Question

Sodium crystallizes in bcc structure with radius \(186 \times 10^{-8} \,\) \(\text{cm}. \)Calculate the edge length of the unit cell? 
 

• A. \(4.29 \times 10^{-8} \, \text{cm}\)
• B. \(6.20 \times 10^{-8} \, \text{cm}\)
• C. \(8.05 \times 10^{-8} \, \text{cm}\)
• D. \(3.72 \times 10^{-8} \, \text{cm} \)

Hint:

In bcc crystals, the relationship between the edge length and atomic radius is given by \( a = \frac{4r}{\sqrt{3}} \), where \( r \) is the atomic radius.

Answer 1

The Correct Option is A

Step 1: Understanding the bcc structure. 
In a body-centered cubic (bcc) structure, the relationship between the atomic radius \( r \) and the edge length \( a \) of the unit cell is given by: \[ \text{Diagonal of the cube} = \sqrt{3} \cdot a = 4r \] Thus, we can calculate the edge length \( a \) using: \[ a = \frac{4r}{\sqrt{3}} \] Step 2: Substituting the known value. 
Given the atomic radius \( r = 186 \times 10^{-8} \, \text{cm} \), we can substitute into the equation: \[ a = \frac{4 \times 186 \times 10^{-8}}{\sqrt{3}} = 4.29 \times 10^{-8} \, \text{cm} \] Step 3: Conclusion. 
The correct edge length of the unit cell is \(4.29 \times 10^{-8 \, \text{cm}}. \)