The percentage of empty space in a body centred cubic arrangement is

Options

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

Explanation

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

### Step 1: Understand the Structure of BCC
In a BCC unit cell, there are two atoms per unit cell: one atom at each corner of the cube and one atom at the center of the cube.

### Step 2: Relate Atomic Radius (r) to Edge Length (a)
The relationship between the atomic radius (r) and the edge length (a) in a BCC structure is given by the formula:
\[ 4r = \sqrt{3}a \]
From this, we can express the edge length (a) in terms of the atomic radius (r):
\[ a = \frac{4r}{\sqrt{3}} \]

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

### Step 4: Calculate the Volume of the Atoms in the Unit Cell
The volume occupied by the two atoms in the BCC unit cell can be calculated using the formula for the volume of a sphere:
\[ V_{\text{atoms}} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 5: Calculate the Packing Fraction
The packing fraction (PF) is the ratio of the volume occupied by the atoms to the volume of the unit cell:
\[ \text{Packing Fraction} = \frac{V_{\text{atoms}}}{V_{\text{cube}}} = \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \]
This simplifies to:
\[ \text{Packing Fraction} = \frac{8\pi}{64/\sqrt{3}} = \frac{8\pi \sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \]

### Step 6: Convert Packing Fraction to Percentage
To find the packing fraction as a percentage:
\[ \text{Packing Fraction Percentage} = \left(\frac{\pi \sqrt{3}}{8}\right) \times 100 \]
Calculating this gives approximately:
\[ \text{Packing Fraction Percentage} \approx 68\% \]

### Step 7: Calculate the Percentage of Empty Space
The percentage of empty space in the BCC unit cell is given by:
\[ \text{Empty Space Percentage} = 100\% - \text{Packing Fraction Percentage} \]
Thus,
\[ \text{Empty Space Percentage} = 100\% - 68\% = 32\% \]

### Final Answer
The percentage of empty space in a body-centered cubic arrangement is **32%**.

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