Percentage of free space in cubic in a body- centred cubic unit cell is .

Options

A. 74
B. 68
C. 32
D. 26

Explanation

To find the percentage of free space in a body-centered cubic (BCC) unit cell, we can follow these steps:

### Step 1: Understand the Structure of BCC
A body-centered cubic unit cell has one atom at each of the eight corners of the cube and one atom at the center of the cube. Each corner atom contributes 1/8 of its volume to the unit cell.

### Step 2: Calculate the Total Number of Atoms in BCC
In a BCC unit cell:
- There are 8 corner atoms, each contributing 1/8:
\(8 \times \frac{1}{8} = 1\) atom from corners.
- There is 1 atom at the center:
\(1\) atom from the center.

Thus, the total number of atoms in a BCC unit cell is:
\[ \text{Total atoms} = 1 + 1 = 2 \text{ atoms} \]

### Step 3: Calculate the Volume Occupied by Atoms
The volume occupied by the atoms can be calculated using the formula for the volume of a sphere, \(V = \frac{4}{3} \pi r^3\).

Assuming the radius of the atom is \(r\), the total volume occupied by the two atoms in the BCC unit cell is:
\[ \text{Volume occupied} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 4: Calculate the Volume of the Unit Cell
The edge length \(a\) of the BCC unit cell can be related to the radius \(r\) of the atoms. For BCC, the relationship is:
\[ a = \frac{4r}{\sqrt{3}} \]

The volume of the unit cell is:
\[ \text{Volume of unit cell} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

### Step 5: Calculate the Efficiency
The efficiency of the packing can be calculated as the ratio of the volume occupied by the atoms to the volume of the unit cell:
\[ \text{Efficiency} = \frac{\text{Volume occupied}}{\text{Volume of unit cell}} \times 100 \]

Substituting the volumes we calculated:
\[ \text{Efficiency} = \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \times 100 \]

This simplifies to:
\[ \text{Efficiency} = \frac{8\pi \sqrt{3}}{64} \times 100 = \frac{\pi \sqrt{3}}{8} \times 100 \approx 68\% \]

### Step 6: Calculate the Percentage of Free Space
The percentage of free space is given by:
\[ \text{Free space} = 100\% - \text{Efficiency} \]
\[ \text{Free space} = 100\% - 68\% = 32\% \]

### Final Answer
The percentage of free space in a body-centered cubic unit cell is **32%**.

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