Problem 80 - Chemistry and Chemical Reactivity

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.)

Step-by-Step Solution

Answer

The water level will be the same in both pools.

Step 1: Understanding the Problem

We have two swimming pools, both filled with ice spheres packed as closely as possible. The spheres in the first pool are small (like grains of sand), whereas those in the second pool are larger (like oranges). We need to determine in which pool the water level will be higher after all the ice melts.

Step 2: Analyzing Packing Efficiency

Both arrangements use spheres, so each pool's packing efficiency is limited by the nature of the close packing, typically meaning the spheres occupy about 74% of the pool's volume, regardless of their size. This is due to the nature of closest sphere packing, such as hexagonal close packing or face-centered cubic arrangements.

Step 3: Volume and Density Considerations

The total volume of ice in both pools will be proportional to 74% of the pool's volume. When the ice melts, it will occupy the same volume of water as its original ice volume, since the density ratio to water leads to increased space between molecules in frozen state but compensates to original volume when melted.

Step 4: Conclusion on Water Levels

Since both pools have the same packing efficiency and the ice melts to occupy the same volume as it originally took up in ice form, the water level in both pools will be the same. Sphere size does not affect the final water level after complete melting.

Key Concepts

Sphere Packing

Sphere packing is the arrangement of equal-sized spheres in a way that fills space most efficiently. This concept is important in various fields, including crystallography and materials science. In our exercise, the swimming pools are filled with spheres of ice packed as closely together as possible, also known as close packing.

• The most efficient packing method is called the "face-centered cubic" or "hexagonal close packing," which means the spheres occupy approximately 74% of the available volume.
• This applies regardless of the size of the spheres, whether they are as small as grains of sand or as large as oranges.

Understanding sphere packing helps us conclude that the volume occupied by spheres in each pool is the same percentage of the total pool volume, regardless of sphere size.

Ice Melting

Melting ice is a phase change from solid to liquid. During this process, ice transforms into water without changing the overall mass.

• When ice melts, it occupies the same volume as liquid water due to density differences between ice and water.
• The molecular structure of ice is more spread out compared to liquid water, which is why it is less dense.

For our pools, as the ice spheres melt, they convert directly into an equal amount of water. This is because any volume differences due to the structure of ice normalize once it turns to liquid water.

Volume

Volume refers to the amount of space an object or substance occupies. In our exercise, both pools are initially filled with ice spheres constituting about 74% of the pool's volume, thanks to efficient packing.

• Despite different sphere sizes, both pools will have the same volume of ice due to identical packing.
• After melting, this ice turns into water, maintaining the same volume because water and ice have a consistent mass-to-volume relationship in these conditions.

Therefore, understanding volume in this context shows why the water level remains unchanged between pools despite the size of ice spheres.

Water Level

The water level in a container is determined by the volume of liquid it contains. In our scenario, both pools start with the same efficient packing of ice spheres.

• Since both pools have spheres occupying 74% of the pool's volume and melting ice becomes water occupying the same volume, the water level will not differ between pools.
• Spoon size, whether as small as sand grains or as large as oranges, does not impact the water level after melting.

Thus, although one might assume larger spheres would displace more water, the packing efficiency ensures both pools reach the same water level.