Question

Consider three boxes of equal volume. One is filled with tennis balls, another with golf balls, and the third with marbles. If a closest-packing arrangement is used in each box, which one has the most occupied space? Which one has the least occupied space? (Disregard the difference in filling space at the walls, bottom, and top of the box.)

Short answer

All boxes have the same occupied space efficiency at 74%.

Step-by-step solution

Understanding Closest Packing

The closest packing of spheres refers to the arrangement where spheres are packed as densely as possible. There are two types of closest packing: face-centered cubic (FCC) and hexagonal close-packed (HCP). Both arrangements have the same packing efficiency of about 74%, meaning that approximately 74% of the volume is occupied by the spheres.

Considering Different Sphere Sizes

The volume of the box remains the same across all cases, but the size of the spheres differs. With closest packing, each box's packing efficiency should be independent of the sphere size since both FCC and HCP have the same efficiency. Thus, the percentage of occupied space in each box, given closest packing, remains constant at 74%.

Conclusion of Occupied Space Comparison

Since the packing efficiency is the same regardless of the sphere's size, the occupied space for tennis balls, golf balls, and marbles is the same at 74%. This means that even though the number of spheres will differ in each box due to different sphere sizes, the percentage of space they occupy in each respective box does not differ.

Key concepts

Face-Centered Cubic

The face-centered cubic (FCC) arrangement is one way to achieve the closest packing of spheres, maximizing the occupancy of space. In the FCC structure, each sphere is surrounded by twelve others, forming a 3D pattern. To visualize this, imagine a cube where each face has a sphere centered on it, plus one in each corner.
This configuration creates layers of spheres in a repeating pattern. The bottom and top layers provide stability, while the middle layers fit snugly into the gaps of the ones below, allowing for the highest possible packing efficiency.

• The structure ensures that spheres contact each other tightly, minimizing gaps.
• Elements like copper, silver, and gold often exhibit FCC structures.
• It is one of the two closest packing methods, offering a packing efficiency of 74%.

Hexagonal Close-Packed

The hexagonal close-packed (HCP) structure offers an alternative to the face-centered cubic arrangement. Here, spheres are stacked in a layered formation, creating a hexagonal lattice that ensures tight packing. The repeating unit in HCP consists of layers identifiable in the sequence 'ABAB.'
In this pattern, every sphere in one layer rests in the dimple of the two layers above and below, achieving optimal density.

• The HCP arrangement results in the same packing efficiency as FCC, which is 74%.
• This arrangement is seen in metals such as magnesium and titanium.
• HCP's structure contributes to material properties like low ductility and high strength, useful in many practical applications.

Packing Efficiency

Packing efficiency is key to understanding closest packing of spheres. It quantifies how fully the spheres fill a given volume. Both FCC and HCP arrangements boast a packing efficiency of 74%, the highest possible for such structures.
Packing efficiency is calculated using the total volume occupied by the spheres divided by the overall volume. A 74% efficiency means that space is maximally utilized without leaving large gaps between spheres.

• Both FCC and HCP structures achieve this maximum due to their regular, repeating patterns.
• Packing efficiency is invariant to sphere size, meaning whether you're packing tennis balls or marbles, the space occupied remains proportional when using closest packing.
• This concept is crucial in material science and chemistry, impacting how materials stack, their density and how space can be used efficiently.