Question

Referring to Figure 11.28 , describe all the phase changes that would occur in each of the following cases: (a) Water vapor originally at 0.005 atm and \(-0.5^{\circ} \mathrm{C}\) is slowly compressed at constant temperature until the final pressure is 20 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 atm and \(-0.5^{\circ}\mathrm{C}\) is slowly compressed at a constant temperature until the pressure reaches 20 atm. There is no phase change in this case, as the water remains in the vapor phase throughout the compression. In case (b), water initially at \(100^{\circ}\mathrm{C}\) and 0.50 atm is cooled at constant pressure until the temperature is \(-10^{\circ}\mathrm{C}\). There is one phase change in this case: liquid water changes into ice as it cools down at constant pressure.

Step-by-step solution

Initial state: Water vapor at 0.005 atm and -0.5°C

The given starting conditions are 0.005 atm pressure and -0.5°C temperature, which is below the freezing point of water at this pressure. Thus, it is in the vapor phase.

Final state: Compression at constant temperature until 20 atm

As the water vapor is slowly compressed, its pressure increases while temperature remains constant at -0.5°C. When the pressure reaches 20 atm, the temperature-which is -0.5°C, is still well below the boiling point of water at this pressure. Thus, the water remains in the vapor phase. In this case, there is no phase change. #Case (b)#

Initial state: Water at 100.0°C and 0.50 atm

The given starting conditions are 100.0°C temperature and 0.50 atm pressure. This corresponds to the boiling point of water at this pressure, which indicates that water is in the liquid phase.

Final state: Cooling at constant pressure until -10°C

As the water is cooled at a constant pressure of 0.50 atm, its temperature decreases. When the temperature reaches 0°C, the water starts freezing and turns into ice. The cooling continues, and the temperature decreases further until it reaches -10°C. The final state will be ice.

Summary of phase changes for case (b)

For this case, there is only one phase change: Liquid water changes into ice as it cools down at constant pressure.

Key concepts

Water Vapor Compression

Understanding the behavior of water vapor during compression is crucial in various applications such as refrigeration and climate control systems. In the given exercise, water vapor originally at 0.005 atm and -0.5°C is compressed at constant temperature until the pressure reaches 20 atm. The process is conducted without allowing the temperature to rise or fall, which is a key characteristic of isothermal processes.

During the compression of water vapor at constant temperature, the gas particles are forced closer together, which increases the pressure. As per the ideal gas law, for a fixed amount of gas at constant temperature, the product of pressure and volume is constant. Consequently, if the volume decreases due to compression, the pressure must increase proportionally. Despite the significant increase in pressure from 0.005 atm to 20 atm, no phase change occurs because the temperature is maintained at a constant value that is below the boiling point of water at the new pressure.

In real-world applications, water vapor compression is utilized in heat pumps and refrigeration cycles where vapor is compressed to raise its temperature and then allowed to release its heat at a higher pressure. This exercise highlights the resilience of the vapor phase under isothermal compression and the importance of pressure-temperature correlation in phase changes.

Constant Temperature Processes

The exercise illustrates a scenario where water vapor is being compressed while maintaining a constant temperature, characteristic of an isothermal process. At constant temperature, the energy added or removed from a substance during a phase change does not change its temperature, but rather effects a change in state or phase.

In the context of phase changes, constant temperature processes are critical to comprehend because they represent conditions where substances boil or condense. For water vapor, maintaining a constant temperature during compression implies that we are moving along an isothermal line on a phase diagram, where the pressure can increase significantly without changing the phase, as long as the substance remains below the boiling point at given pressures.

Furthermore, in practical applications like refrigeration, isothermal compression is vital for efficiency. For instance, when refrigerant vapor is compressed isothermally in a refrigerator's compressor, it ensures that the refrigerant can absorb maximal heat from the interior before being circulated through the coils to dissipate the absorbed heat to the environment.

Cooling of Water at Constant Pressure

The cooling of water at constant pressure, as outlined in case (b) of the exercise, demonstrates a very typical experience: the freezing of water. In this scenario, water at 100.0°C and 0.50 atm is cooled, keeping the pressure unaltered, until it reaches -10°C. This progression through temperatures covers a change in phase from liquid to solid.

Cooling water at a constant pressure allows the heat energy to be released at a stable rate. When the temperature of water descends to its freezing point at the given pressure, the water undergoes a phase change from liquid to solid, forming ice. The temperatures at which these phase changes occur are determined by the specific pressure; at 0.50 atm, water freezes at 0°C.

Importantly, the release of heat during the transition from liquid to solid is known as the latent heat of fusion, where the substance releases a considerable amount of energy without a change in temperature. In everyday life, this process is evident when ice forms in a freezer or when frost develops on surfaces during winter—all at constant atmospheric pressure. This exercise effectively captures the crucial concept of heat transfer during phase changes and the conditions necessary for the transition between liquid and solid states of water.