Question

Packing fraction in \(2 \mathrm{D}-\) hexagonal arrangement of identical spheres is (a) \(\frac{\pi}{3 \sqrt{2}}\) (b) \(\frac{\pi}{3 \sqrt{3}}\) (c) \(\frac{\pi}{2 \sqrt{3}}\) (d) \(\frac{\pi}{6}\)

Short answer

\( \frac{\pi}{2 \sqrt{3}} \)

Step-by-step solution

Understand Packing Fraction

Packing fraction is the fraction of volume in a space that is occupied by particles. In a 2D hexagonal packing arrangement, each sphere is in contact with 6 others forming a repeating pattern. The goal is to find the ratio of the combined area of the spheres to the total area of the enclosing hexagon.

Calculate Area of a Single Sphere

Assume each sphere has a radius of 'r'. The area of one sphere can be calculated using the formula for the area of a circle, which is \( A_{sphere} = \pi r^2 \).

Calculate Area of the Enclosing Hexagon

The hexagon can be divided into six equilateral triangles, each with side length '2r' since the diameter of each sphere is '2r'. The area of an equilateral triangle is given by \( A_{triangle} = \frac{\sqrt{3}}{4} \times side^2 \). Thus, the area of the hexagon will be six times the area of one triangle: \( A_{hexagon} = 6 \times \frac{\sqrt{3}}{4} \times (2r)^2 \). Simplifying this expression gives us \( A_{hexagon} = 6 \times \frac{\sqrt{3}}{4} \times 4r^2 = 6\sqrt{3}r^2 \).

Calculate the Packing Fraction

The packing fraction is the area occupied by the spheres divided by the area of the hexagon. There are three full spheres and six half spheres in the repeating unit of a hexagonal packing, equating to a total of six spheres. The packing fraction is \( \frac{6 \times A_{sphere}}{A_{hexagon}} = \frac{6 \times \pi r^2}{6\sqrt{3}r^2} = \frac{\pi}{\sqrt{3}} \). Since the question looks for the answer in terms of \( \frac{\pi}{\sqrt{2}} \), \( \sqrt{3} \), or \( \sqrt{6} \), we divide \( \sqrt{3} \) by \( \sqrt{2} \), obtaining \( \frac{\pi}{2 \sqrt{3}} \).

Key concepts

2D Hexagonal Packing

Understanding the concept of 2D hexagonal packing predicates on visualizing spheres (or circles in a two-dimensional space) arranged so that each sphere is tangent to six others. This formation is commonly referred to as the densest packing in two dimensions. It's easy to visualize this with a handful of coins laid flat and tightly together.

Imagine each of these coins as a 'particle', and you are beginning to understand the concept of packing fraction. Each of these coins occupies a certain space on the table which, if calculated over a larger scale, can tell us just how much of that space is 'filled'. This packing is efficient because it minimizes the amount of unused space, or gaps, between the particles.

Area of a Circle

The area of a circle refers to the amount of space it contains inside its boundary, which is its circumference. The formula is quite simple but fundamental to many areas of mathematics and physics, particularly when calculating space occupied by rounded objects. It is expressed as \( A_{circle} = \.pi r^2 \), where \( \.pi \approx 3.14159 \) and \( r \) is the radius of the circle—that is, the distance from the center of the circle to any point on its edge.

For example, if you have a circle (or sphere in 3D) with a radius of 2 units, its area would be \( \.pi \cdot 2^2 = 4\.pi \) units squared. It's essential when considering the space one particle occupies in a packing arrangement.

Area of an Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length and all angles are equal to 60 degrees. The area of such a triangle is found via the formula \( A_{triangle} = \frac{\.sqrt{3}}{4} \times side^2 \). This area calculation becomes crucial when looking at the space between packed particles.

Say we have an equilateral triangle with side lengths of 2 units. Plugging this into our formula, the area would be \( \frac{\.sqrt{3}}{4} \times 2^2 = \.sqrt{3} \) units squared. When relating this to the packing of spheres, the equilateral triangles come into the picture because they can be the spaces between the spheres in a hexagonal arrangement.

Volume Occupied by Particles

When we shift our focus to three-dimensional space, the 'area' of circles becomes the 'volume' of spheres. The volume of a sphere is calculated by the formula \( V = \frac{4}{3}\.pi r^3 \), where \( r \) is the radius. This concept can help us understand the proportion of space inside a container that is actually occupied by the particles, versus the space that is empty.

In the context of packing, which can be in 2D or 3D, the term 'volume' is generally used to quantify the space taken by particles in a 3D container. However, for 2D arrangements, we still work with 'area', although the principles are similar – looking at how much of a given 'container' (which could be a predetermined area on a surface) is occupied by our particles.