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: Setup
Let the total volume be $V$. The atom volume is $\frac{\pi}{6}V$ since one atom of radius $a/2$ sits in a cube of side $a$.

Step 2: Void equation
Void volume = total - atom volume = $V(1 - \pi/6) = 0.4764\,V$.

Step 3: Solve
$V = 2.0\times10^{-23}/0.4764 = 4.20\times10^{-23}\ \text{cm}^3$.

Step 4: Cross-check
The atom volume is then $0.5236 \times 4.2\times10^{-23} = 2.2\times10^{-23}$ cm$^3$, and adding the void volume of $2.0\times10^{-23}$ gives back $4.2\times10^{-23}$ cm$^3$.

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