Question

Two identical swimming pools are filled with uniform spheres of ice packed as closely as possible. The spheres in the first pool are the size of grains of sand; those in the second pool are the size of oranges. The ice in both pools melts. In which pool, if either, will the water level be higher? (Ignore any differences in filling space at the planes next to the walls and bottom.)

Short answer

The water level will be the same in both pools.

Step-by-step solution

Understand the Problem Context

We have two identical swimming pools filled with packed spheres of ice: one filled with smaller spheres (like grains of sand) and the other with larger spheres (like oranges). The spheres will melt, and we need to determine in which pool the water level will be higher after melting.

Consider Sphere Packing Density

When spheres are packed, there is a packing density that determines how much of the space is actually occupied by the spheres. The most efficient packing for spheres is around 74.04% of the volume being filled with sphere material, regardless of the size of the spheres.

Reason About the Melting Process

Since the spheres are made of ice, they will melt into water and take up the same volume as their solid form. However, ice has a slightly lower density than water, meaning the melted water will occupy slightly less space than the ice did.

Apply Uniform Density for Spheres

Since the packing density is uniform for both sizes of spheres, and because melting will not change this relative packing density, the water produced from the melted ice will have the same attitude of filling only about 74% of the pool’s original filled volume.

Analyze the Pool Context

Given that both pools are identical and filled with the same packing density, the volume of water melted from the total number of spheres will be the same, regardless of sphere size. Thus, the final water level is expected to be the same in both pools.

Key concepts

Density of Ice

The density of ice refers to how tightly packed the molecules are in the solid form of water. Ice is less dense than water, which is why it floats on lakes and oceans. This density difference occurs because the structure of ice is formed by a crystalline matrix, where water molecules are arranged in a pattern that takes more space than when molecules are in liquid form. As a result, the density of ice is approximately 0.92 grams per cubic centimeter, compared to water's density of about 1 gram per cubic centimeter. This concept plays a crucial role in understanding how ice melts and transforms into water, affecting its volume during the melting process.

Melting Process

The melting process involves the conversion of ice to water at 0 degrees Celsius under normal atmospheric pressure. As ice absorbs heat, its molecules begin to move more vigorously, breaking apart the crystalline structure of the solid form. This process does not change the mass of the ice but changes its state from a solid to liquid. During melting, the volume of water produced is slightly less than the ice it came from because of the density difference between ice and water. One intriguing aspect of this process in sphere packing is that the melting does not affect the packing density — the spheres' configuration remains consistent, as does the percentage of space they occupy.

Volume Displacement

Volume displacement in the context of melting ice spheres refers to how much space the resultant water takes up in the swimming pool once the ice melts. Because ice is less dense than water, the volume of the water generated after melting will occupy slightly less space. For example, when ice melts, every unit of volume slightly shrinks, according to the difference in densities. As it pertains to sphere packing, the initially occupied volume by the ice spheres remains affected only by the proportion of the pool covered by the packed spheres, rather than their size, leading to the same volume of water being displaced regardless of sphere size.

Ice Spheres Melting

Ice spheres melting in this context analyzes the effect of melting on spheres of varying sizes. Regardless of whether the spheres are the size of grains of sand or oranges, the percentage of the pool occupied by these packed spheres remains consistent due to uniform packing density. As these spheres melt, the resultant water will uniformly fill the same percentage of the pool's volume that the spheres initially occupied due to their consistent packing density of about 74%. This is why the final water level in two identical pools, filled with spheres of different sizes, remains the same. The water volume from the melting spheres will correlate directly with the initial volume the spheres occupied, ensuring that sphere size does not lead to a difference in final water level.