Question

A metal crystallizes to form simple cubic unit cell. Calculate the void volume of unit cell. [\(a = 3.36\times 10^{-8}\) cm]

• A. \(1.807\times 10^{-23}\) cm\(^3\).
• B. \(1.214\times 10^{-23}\) cm\(^3\).
• C. \(3.662\times 10^{-23}\) cm\(^3\).
• D. \(1.986\times 10^{-23}\) cm\(^3\).

Hint:

Void volume = cell volume \(a^3\) minus atom volume (packing efficiency 52.4 percent).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In a simple cubic cell there is \(1\) atom per cell, and its radius is \(r = a/2\). The packing efficiency is \(52.4\%\), so \(47.6\%\) of the cell is empty.

Step 2: Cell volume:
\[ a^3 = (3.36\times10^{-8})^3 = 37.93\times10^{-24} = 3.793\times10^{-23}\ \text{cm}^3 \]

Step 3: Void volume:
Volume of the one atom: \(\frac{4}{3}\pi r^3 = \frac{4}{3}\pi\left(\frac{a}{2}\right)^3 = 0.5236\,a^3 = 1.986\times10^{-23}\) cm\(^3\).
\[ V_{\text{void}} = 3.793\times10^{-23} - 1.986\times10^{-23} = 1.807\times10^{-23}\ \text{cm}^3 \]
Option D (\(1.986\times10^{-23}\)) is the volume of the atom, not the void. Options B and C do not match either quantity.

Final Answer:
The void volume is \(1.807\times10^{-23}\) cm\(^3\), option (A). \[ \boxed{1.807\times10^{-23}\ \text{cm}^3} \]