The percentage of free space in a body centred cubic unit cell is

Options

A. 0.3
B. 0.32
C. 0.34
D. 0.28

Answer

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
In a body-centered cubic unit cell, there are 2 atoms per unit cell (1 atom at the center and 8 corner atoms, each contributing 1/8 of an atom).

### Step 2: Calculate the Volume Occupied by Atoms
The volume occupied by the atoms in the unit cell can be calculated using the formula for the volume of a sphere:
\[ V_{\text{occupied}} = z \times \frac{4}{3} \pi r^3 \]
where \( z = 2 \) for BCC.

### Step 3: Relate Radius (r) to Edge Length (a)
In a BCC unit cell, the relationship between the radius \( r \) of the atom and the edge length \( a \) of the cube is given by:
\[ r = \frac{\sqrt{3}}{4} a \]

### Step 4: Substitute Radius in the Volume Formula
Substituting the expression for \( r \) into the volume formula:
\[ V_{\text{occupied}} = 2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3 \]

### Step 5: Calculate the Total Volume of the Unit Cell
The total volume of the unit cell is given by:
\[ V_{\text{total}} = a^3 \]

### Step 6: Calculate Packing Efficiency
Now, we can calculate the packing efficiency:
\[ \text{Packing Efficiency} = \frac{V_{\text{occupied}}}{V_{\text{total}}} \]
Substituting the values we calculated:
\[ \text{Packing Efficiency} = \frac{2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3}{a^3} \]

### Step 7: Simplify the Expression
After simplifying, we find:
\[ \text{Packing Efficiency} = \frac{\sqrt{3} \pi}{8} \]

### Step 8: Calculate the Numerical Value of Packing Efficiency
Calculating the numerical value:
\[ \text{Packing Efficiency} \approx 0.68 \]

### Step 9: Calculate Free Space
The free space in the unit cell can be calculated as:
\[ \text{Free Space} = 100\% - \text{Packing Efficiency} \]
\[ \text{Free Space} = 100\% - 68\% = 32\% \]

### Conclusion
Thus, the percentage of free space in a body-centered cubic unit cell is **32%**.

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