All real structures are three dimensional structures. They can be obtained by stacking two dimensional layers one above the other while placing the second square dose packed layer above the first we follow the same rule that was followed when one row was placed adjacent to the other. The second layer is placed over the first layer such that the spheres of the upper layer are exactly above there of the first layer. In his arrangement spheres of both the layers are perfectly aligned horizontally as well as vertically. A metallic element crystallise into a lattice having a ABC ABC pattern and packing of spheres leaves out voids in the lattice.
What is the total volume of atoms in a face centred cubic unit cell of a metal? (r is atomic radius).

Options

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

Explanation

To find the total volume of atoms in a face-centered cubic (FCC) unit cell of a metal, we can follow these steps:

### Step 1: Determine the number of atoms in an FCC unit cell
In a face-centered cubic unit cell, there are:
- 8 corner atoms, each contributing 1/8 of an atom to the unit cell (since each corner atom is shared among 8 unit cells).
- 6 face-centered atoms, each contributing 1/2 of an atom to the unit cell (since each face-centered atom is shared between 2 unit cells).

Calculating the total number of atoms:
\[ \text{Total atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 2: Calculate the volume of one atom
The volume \( V \) of a single atom 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 the FCC unit cell
Since there are 4 atoms in the FCC unit cell, the total volume of atoms in the unit cell is:
\[ \text{Total volume of atoms} = \text{Number of atoms} \times \text{Volume of one atom} \]
Substituting the values:
\[ \text{Total volume of atoms} = 4 \times \left(\frac{4}{3} \pi r^3\right) = \frac{16}{3} \pi r^3 \]

### Final Answer
The total volume of atoms in a face-centered cubic unit cell of a metal is:
\[ \frac{16}{3} \pi r^3 \]

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