The space occupied by b.c.c. arrangement is approximately

Options

A. `50%`
B. `68%`
C. `74%`
D. `56%`

Explanation

To find the space occupied by a body-centered cubic (BCC) arrangement, we need to calculate the packing efficiency. Here’s a step-by-step solution:

### Step 1: Understand the BCC Structure
In a BCC unit cell, there are 2 atoms per unit cell. This is denoted by the variable \( z \).

### 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:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Thus, for 2 atoms:
\[ \text{Total volume occupied by atoms} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 3: Determine the Edge Length of the BCC Unit Cell
The relationship between the radius \( r \) of the atom and the edge length \( a \) of the BCC unit cell is given by:
\[ a = \frac{4r}{\sqrt{3}} \]

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

### Step 5: Calculate Packing Efficiency
The packing efficiency (PE) is the ratio of the volume occupied by the atoms to the volume of the unit cell, expressed as a percentage:
\[ \text{Packing Efficiency} = \left(\frac{\text{Volume occupied by atoms}}{\text{Volume of the cube}}\right) \times 100 \]
Substituting the values we calculated:
\[ \text{Packing Efficiency} = \left(\frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}}\right) \times 100 \]
Simplifying this:
\[ \text{Packing Efficiency} = \left(\frac{8 \pi \sqrt{3}}{64}\right) \times 100 = \left(\frac{\pi \sqrt{3}}{8}\right) \times 100 \]

### Step 6: Calculate the Numerical Value
Using the approximate value of \( \pi \approx 3.14 \) and \( \sqrt{3} \approx 1.732 \):
\[ \text{Packing Efficiency} \approx \left(\frac{3.14 \times 1.732}{8}\right) \times 100 \approx 68\% \]

### Final Answer
The space occupied by the BCC arrangement is approximately **68%**.

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