In polonium the total space occupied by the atoms in its crystal lattice is 

Options

A. `52%`
B. `68%`
C. `74%`
D. `32%`

Answer

To solve the question regarding the total space occupied by atoms in the crystal lattice of polonium, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Crystal Structure**:
Polonium crystallizes in a simple cubic unit cell. In a simple cubic structure, atoms are located at the corners of the cube.

2. **Determine the Number of Atoms in the Unit Cell**:
In a simple cubic unit cell, there are 8 corner atoms. Each corner atom contributes \( \frac{1}{8} \) of its volume to the unit cell. Therefore, the total number of atoms in the unit cell can be calculated as:
\[ \text{Number of atoms} = 8 \times \frac{1}{8} = 1 \]

3. **Relate Edge Length to Atomic Radius**:
Let \( a \) be the edge length of the unit cell and \( r \) be the radius of the atom. In a simple cubic structure, the edge length \( a \) is equal to twice the radius of the atom:
\[ a = 2r \]

4. **Calculate the Volume Occupied by Atoms**:
The volume occupied by a single atom can be calculated using the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Since there is 1 atom in the unit cell, the total volume occupied by atoms in the unit cell is:
\[ \text{Total volume occupied} = \frac{4}{3} \pi r^3 \]

5. **Calculate the Volume of the Unit Cell**:
The volume of the unit cell can be calculated using the edge length:
\[ \text{Volume of unit cell} = a^3 = (2r)^3 = 8r^3 \]

6. **Calculate the Packing Fraction**:
The packing fraction (or the percentage of volume occupied by the atoms) is given by the ratio of the volume occupied by the atoms to the volume of the unit cell:
\[ \text{Packing fraction} = \frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} = \frac{\frac{4}{3} \pi r^3}{8r^3} \]
Simplifying this gives:
\[ \text{Packing fraction} = \frac{\frac{4}{3} \pi}{8} = \frac{\pi}{6} \]

7. **Convert Packing Fraction to Percentage**:
To express the packing fraction as a percentage, we multiply by 100:
\[ \text{Packing fraction percentage} = \frac{\pi}{6} \times 100 \approx 52.36\% \]

8. **Final Answer**:
Therefore, the total space occupied by the atoms in the crystal lattice of polonium is approximately **52%**.