Question

If a is the length of the side of a cube , the distance between the body centred atom and one corner atom in the cube will be

• A. \(\frac {4}{\sqrt 3}a\)
• B. \(\frac {\sqrt 3}{4}a\)
• C. \(\frac {\sqrt 3}{2}a\)
• D. \(\frac {2}{\sqrt 3}a\)

Answer 1

The Correct Option is C

The question relates to the structure of a cubic lattice in a solid-state chemistry context. In a cubic lattice, specifically in a body-centered cubic (BCC) structure, there is an atom at the center of the cube and one atom located at each corner.

To determine the distance from a corner atom to the body-centered atom in a cube of side length \(a\), we perform the following calculations:

1. Identify the diagonal of the cube that connects two opposite corners. This is also known as the space diagonal of the cube.
2. The formula to find the space diagonal (d) of a cube with side \(a\) is given by: \(d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3}\).
3. The body-centered atom is located at the midpoint of this space diagonal.
4. The distance between one corner atom and the body-centered atom is half of the space diagonal: \( \frac{a\sqrt{3}}{2}\).

Therefore, the distance between the body-centered atom and one corner atom in the cube is \(\frac{\sqrt{3}}{2} a\).

This matches the correct option given:

• Option: \(\frac{\sqrt{3}}{2}a\)

This calculation aligns with the properties of a body-centered cubic lattice structure where inter-atomic distances are vital for understanding material geometry.