Question

Calculate the volume occupied by a particle in simple cubic unit cell if the total volume of unit cell is \(4.0 \times 10^{-23} \mathrm{~cm}^3\).

• A. \(1.5\times 10^{-23}\text{ }\mathrm{cm}^{3}\)
• B. \(2.9\times 10^{-23}\text{ }\mathrm{cm}^{3}\)
• C. \(2.1\times 10^{-23}\text{ }\mathrm{cm}^{3}\)
• D. \(3.7\times 10^{-23}\text{ }\mathrm{cm}^{3}\)

Hint:

Use the packing efficiency of a simple cubic lattice, which is 52.4 percent.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Volume occupied by particles in a unit cell is the packing efficiency multiplied by the unit cell volume.

Step 2: Packing efficiency
For a simple cubic unit cell, there is 1 atom and \(a = 2r\).
Packing efficiency \(= \dfrac{\frac{4}{3}\pi r^3}{(2r)^3} = \dfrac{\pi}{6} = 0.524\).

Step 3: Calculation
Volume occupied \(= 0.524 \times 4.0 \times 10^{-23} = 2.096 \times 10^{-23}\ \text{cm}^3\).

Step 4: Compare the options
This is about \(2.1 \times 10^{-23}\ \text{cm}^3\).

Final Answer:
The particle occupies about 2.1 x 10^-23 cubic centimetres. \[ \boxed{\text{(C) }2.1\times 10^{-23}\text{ cm}^{3}} \]