Question

The empty space left in a hexagonal close packing of spheres in three dimensions is (a) \(64 \%\) (b) \(26 \%\) (c) \(14 \%\) (d) \(52.4 \%\)

Short answer

The empty space left in hexagonal close packing is 26%, which is option (b).

Step-by-step solution

Define the Concept of Packing Efficiency

Hexagonal close packing (HCP) is one way to arrange spheres in a tightly packed manner. The formula for packing efficiency gives the proportion of the volume occupied by the spheres in a unit cell. Packing efficiency is calculated using the formula: \( \text{Packing Efficiency} = \frac{\text{Volume of spheres in the unit cell}}{\text{Total volume of the unit cell}} \times 100\% \).

Calculate Packing Efficiency for HCP

For hexagonal close packing, the packing efficiency is known to be approximately 74%. This means that 74% of the space in a hexagonal close packed structure is occupied by spheres.

Determine Empty Space Percentage

Given that the packing efficiency for HCP is 74%, the empty space will be the remaining space, calculated as: \( 100\% - 74\% = 26\% \). Thus, the empty space in a hexagonal close packing of spheres is 26%.

Select the Correct Answer

Based on the calculation, the percentage of empty space in hexagonal close packing is 26%. Therefore, the correct answer is option (b) \(26\%\).

Key concepts

Hexagonal Close Packing

Hexagonal close packing (HCP) refers to the most efficient arrangement of spheres in a three-dimensional space, specifically in layers. In this arrangement, each sphere is surrounded closely by other spheres. The spheres are aligned in a way that fits them together as tightly as possible, minimizing the space between them. This method is commonly seen in metals and minerals in nature. It involves layering the spheres in a particular pattern where each layer is arranged in a hexagonal lattice.
- Each sphere in one layer sits in the indentations formed by the previous layer, maximizing contact. - The sequence of layers is generally written as ABAB, indicating how each alternate layer replicates itself.
This setup optimizes the use of space, resulting in a notably high packing efficiency.
The key is the arrangement: always align the spheres to occupy the space optimally, thereby reducing voids that lead to empty spaces.

Empty Space Percentage

In the context of packed spheres, the empty space percentage refers to the amount of space not occupied by the spheres within the structure. Hexagonal close packing, known for its high packing efficiency, leaves less space unfilled compared to other forms of sphere packing.
To calculate the empty space percentage, subtract the packing efficiency from 100%. As noted in hexagonal close packing:

• The packing efficiency is approximately 74%.
• This leaves an empty space percentage of 26%.

Understanding this concept helps illustrate how tight the packing is and how much space is still available for other functions.
In industrial applications, or when studying natural materials, knowing this space can help infer other properties like density or potential for impurities.

Three Dimensional Packing

Three dimensional packing involves arranging spheres (or any atoms, particles) in space to occupy minimal total volume.
This concept is crucial in fields ranging from materials science to chemistry as it dictates how densely atoms in a solid are arranged.
When spheres are packed in three dimensions, the goal is to reach the most efficient arrangement, hence minimizing wasted space. Two common three-dimensional packing arrangements are:

• Cubic Close Packing (CCP): Also known as face-centered cubic, it has a similar packing efficiency to HCP at about 74%.
• Hexagonal Close Packing (HCP): Stacking layers in an ABAB sequence which we discussed above.

Both methods aim to balance occupied space and voids, as this balance influences how materials conduct electricity, resist compression, and interact with other materials.
Utilizing these packing methods efficiently can dictate properties in manufactured and natural materials alike.