Problem 49 - Chemistry and Chemical Reactivity

Question

Imagine the following experiment: You have a large graduated cylinder containing 100. mL of liquid water at \(0^{\circ} \mathrm{C}\). You drop an ice cube with a volume of \(25 \mathrm{cm}^{3}\) into the cylinder. Ice has a density of \(0.92 \mathrm{g} / \mathrm{cm}^{3},\) less than the density of liquid water, so it floats with \(92 \%\) being under water. (a) To what level will the water in the graduated cylinder rise after adding the ice? (b) Allow the ice to melt. What volume will now be occupied by the liquid water? (One consequence of global warming will be a rise in sea level as ice in the northern and southern regions of the planet melts. However, the effect relates only to melting of ice on land. Melting of floating ice will have no effect on sea levels.

Step-by-Step Solution

Answer

(a) 123 mL; (b) 123 mL.

Step 1: Calculate Volume of Submerged Ice

We know the ice floats, with 92% submerged. Thus, the submerged volume is given by:\[ V_{submerged} = 0.92 \times 25 \ \mathrm{cm}^3 = 23 \ \mathrm{cm}^3. \]

Step 2: Determine Initial Water Level

Initially, the water is at \(100\ \mathrm{mL}\). The submerged ice displaces water equivalent to its volume, causing the water level to rise by 23 mL, so it becomes:\[ 100\ \mathrm{mL} + 23\ \mathrm{mL} = 123\ \mathrm{mL} .\]

Step 3: Ice Melting and Final Volume

When the ice melts, it turns completely into liquid water. The total mass of the ice will turn into water occupying the same volume, since density and mass (in terms of floating ice) counteract volume displacement:\[ V_{water} = (100\ \mathrm{mL} + 0\ \mathrm{mL}) = 123\ \mathrm{mL}. \] The noticeable change: none—water level remains 123 mL, since displacing effects essentially balance addition from ice-melt conversion.

Key Concepts

Ice melting experiment

The ice melting experiment is a fascinating way to explore the concepts of density and buoyancy. In this scenario, an ice cube is dropped into a container of water. Initially, you observe the ice cube floating, with a portion submerged and a portion sitting above the water surface. This setup provides a simple but powerful way to study how objects interact with liquids based on their density.

When the ice cube melts, all of its water content gets added to the water in the container. Since the ice was floating, melting does not change the overall water level. This is because the water displaced by the ice while floating is equal to the volume of water it turns into after melting.

Through such experiments, students can vividly see the implications of principles that govern fluid dynamics and density, providing visual and tangible understanding of these essential scientific concepts.

Volume displacement

Volume displacement is an integral part of understanding buoyancy and how objects behave in fluids. Displacement occurs when an object is placed in a fluid, like water, and causes the fluid level to rise in accordance with the volume of the submerged part of the object. This is precisely what happens with an ice cube in our experiment.

Here, the ice cube displaces water based on its submerged volume. Given that 92% of the ice is submerged, this specific proportion directly pushes on the water, causing a rise proportional to the submerged volume. This rise in water level is calculable by determining the submerged volume of the ice, which exemplifies how volume displacement allows us to understand the space taken up within a container by a submerged object.

Thus, volume displacement allows us to observe a visible change in water levels, providing a straightforward way to apply mathematical principles to physical observations in buoyancy.

Effect of density on buoyancy

Density plays a crucial role in determining whether an object will float or sink when placed in a fluid. In our experiment, the ice cube remains afloat because its density is lower than that of liquid water. Specifically, ice has a density of 0.92 g/cm³, which is less than water's density of about 1.00 g/cm³.

This difference in density means the ice cube is less dense and, therefore, buoyant enough to float. Because the ice's density is 92% that of water, it submerges just enough to displace an equivalent weight of water, a perfect demonstration of Archimedes' principle of buoyancy. In essence, the principle states that a floating object displaces a volume of fluid equal to its weight.

Understanding how differences in density affect buoyancy helps explain why certain materials can float while others sink, as well as the implications for natural phenomena, like sea level rise due to ice melting.