Question

A 250 -mL glass bottle was filled with \(242 \mathrm{~mL}\) of water at \(20^{\circ} \mathrm{C}\) and tightly capped. It was then left outdoors overnight, where the average temperature was \(-5^{\circ} \mathrm{C}\). Predict what would happen. The density of water at \(20^{\circ} \mathrm{C}\) is \(0.998 \mathrm{~g} / \mathrm{cm}^{3}\) and that of ice at \(-5^{\circ} \mathrm{C}\) is \(0.916 \mathrm{~g} / \mathrm{cm}^{3} .\)

Short answer

The glass bottle will likely crack or shatter.

Step-by-step solution

Determine Initial Mass of Water

First, calculate the mass of the water at \(20^{\circ} \, \mathrm{C}\) using the given volume of water and the density. The volume of water is \(242 \, \mathrm{mL}\) and the density is \(0.998 \, \mathrm{g/cm^3}\). The mass \(m\) is calculated as: \[m = ext{volume} \times ext{density} = 242 \, \mathrm{mL} \times 0.998 \, \mathrm{g/cm^3} = 241.516 \, \mathrm{g}.\]

Calculate Volume of Water When Frozen

When water freezes, it turns into ice, which is less dense. Using the density of ice at \(-5^{\circ} \, \mathrm{C}\), which is \(0.916 \, \mathrm{g/cm^3}\), calculate the new volume \(V\) of the ice: \[V = \frac{m}{\text{density of ice}} = \frac{241.516 \, \mathrm{g}}{0.916 \, \mathrm{g/cm^3}} \approx 263.7 \, \mathrm{cm^3}.\]

Compare Ice Volume With Bottle Volume

The bottle's capacity is \(250 \, \mathrm{mL}\). Compare this with the calculated ice volume. The ice volume \((263.7 \, \mathrm{cm^3})\) is greater than the bottle's volume \((250 \, \mathrm{cm^3})\), indicating the bottle cannot contain the expanded ice.

Predict the Consequence

Since the volume of the ice exceeds the bottle's capacity, the increase in volume will exert pressure on the bottle. As a result, the glass bottle is likely to crack or shatter due to the expansion of ice beyond the available space.

Key concepts

Density Calculations

Density is a crucial concept in thermodynamics, particularly when assessing the behavior of materials under temperature changes. It is essentially the mass per unit volume, typically expressed in grams per cubic centimeter \( \text{g/cm}^3 \). To determine the density, you need to know both the mass and volume of the substance involved.

For the problem at hand, we calculate the mass of water at \(20^{\circ} \mathrm{C}\) using its density and volume. Given that the density of water is \(0.998 \, \text{g/cm}^3\) and the volume is \(242 \, \text{mL}\), we multiply these values to find the mass:
\[ m = 242 \, \text{mL} \times 0.998 \, \text{g/cm}^3 = 241.516 \, \text{g}.\]

This mass calculation is foundational because it remains constant despite temperature changes, only the volume changes as water changes state.

Volume Expansion

Volume expansion occurs when a substance increases in size due to a change in conditions, like temperature. This concept is especially important when dealing with materials transitioning from one state to another, such as water freezing into ice. When water freezes, it expands because ice is less dense than liquid water.

Using the calculated mass of water, the volume of water at an initial temperature can be compared with its volume when turned into ice. At \(-5^{\circ} \mathrm{C}\), the density of ice is \(0.916 \, \text{g/cm}^3\). By having the previously calculated mass of water \(241.516 \, \text{g}\), we find the volume of ice:
\[ V = \frac{241.516 \, \text{g}}{0.916 \, \text{g/cm}^3} \approx 263.7 \, \text{cm}^3. \]

This equation demonstrates that the volume of water increases when it becomes ice, as \(263.7 \, \text{cm}^3\) of ice tries to fit into a \(250 \, \text{cm}^3\) space. The result is an overflow or an applied pressure on the containing vessel.

State Change of Water

Water is an interesting substance in thermodynamics because it behaves uniquely during state changes. When water cools to \(0^{\circ} \mathrm{C}\) and further to \(-5^{\circ} \mathrm{C}\), it undergoes a phase transition from liquid to solid, becoming ice.

This transition is significant because unlike most substances, water expands as it freezes. Ice is less dense than liquid water, which is why ice floats. This property causes significant changes in volume that can exert pressure on containers if the transition occurs in a constrained environment.

• At \(20^{\circ} \mathrm{C}\), water molecules are packed closely, yet freely, allowing the volume to be \(242 \, \text{mL}\).
• Upon freezing, the crystallized structure of ice causes the water molecules to occupy more space, despite having the same mass.

This understanding of state change allows us to predict physical changes, such as a glass bottle cracking due to the expansion from water turning into ice.