packing fraction of a unit cell is drfined as the fraction of the total volume of the unit cell occupied by the atom(s).
`P.E=("Volume of the atoms(s) present in a unit cell")/("Volume of unit cell")=(Zxx(4)/(3)pir^(3))/(a^(3))`
and `%` of empty space = `100- P.F.xx100 `
where Z= effective number of stoms in s cube .
r= radius of a an atoms
a = edge lenght of the cube
`%` empty space in body centered cubic cell unit is nearly :

Options

A. 52.36
B. 47.6
C. 32
D. 26

Answer

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

### Step 1: Understand the structure of the BCC unit cell
In a BCC unit cell, there are atoms located at the eight corners of the cube and one atom at the center of the cube.

### Step 2: Determine the effective number of atoms (Z)
The effective number of atoms (Z) in a BCC unit cell can be calculated as follows:
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell (since each corner atom is shared by 8 unit cells).
- There are 8 corner atoms, contributing \( 8 \times \frac{1}{8} = 1 \) atom.
- The center atom contributes 1 atom.
- Thus, the total effective number of atoms \( Z \) in a BCC unit cell is:
\[ Z = 1 + 1 = 2 \]

### Step 3: Relate the edge length (a) to the atomic radius (r)
For a BCC unit cell, the relationship between the edge length \( a \) and the atomic radius \( r \) is given by the body diagonal:
\[ \text{Body diagonal} = \sqrt{3}a = 4r \]
From this, we can derive:
\[ a = \frac{4r}{\sqrt{3}} \]

### Step 4: Calculate the volume of the atoms in the unit cell
The volume of the atoms present in the unit cell can be calculated using the formula:
\[ \text{Volume of atoms} = Z \times \left(\frac{4}{3} \pi r^3\right) \]
Substituting \( Z = 2 \):
\[ \text{Volume of atoms} = 2 \times \left(\frac{4}{3} \pi r^3\right) = \frac{8}{3} \pi r^3 \]

### Step 5: Calculate the volume of the unit cell
The volume of the unit cell is given by:
\[ \text{Volume of unit cell} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

### Step 6: Calculate the packing fraction (PF)
The packing fraction is defined as:
\[ PF = \frac{\text{Volume of atoms}}{\text{Volume of unit cell}} = \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \]
Simplifying this:
\[ PF = \frac{8\pi}{64/\sqrt{3}} = \frac{8\pi \sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \approx 0.524 \]

### Step 7: Calculate the percentage of empty space
The percentage of empty space in the unit cell can be calculated as:
\[ \text{Percentage of empty space} = 100 - (PF \times 100) \]
Substituting \( PF \approx 0.524 \):
\[ \text{Percentage of empty space} = 100 - (0.524 \times 100) = 100 - 52.4 = 47.6\% \]

### Final Answer
The percentage of empty space in a body-centered cubic unit cell is approximately **47.6%**.

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