Water Volume: Newsletter Challenge (11/08/05)

The ice cubes will float to the top of the beaker. Although the ice is less dense than the water, since it's floating on top, the water line will remain the same after melting. Now once you put marbles in the ice, the ice sinks to the bottom. The ice now displaces more water than it will after it melts, therefore I think the water level will go down after melting.

I think the water level will drop. Since water expands when it freezes, you have some expanded water in the container, when it melts it will "shrink" back to "normal" size, and lowering the waterline. As far as the marbles inside of the cubes, they will not melt, unless they are ice marbles, so the volume of water they displace remains constant, and will not affect the water line after the ice melts, so the water line will still go down, just not as much as without the marbles.

For the first half of the question the water level will remain the same before and after the ice has melted (in ideal theory). The density of ice is about 9/10 of liquid water (remember that water expands when frozen, which is one of the unique properties of water) and will float on water. The volume of ice that is submerged represents the total volume of the frozen ice after it has melted. QED.

The problem with theory is that it does not always take all factors into effect. If the column of ice extends to and touches the bottom of the cylinder, then there will be more frozen volume of ice than would normally be suspended in liquid water above the water line. In this case the water level would rise. I will assume that this is not the case, but I wanted to cover my bases.

If a marble is added to each cube, the average density of each cube will change. Marble has a density just slightly 2.5 times that of water. If the marbles are actually made of glass, as are most marbles, the density for glass is also about 2.5 times that of water. The coefficient of expansion for glass and marble is negligible for the difference of temperature in this experiment, so we can leave that out of the equation since the question requires pretty much a digital answer. So the volume the marble represents is about the same for the initial state of the experiment and the final state.

What the marble will do is increase the mass of the ice. The ice will submerge deeper into the water and displace more volume in the cylinder in its frozen state than in its liquid state. When the cylinder reaches thermal equilibrium (the ice melts for those regular guys) the water level in the cylinder that contains marbles will drop compared to its initial state.

The reason the water level drops is that the ice cubes are artificially (due to the higher density glass marble) submerged deeper into the water and displace more liquid water than normal. When the ice melts it contracts and the volume that it displaced in the frozen state is smaller than the liquid state. The water level in the cylinder will drop. QED.

The second to the last sentence is in error (writing too fast). It should read:

"When the ice melts it contracts and the volume that it displaced in the frozen state is larger than in the liquid state."

I can now sleep tonight.

The water level will become lower. When the water is froozen to become ice cubes, there is always air bubble trap inside the ice cube and causes the water to expand. Thus the mass or the volume of the ice cube increases. By putting in a marble in to the ice cube, which is a solid mass as to help to hold the ice cubes at the bottom of the cylinder, the total mass or the volume of the ice cubes will be, the water + the air bubble + marble. When we put the ice cube at the bottom of the volumetric cylinder and top up to the level and after evaporation with all the ice melt down, the air bubble will escape from the ice cubes leaving the empty space in the water which is the (water + the marble) - (the air bubble) = less mass or volume inside the water, therefore, the water will take up the empty space and thus the water level become lower.

If the cubes are floating, the liquid level does not change. This is because ice cubes, SG less than 1 (water) float partly above water level. Mass of water displaced by that part of the cube below water = total mass of cube (Archimedes). So when the cube melts the water formed (same SG as the rest of the water) fits exactly into the space left below water level. Note this is true irrespective of actual amount of expansion on freezing and hence SG of ice. If the cubes are resting on the bottom, the water level rises on melting, for obvious reasons if we consider cubes "just" floating, and then add some more. If the cubes contain marbles the level falls on melting, whether the ratio of ice to marble causes the cube to float or sink. If it sinks, it's clear that water formed occupies less volume than the ice, while the marble doesn't change. If the cubes float, it's a similar situation to a swimming pool with a boat carrying a metal weight. If the weight is thrown overboard, the pool level falls. Because when in the boat, the weight displaces its own weight of water, with volume greater than that of the weight, but when in the pool it only displaces its own volume.