The packing efficiency in a simple cubic cell system of lattice is

Options

A. `68%`
B. `52%`
C. `74%`
D. `92%`

Answer

To find the packing efficiency in a simple cubic cell system, we will follow these steps:

### Step 1: Understand the structure of a simple cubic unit cell
In a simple cubic unit cell, atoms are located at the corners of the cube. Each corner atom is shared by eight adjacent unit cells, so effectively, each unit cell contains one atom.

### Step 2: Define the edge length in terms of atomic radius
Let the radius of each atom be denoted as \( r \). In a simple cubic structure, the edge length \( a \) of the cube is equal to twice the radius of the atom:
\[ a = 2r \]

### Step 3: Calculate the volume of the cube
The volume \( V_{\text{cube}} \) of the cube can be calculated using the formula for the volume of a cube, which is given by:
\[ V_{\text{cube}} = a^3 \]
Substituting \( a = 2r \):
\[ V_{\text{cube}} = (2r)^3 = 8r^3 \]

### Step 4: Calculate the volume of a single atom (sphere)
The volume \( V_{\text{sphere}} \) of a single atom can be calculated using the formula for the volume of a sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

### Step 5: Calculate the packing efficiency
Packing efficiency is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell, expressed as a percentage:
\[ \text{Packing Efficiency} = \left( \frac{V_{\text{sphere}}}{V_{\text{cube}}} \right) \times 100 \]
Substituting the volumes we calculated:
\[ \text{Packing Efficiency} = \left( \frac{\frac{4}{3} \pi r^3}{8r^3} \right) \times 100 \]

### Step 6: Simplify the expression
We can simplify the expression by canceling \( r^3 \):
\[ \text{Packing Efficiency} = \left( \frac{\frac{4}{3} \pi}{8} \right) \times 100 \]
\[ = \left( \frac{4 \pi}{24} \right) \times 100 \]
\[ = \left( \frac{\pi}{6} \right) \times 100 \]

### Step 7: Substitute the value of \( \pi \)
Using \( \pi \approx 3.14 \):
\[ \text{Packing Efficiency} \approx \left( \frac{3.14}{6} \right) \times 100 \approx 52.33\% \]

### Step 8: Round off the result
Rounding off gives us approximately \( 52\% \).

### Conclusion
Thus, the packing efficiency in a simple cubic cell system is approximately **52%**.

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