Question

Refer to Figure \(11.27(\mathrm{a})\), and describe all the phase changes that would occur in each of the following cases: (a) Water vapor originally at \(0.005 \mathrm{~atm}\) and \(-0.5^{\circ} \mathrm{C}\) is slowly compressed at constant temperature until the final pressure is \(20 \mathrm{~atm} .\) (b) Water originally at \(100.0^{\circ} \mathrm{C}\) and \(0.50 \mathrm{~atm}\) is cooled at constant pressure until the temperature is \(-10^{\circ} \mathrm{C}\).

Short answer

In case (a), water vapor initially at \(0.005 \mathrm{~atm}\) and \(-0.5^{\circ}\mathrm{C}\) is compressed at constant temperature, crossing the vapor-liquid boundary and condensing into liquid water. Upon reaching \(20 \mathrm{~atm}\), the liquid water crosses the liquid-solid boundary, resulting in solid ice. In case (b), water initially at \(100.0^{\circ}\mathrm{C}\) and \(0.50 \mathrm{~atm}\) is cooled, crossing the liquid-vapor boundary and evaporating into water vapor before crossing the vapor-solid boundary and forming solid ice at \(-10^{\circ}\mathrm{C}\).

Step-by-step solution

Understand and Identify the Initial Conditions

In case (a), we have water vapor originally at \(0.005 \mathrm{~atm}\) and \(-0.5^{\circ}\mathrm{C}\). This corresponds to a point on the Figure \(11.27(a)\) from which we can start tracing the phase changes. In case (b), we have water originally at \(100.0^{\circ}\mathrm{C}\) and \(0.50 \mathrm{~atm}\). This also corresponds to a specific point on the diagram. Mark these points as starting points for both cases.

Trace the Path of the Phase Changes for Case (a)

In case (a), we are slowly compressing the water vapor at a constant temperature of \(-0.5^{\circ}\mathrm{C}\) until the final pressure is \(20\mathrm{~atm}\). This means we'll be moving horizontally towards the right in the phase diagram. Keep tracing the path on the phase diagram by maintaining the temperature constant at \(-0.5^{\circ}\mathrm{C}\) and increasing the pressure from \(0.005 \mathrm{~atm}\) to \(20 \mathrm{~atm}\). Take note of any phase boundaries crossed during this process.

Describe the Phase Changes for Case (a)

As we trace the path horizontally along the constant temperature line \(-0.5^{\circ}\mathrm{C}\) in the phase diagram, we'll observe the water vapor crossing the vapor-liquid phase boundary. Once we reach this point, the vapor will start to condense into liquid water. As we continue to increase the pressure, the water will eventually cross the liquid-solid phase boundary, and it starts turning into ice. The final state at \(20\mathrm{~atm}\) should be solid ice.

Trace the Path of the Phase Changes for Case (b)

In case (b), we are cooling the water at a constant pressure of \(0.50 \mathrm{~atm}\) until the temperature is \(-10^{\circ}\mathrm{C}\). Here, we'll be moving vertically downwards on the phase diagram. Starting from the point corresponding to \(0.50 \mathrm{~atm}\) and \(100.0^{\circ}\mathrm{C}\), decrease the temperature along the constant pressure line until we reach \(-10^{\circ}\mathrm{C}\). Again, take note of any phase boundaries crossed during this process.

Describe the Phase Changes for Case (b)

As we trace the path vertically downwards along the constant pressure line of \(0.50\mathrm{~atm}\) in the phase diagram, we'll observe the liquid water crossing the liquid-vapor phase boundary. Once we reach this point, the liquid water will start to evaporate into water vapor. However, as we continue to decrease the temperature, the water vapor will cross the vapor-solid phase boundary and will start turning into ice. The final state at \(-10^{\circ}\mathrm{C}\) should be solid ice.

Key concepts

Water Vapor

Water vapor is the gaseous phase of water and is a critical component in various atmospheric processes, such as weather patterns and humidity levels. It is crucial to understand water vapor within the context of phase changes. These phase changes occur when water transforms from one state to another.

• Initial State: Water vapor often forms when water evaporates, meaning it absorbs heat and transitions from a liquid to a gas.
• Water vapor is present in the atmosphere, contributing to cloud formation and precipitation.
• Understanding water vapor's role is essential for comprehending larger hydrological cycles and weather predictions.

When dealing with problems involving water vapor, it helps to refer to a phase diagram to determine under what conditions water vapor might condense or lead to further phase transitions. In the exercise, a change from water vapor at a low pressure eventually leads to condensation as the pressure increases while the temperature remains constant.

Phase Diagram

A phase diagram is a valuable tool representing the states of matter (solid, liquid, gas) of a substance in relation to pressure and temperature. For water, it helps us understand where transitions between these states occur.

• Axes of a phase diagram: Temperature is typically plotted on the x-axis while pressure is on the y-axis.
• Boundaries between phases show the conditions under which different states exist.
  • The line separating liquid and vapor indicates the boiling point at different pressures.
  • Similarly, the line between solid and liquid marks the melting point under various conditions.
• The critical point and triple point are key features.
  • Critical point: Beyond this point, the liquid and gas phases can't be distinguished.
  • Triple point: The condition where all three phases coexist in equilibrium.

In the exercise, analyzing the phase diagram for water at given temperatures and pressures was pivotal in tracing the paths of phase changes, such as observing how an increase in pressure causes water vapor to condense and eventually freeze.

Condensation

Condensation is an essential phase change where water vapor transforms into liquid water. It occurs when the vapor cools or is compressed, reaching a point where it can't hold as much moisture effectively in the gaseous state.

• Triggers of condensation: This process typically happens when there's a decrease in temperature or an increase in pressure.
  • In atmospheric conditions, when warm, moist air rises, it cools, leading to cloud formation due to condensation.
• During the exercise, increasing pressure at a constant low temperature induced condensation in case (a). Water vapor at \(0.005 \text{ atm}\) and \(-0.5^{\circ} \mathrm{C}\) condensed as pressure increased towards \(20 \text{ atm}\) on the phase diagram.

This phase change is not only crucial in everyday life by influencing weather—as seen in clouds but also in industrial processes where moisture control is critical. It's the basis for phenomena like dew formation and essential in systems such as air conditioning.

Sublimation

Sublimation is a fascinating process where a substance transitions directly from a solid to a gas without passing through the liquid phase. This occurs under specific temperature and pressure conditions where the solid's vapor pressure is higher than that of the liquid phase.

• Common examples of sublimation:
  • Dry ice (solid carbon dioxide) sublimating at room temperature, bypassing the liquid state entirely.
  • In nature, snow or ice can sublimate in cold, dry conditions without melting, a crucial process in the water cycle in polar regions.
• In the context of the exercise, even though sublimation specifically was not encountered within the described pathways on the phase diagram, understanding sublimation's existence enriches comprehension of phase transitions overall.
• Sublimation is utilized in freeze-drying processes, preserving food and pharmaceuticals by removing moisture while maintaining structure and quality.

Grasping the concept of sublimation enhances your broader knowledge of thermodynamics and phase changes, illustrating nature's diverse mechanisms in transitioning between material states.