Problem 143 - A Complete Resource Book in Chemistry for JEE Main

Question

Percentage of free space in cubic close packed structure and in body centered packed structure are respectively [2010] (a) \(30 \%\) and \(26 \%\) (b) \(26 \%\) and \(32 \%\) (c) \(32 \%\) and \(48 \%\) (d) \(48 \%\) and \(26 \%\)

Step-by-Step Solution

Answer

Option (b): 26% for CCP and 32% for BCC

Step 1: Understanding the Problem

We need to find the percentage of free space in two different packing structures: cubic close-packed (ccp) and body-centered cubic (bcc). The options provided give different free space percentages for both structures.

Step 2: Identifying Formula for Packing Efficiency

Packing efficiency is the fraction of volume in a crystal structure occupied by constituent particles. The packing efficiency of ccp is approximately 74% and for bcc is about 68%. The percentage of free space can be calculated as 100% minus the packing efficiency.

Step 3: Calculate Free Space for CCP

For cubic close packed, we start with a packing efficiency of 74%. The free space is calculated as:\[ 100 ext{ ext %} - 74 ext{ ext %} = 26 ext{ ext %} \]

Step 4: Calculate Free Space for BCC

For body centered cubic, we have a packing efficiency of 68%. Thus, the free space is:\[ 100 ext{ ext %} - 68 ext{ ext %} = 32 ext{ ext %} \]

Step 5: Compare with Options

From the calculations, the free space percentages for ccp and bcc are 26% and 32% respectively. Comparing this with the options provided only option (b) matches: 26% for ccp and 32% for bcc.

Key Concepts

Packing Efficiency

Packing efficiency is a key concept when discussing crystal structures. It refers to how tightly the atoms or molecules are packed together in a given volume. Simply put, it's the fraction of the total volume of a crystal that is actually occupied by atoms. Think of it like packing suitcases in a car trunk — the tighter you pack, the more efficient you are with space.

To calculate packing efficiency, you take the volume occupied by the atoms in your structure and divide it by the total volume of the unit cell. This result is usually expressed as a percentage, indicating how much of the space is filled. For example, if the packing efficiency is 74%, it means that 74% of the structure is occupied by atoms, and the remaining 26% is empty space.

• High packing efficiency means atoms are closely packed with minimal gaps.
• Different crystal structures have varying efficiencies due to the arrangement of atoms.

Understanding packing efficiency helps us determine properties like density and stability of materials, which are crucial in material science.

Cubic Close-Packed Structure

The cubic close-packed (ccp) structure is one of the most efficient ways to pack atoms. It's also known as face-centered cubic (fcc) and is prevalent in many metals and minerals.

In ccp structures, each atom is surrounded by 12 other atoms, creating layers that overlap in a way that maximizes space usage. Thanks to this arrangement, ccp structures achieve a packing efficiency of approximately 74%, which means there is only about 26% free space.

• Atoms arrange themselves in layers, where each layer fills the gaps of the previous one.
• This close-packed arrangement contributes to the material's density and strength.

Due to their high packing efficiency, materials with ccp structures, such as copper and aluminum, are dense and exhibit high degrees of ductility and thermal conductivity.

Body-Centered Cubic Structure

The body-centered cubic (bcc) structure features atoms arranged at each corner of a cube, with a single atom in the center of the cube. Despite being less densely packed than the ccp structure, bcc is a strong and noteworthy arrangement.

With a packing efficiency of about 68%, the bcc structure has around 32% free space. This means that while the atoms are efficiently packed, there is more empty space compared to ccp structures.

• Bcc structures accommodate slip systems that help in deformation, giving such materials strength.
• Common metals like iron and chromium exhibit bcc structures at certain temperatures.

Interestingly, this arrangement can contribute to the material's magnetic properties and impact hardness. Despite having more free space, materials with a bcc structure such as tungsten and molybdenum are very strong.