Step 1: Understanding the Concept:
In a face-centered cubic (fcc) unit cell, the particles (atoms) occupy exactly 74% of the total volume of the unit cell. This is known as its packing efficiency. Step 2: Key Formula or Approach:
Packing efficiency of fcc = 0.74.
Number of particles in fcc unit cell (\(Z\)) = 4.
Volume occupied by all particles = \(0.74 \times \text{Volume of unit cell}\).
Volume of one particle = \(\frac{\text{Total volume occupied by particles}}{\text{Number of particles (Z)}}\). Step 3: Detailed Explanation:
Given volume of unit cell (\(V_{\text{cell}}\)) = \(1.6 \times 10^{-23} \text{ cm}^3\).
Total volume occupied by all 4 particles = \(0.74 \times 1.6 \times 10^{-23} \text{ cm}^3 = 1.184 \times 10^{-23} \text{ cm}^3\).
Volume occupied by one particle = \(\frac{1.184 \times 10^{-23}}{4} \text{ cm}^3 = 0.296 \times 10^{-23} \text{ cm}^3 = 2.96 \times 10^{-24} \text{ cm}^3\). Step 4: Final Answer:
The volume occupied by a particle is \(2.96 \times 10^{-24} \text{ cm}^3\).