Question

What is the radius of sodium atom if it crystallizes in fcc structure with edge length of unit cell \( 4.29 \times 10^{-8} \, \text{cm} \)?

• A. \( 2.30 \times 10^{-8} \, \text{cm} \)
• B. \( 6.19 \times 10^{-9} \, \text{cm} \)
• C. \( 1.85 \times 10^{-8} \, \text{cm} \)
• D. \( 1.61 \times 10^{-8} \, \text{cm} \)

Hint:

The atomic radius in an fcc structure is related to the edge length of the unit cell by the formula \( a = 2\sqrt{2} \times r \).

Answer 1

The Correct Option is C

Step 1: Using the relationship for fcc crystals.
For an fcc structure, the edge length \(a\) is related to the atomic radius \(r\) by the equation: \[ a = 2\sqrt{2} \times r \] Given that the edge length \(a = 4.29 \times 10^{-8} \, \text{cm}\), we can solve for \(r\): \[ r = \frac{a}{2\sqrt{2}} = \frac{4.29 \times 10^{-8}}{2\sqrt{2}} = 1.85 \times 10^{-8} \, \text{cm} \] Step 2: Analyzing the options.
(A) \( 2.30 \times 10^{-8} \, \text{cm} \): Incorrect. This is too large.
(B) \( 6.19 \times 10^{-9} \, \text{cm} \): Incorrect. This is too small.
(C) \( 1.85 \times 10^{-8} \, \text{cm} \): Correct — This is the correct atomic radius for sodium in fcc structure.
(D) \( 1.61 \times 10^{-8} \, \text{cm} \): Incorrect. This is not the correct radius.
Step 3: Conclusion.
The correct answer is (C) \( 1.85 \times 10^{-8} \, \text{cm} \).