Question

Calculate the volume of fcc unit cell in cm\(^3\) if void volume of it is \( 4.16 \times 10^{-24} \) cm\(^3\).

• A. \( 1.3 \times 10^{-23} \)
• B. \( 1.6 \times 10^{-23} \)
• C. \( 4.1 \times 10^{-23} \)
• D. \( 5.8 \times 10^{-23} \)

Hint:

In fcc structures, the void volume represents the unoccupied space, and it is typically around 26% of the total unit cell volume.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The void volume of a face-centered cubic (fcc) unit cell is given. We need to find the total volume of the unit cell based on its packing efficiency.
Step 2: Key Formula or Approach:
The packing efficiency of an fcc unit cell is \( 74% \), meaning the volume occupied by atoms is \( 74% \) of the total volume.
Therefore, the percentage of empty space (void volume) is \( 100% - 74% = 26% \).
\[ \text{Total Volume} = \frac{\text{Void Volume}}{\text{Void Fraction}} \] Step 3: Detailed Explanation:
Given:
Void Volume = \( 4.16 \times 10^{-24} \text{ cm}^3 \)
Void Fraction = \( 0.26 \)
Now, substitute the values into the formula:
\[ \text{Total Volume} = \frac{4.16 \times 10^{-24}}{0.26} \] \[ \text{Total Volume} = 16 \times 10^{-24} \text{ cm}^3 \] \[ \text{Total Volume} = 1.6 \times 10^{-23} \text{ cm}^3 \] Step 4: Final Answer:
The total volume of the fcc unit cell is \( 1.6 \times 10^{-23} \text{ cm}^3 \).