The percentage of empty space in a simple cubic arrangement is :

Options

A. 74
B. 68
C. 32
D. 48

Answer

To find the percentage of empty space in a simple cubic arrangement, we will follow these steps:

### Step 1: Understand the Simple Cubic Unit Cell
A simple cubic unit cell consists of atoms located only at the corners of a cube. Each corner atom contributes 1/8th of its volume to the unit cell.

### Step 2: Determine the Number of Atoms in the Unit Cell
In a simple cubic structure, there are 8 corner atoms. The contribution of these atoms to the unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} = 1 \]
Thus, the number of atoms per unit cell (Z) is 1.

### Step 3: Calculate the Edge Length
Let the radius of the atom be \( r \). In a simple cubic arrangement, the edge length \( a \) of the cube is given by:
\[ a = 2r \]

### Step 4: Calculate the Volume of the Sphere
The volume \( V \) of a single atom (considered as a sphere) is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 5: Calculate the Volume of the Cube
The volume of the cube is given by:
\[ \text{Volume of cube} = a^3 = (2r)^3 = 8r^3 \]

### Step 6: 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:
\[ \text{Packing Fraction} = \frac{\text{Volume of atoms}}{\text{Volume of cube}} = \frac{\frac{4}{3} \pi r^3}{8r^3} \]
Simplifying this gives:
\[ \text{Packing Fraction} = \frac{4\pi}{24} = \frac{\pi}{6} \]
Calculating this numerically:
\[ \text{Packing Fraction} \approx 0.524 \]

### Step 7: Calculate Packing Efficiency
The packing efficiency is given by:
\[ \text{Packing Efficiency} = \text{Packing Fraction} \times 100 \approx 52.4\% \]

### Step 8: Calculate the Percentage of Empty Space
The percentage of empty space (voids) in the unit cell is calculated as:
\[ \text{Percentage of voids} = 100 - \text{Packing Efficiency} \]
Substituting the values:
\[ \text{Percentage of voids} = 100 - 52.4 = 47.6\% \]
This can be approximated to:
\[ \text{Percentage of voids} \approx 48\% \]

### Conclusion
Thus, the percentage of empty space in a simple cubic arrangement is approximately **48%**.

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