Question

You have two 1-liter containers connected to each other by a valve which is closed. In one container, you have liquid water in equilibrium with water vapor at \(25^{\circ} \mathrm{C}\). The other container contains a vacuum. Suddenly, you open the valve. Discuss the changes that take place, assuming temperature is constant with regard to (a) the vapor pressure, (b) the concentration of the water molecules in the vapor, (c) the number of molecules in the vapor state.

Short answer

In summary, when the valve between the two 1-liter containers is opened, the vapor pressure will decrease and reach a lower equilibrium value in the 2-liter system. The concentration of water molecules in the vapor will be reduced by half due to the expansion into the vacuum container. Meanwhile, the number of molecules in the vapor state will increase as more liquid water evaporates to reach the new equilibrium in the larger system.

Step-by-step solution

Understand vapors and vacuum conditions

At 25°C, the liquid water is in equilibrium with its vapor in the first container, meaning that the number of water molecules evaporating is equal to the number of water molecules condensing. When the valve is opened, the water vapor (a gas) has a chance to move freely between both containers, including the container that initially has a vacuum, which will cause the equilibrium to shift.

Analyze the change in vapor pressure

Initially, the vapor pressure of water in the first container is equal to its saturated vapor pressure at 25°C, which is about 3.2 kPa. When the valve is opened, the water vapor will spread into the vacuum container and will no longer be in equilibrium with the liquid water. Therefore, as the water vapor expands into the second container, the vapor pressure will decrease, and it will eventually come to a new equilibrium between both containers at a lower vapor pressure value.

Analyze the change in concentration of water molecules in the vapor

Initially, the concentration of water molecules in the vapor in the first container is determined by the saturated vapor pressure at 25°C. After the valve is opened, the water vapor expands and will cover twice the volume it initially occupied (2 L in total). Since the temperature remains constant, the concentration of water molecules in the vapor will be reduced by half.

Analyze the change in the number of molecules in the vapor state

Initially, the number of water molecules in the vapor state is defined by the equilibrium in the first container. When the valve is opened, more water molecules will evaporate from the liquid to reach a new equilibrium in the 2-liter system. This means that the total number of water molecules in the vapor state will increase as more liquid water evaporates, and the number of water molecules returning to the liquid state will also increase to achieve the new equilibrium. In conclusion, when the valve is opened: (a) The vapor pressure will decrease, reaching a lower equilibrium value in the 2-liter system. (b) The concentration of water molecules in the vapor will be reduced by half. (c) The number of molecules in the vapor state will increase as more liquid water evaporates to reach the new equilibrium.

Key concepts

Vapor Pressure

Vapor pressure is an important concept in understanding the behavior of liquids and gases. It refers to the pressure exerted by a vapor in thermodynamic equilibrium with its liquid phase at a particular temperature.
When a liquid is placed in a closed container, molecules from the liquid evaporate into the gas phase, creating vapor pressure.

• At 25°C, water's vapor pressure is about 3.2 kPa.
• This pressure represents the tendency of water molecules to leave the liquid for the gaseous phase.
• The vapor pressure is determined by the temperature and intrinsic properties of the liquid, not by the volume of the container.

When the valve between the two containers is opened, the water vapor from the first container moves into the second, initially empty, container. This expansion lowers the vapor pressure since it is now distributed over a larger space. Eventually, the system reaches a new equilibrium with a lower vapor pressure spread across both containers.

Equilibrium

Equilibrium in a thermodynamic system refers to a state where opposing processes occur at the same rate. For a liquid and its vapor, this means the rate of evaporation equals the rate of condensation. In our scenario, before opening the valve, the liquid water is in equilibrium with its vapor at 25°C. - No net change in the number of water molecules in the vapor phase occurs during equilibrium.
- When the valve is opened, this equilibrium is disrupted as the vapor spreads into the second container. Over time, the system reaches a new equilibrium where the water vapor is distributed equally between both containers.
A balance is again achieved, but at a lower vapor pressure.
In reaching the new equilibrium, more water evaporates from the liquid to replenish the vapor, maintaining the balance over an increased volume.

Molecular Concentration

Molecular concentration is a measure of the number of molecules present in a given volume. Initially, in the first container, the concentration of water molecules in the vapor phase corresponds to the saturated vapor pressure at 25°C. When the valve is opened:

• The total volume available to the vapor doubles.
• Given constant temperature, the concentration of water vapor molecules halves as they now fill the space of both containers.

The concentration is an important aspect because it affects how molecules collide and react, affecting overall system behavior.
As the vapor spreads out into both containers, it's crucial to note that the actual number of molecules hasn't changed initially—it's the volume they occupy that has increased, leading to the decreased concentration.

Phase Change

A phase change involves a substance transforming from one physical state to another, like liquid to gas. In our scenario, the focus is on the liquid-gas equilibrium of water. - At equilibrium in the first container, water molecules continuously evaporate and condense at the same rate. - Opening the valve causes more water molecules to evaporate into the now larger gas volume in both containers.
- New equilibrium necessitates more molecules in the vapor phase to balance the pressure, resulting in increased evaporation from the liquid water source. This phase change—from liquid to vapor—results due to the necessity for the system to establish a new equilibrium across the larger container volume.
Thus, even if the temperature remains constant, the physical transformation of molecules between phases continues until equilibrium is re-established.