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: Understand the relation between void volume and unit cell volume.
For a face-centered cubic (fcc) unit cell, the void volume is the volume that is not occupied by the atoms. The relation between the void volume and the total unit cell volume is given by:
\[ \text{Void Volume} = \text{Total Unit Cell Volume} - \text{Atomic Volume} \]

Step 2: Use the void volume.
Given that the void volume is \( 4.16 \times 10^{-24} \, \text{cm}^3 \), we can calculate the total unit cell volume. For an fcc structure, the void volume is roughly 0.26 of the total volume.

Step 3: Calculate the total unit cell volume.
The void volume is 26 % of the total volume, so:
\[ \text{Void Volume} = 0.26 \times \text{Total Unit Cell Volume} \] Substitute the given void volume: \[ 4.16 \times 10^{-24} = 0.26 \times \text{Total Unit Cell Volume} \] Solve for the total unit cell volume: \[ \text{Total Unit Cell Volume} = \frac{4.16 \times 10^{-24}}{0.26} = 4.1 \times 10^{-23} \, \text{cm}^3 \]

Step 4: Final conclusion.
Thus, the volume of the fcc unit cell is:
\[ \boxed{4.1 \times 10^{-23} \, \text{cm}^3} \] Therefore, the correct answer is:
\[ \boxed{(3)\ 4.1 \times 10^{-23} \, \text{cm}^3} \]