \text{Total Volume} = \frac{8}{3} \pi r^3 \]

### Conclusion
Thus, the volume of atoms present in a body-centered cubic unit cell of a metal with atomic radius \( r \) is:
\[ \frac{8}{3} \pi r^3 \]