Step 1: Understanding the Concept:
The packing efficiency of a crystal lattice is the fraction of the total volume of the unit cell that is actually occupied by the constituent particles (atoms, ions, or molecules).
For a Face-Centered Cubic (FCC) lattice, the atoms are packed very efficiently.
Step 2: Key Formula or Approach:
The formula for the volume occupied by particles is \(\text{Occupied Volume} = \text{Packing Fraction} \times \text{Total Volume}\).
For an fcc unit cell, the packing fraction is \(0.74\).
This means that \(74%\) of the total volume of the unit cell is occupied by the spheres, and the remaining \(26%\) is empty space.
Step 3: Detailed Explanation:
We are given the total volume of the unit cell:
\(\text{Volume}_{\text{unit cell}} = 1.6 \times 10^{-23} \text{ cm}^3\).
Using the packing fraction for FCC (\(0.74\)), we calculate the occupied volume:
\[ \text{Volume occupied} = 0.74 \times (1.6 \times 10^{-23} \text{ cm}^3) \]
\[ \text{Volume occupied} = 1.184 \times 10^{-23} \text{ cm}^3 \]
Step 4: Final Answer:
The volume occupied by all particles in the given FCC unit cell is \(1.184 \times 10^{-23} \text{ cm}^3\).