Calculate the radius of Xe atom , If the edge of the unit cell (FCC) is 620 pm.

Options

A. 189.37 pm
B. 209.87 pm
C. 219.25 pm
D. 235.16 pm

Answer

To calculate the radius of a xenon (Xe) atom given that the edge length of the face-centered cubic (FCC) unit cell is 620 pm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the FCC Structure**:
In a face-centered cubic (FCC) unit cell, atoms are located at each corner and the centers of all the faces of the cube. Each unit cell effectively contains 4 atoms.

2. **Identify the Relationship Between Edge Length and Atomic Radius**:
For an FCC unit cell, the relationship between the edge length (a) and the atomic radius (r) is given by the formula:
\[ a = 2\sqrt{2}r \]
where \( a \) is the edge length of the unit cell.

3. **Rearranging the Formula to Find the Radius**:
We can rearrange the formula to solve for the atomic radius \( r \):
\[ r = \frac{a}{2\sqrt{2}} \]

4. **Substituting the Given Edge Length**:
Substitute the given edge length \( a = 620 \, \text{pm} \) into the rearranged formula:
\[ r = \frac{620 \, \text{pm}}{2\sqrt{2}} \]

5. **Calculating the Radius**:
First, calculate \( 2\sqrt{2} \):
\[ 2\sqrt{2} \approx 2 \times 1.414 = 2.828 \]
Now, substitute this value back into the equation for \( r \):
\[ r = \frac{620 \, \text{pm}}{2.828} \approx 219.25 \, \text{pm} \]

6. **Final Answer**:
Therefore, the radius of the xenon atom is approximately:
\[ r \approx 219.25 \, \text{pm} \]