Animated Solution for Chemistry - States of Matter: Percentage of free space in cubic close packed structure and in body centred packed structure are respectively

Have you ever tried to pack oranges into a square box? No matter how carefully you arrange them, there are always little gaps—empty pockets of air—between the fruits. This simple everyday observation is the exact foundation of one of the most important concepts in solid-state chemistry: Packing Efficiency.

When we look at a piece of metal, it appears perfectly solid. However, at the atomic level, it is composed of spherical atoms packed together in a repeating three-dimensional lattice. Because spheres cannot perfectly tile a 3D space without leaving gaps, every crystal structure inherently contains some empty volume, known as free space or voids.

The Cubic Close Packed (CCP) structure, which is geometrically identical to the Face-Centered Cubic (FCC) lattice, represents one of the most efficient ways nature can pack spheres. In this arrangement, atoms are located at all eight corners of the cubic unit cell and at the center of all six faces.

Because the atoms touch each other along the face diagonal, we can establish a mathematical relationship between the edge length of the cube ($a$) and the radius of the atom ($r$):

By calculating the volume of the 4 effective atoms in the unit cell and dividing it by the total volume of the cube ($a^3$), we arrive at the packing fraction:

This means that 74% of the total volume is occupied by solid matter. To find the free space, we simply subtract this from the total volume:

Now, let's examine the Body-Centered Cubic (BCC) structure. In this lattice, atoms are located at the eight corners of the cube, with one additional atom sitting perfectly in the center of the body.

Here, the atoms touch along the body diagonal of the cube, leading to a different geometric relationship:

With 2 effective atoms per unit cell, the packing fraction evaluates to:

This tells us that the BCC structure is slightly less dense, occupying only 68% of the available volume. Consequently, the free space is larger:

By comparing our results, we can clearly see the difference in packing densities. The CCP structure leaves 26% of its volume as free space, while the BCC structure leaves 32%.

Matching these derived values with the given options, we find that the correct sequence is 26% and 32%. Memorizing these standard packing efficiencies (74% for FCC/CCP, 68% for BCC, and 52.4% for Simple Cubic) is a massive time-saver for competitive exams, allowing you to bypass lengthy derivations and jump straight to the correct answer.