Total volume of atoms present in a face centred cubic unit cell of a metal is (r=radius of atom)

Options

A. `20/3 pir^3`
B. `24/3 pi r^3`
C. `12/3 pi r^3`
D. `16/3 pi r^3`

Explanation

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

### Step 1: Understand the structure of the FCC unit cell
In a face-centered cubic unit cell, there are atoms located at each of the eight corners and at the center of each of the six faces.

### Step 2: Calculate the contribution of corner atoms
Each corner atom is shared by eight adjacent unit cells. Therefore, the contribution of one corner atom to the unit cell is:
\[ \text{Contribution from one corner atom} = \frac{1}{8} \]
Since there are 8 corners in the unit cell, the total contribution from the corner atoms is:
\[ \text{Total contribution from corner atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

### Step 3: Calculate the contribution of face-centered atoms
Each face-centered atom is shared between two unit cells. Therefore, the contribution of one face-centered atom to the unit cell is:
\[ \text{Contribution from one face-centered atom} = \frac{1}{2} \]
Since there are 6 faces in the unit cell, the total contribution from the face-centered atoms is:
\[ \text{Total contribution from face-centered atoms} = 6 \times \frac{1}{2} = 3 \text{ atoms} \]

### Step 4: Calculate the total number of atoms in the FCC unit cell
Now, we can add the contributions from the corner and face-centered atoms:
\[ \text{Total number of atoms in FCC unit cell} = 1 + 3 = 4 \text{ atoms} \]

### Step 5: 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:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the atom.

### Step 6: 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:
\[ \text{Total volume} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]

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

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