Question

Calculate the total volume of simple cubic unit cell in \(\text{cm}^3\) if void volume is \(2.0\times 10^{-23}\,\text{cm}^3\)

• A. \(4.2\times 10^{-23}\)
• B. \(3.8\times 10^{-23}\)
• C. \(3.4\times 10^{-23}\)
• D. \(4.6\times 10^{-23}\)

Hint:

The packing efficiency of a simple cubic cell is 52.4 percent, so void is 47.6 percent of the total volume.

Answer 1

The Correct Option is A

Step 1: Understand the concept
In a simple cubic unit cell the atoms touch along the edge, so \(a = 2r\). The occupied fraction (packing efficiency) is the volume of the one atom divided by the volume of the cell.

Step 2: Find the packing fraction
\[ \text{Packing efficiency} = \frac{\frac{4}{3}\pi r^3}{(2r)^3} = \frac{\pi}{6} = 0.524 \]
So the void fraction is \(1 - 0.524 = 0.476\).

Step 3: Use the void volume
\[ V_{\text{void}} = 0.476 \times V_{\text{total}} \Rightarrow V_{\text{total}} = \frac{2.0 \times 10^{-23}}{0.476} \]

Step 4: Evaluate
\[ V_{\text{total}} = 4.2 \times 10^{-23}\ \text{cm}^3 \]
The other options correspond to void fractions of 0.53, 0.59 and 0.43, which do not match the simple cubic cell.

Final Answer:
The total volume is 4.2 x 10^-23 cm^3. This is option (A). \[ \boxed{\text{(A) }4.2 \times 10^{-23}\ \text{cm}^3} \]