Question

Find the volume occupied by a particle in a simple cubic unit cell if the radius of a particle in it is 190 pm.

• A. \(6.410\times 10^{-23} \text{cm}^3\)
• B. \(3.625\times 10^{-23} \text{cm}^3\)
• C. \(5.715\times 10^{-23} \text{cm}^3\)
• D. \(2.873\times 10^{-23} \text{cm}^3\)

Hint:

Volume of one sphere is \(\frac43\pi r^3\), with \(r = 190\) pm \(=1.9\times10^{-8}\) cm.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A simple cubic unit cell contains one atom. The volume occupied by the particle is the volume of one sphere of radius \(r\).

Step 2: Key Formula or Approach
\[ V = \frac{4}{3}\pi r^3 \]

Step 3: Detailed Explanation
Convert: \(190\ \text{pm} = 190\times10^{-10}\ \text{cm} = 1.9\times10^{-8}\) cm.
\[ r^3 = (1.9\times10^{-8})^3 = 6.859\times10^{-24}\ \text{cm}^3 \]
\[ V = 4.189 \times 6.859\times10^{-24} = 2.873\times10^{-23}\ \text{cm}^3 \]

Final Answer:
The volume occupied by the particle is \(2.873\times10^{-23}\) cm\(^3\), option (D). \[ \boxed{2.873\times10^{-23}\ \text{cm}^3\ \text{(D)}} \]