Consider a glass of water in a room at 27^∘ C and 97 kPa . If the relative humidity in the room

step 1 Problem number 81 The mole fraction of the water vapor in the air is determined from the saturat

Consider a glass of water in a room at 27^∘ C and 97 kPa ....

Consider a glass of water in a room at $27^{\circ} \mathrm{C}$ and $97 \mathrm{kPa} .$ If the relative humidity in the room is 100 percent and the water and the air are in thermal and phase equilibrium, determine $(a)$ the mole fraction of the water vapor in the air and $(b)$ the mole fraction of air in the water.

Key Concepts

Phase Equilibrium

Phase equilibrium refers to the condition where different phases of a substance, such as liquid water and its vapor, coexist in thermodynamic equilibrium. In this state, the chemical potential of water is the same in both the liquid and vapor phases, meaning there is no net evaporation or condensation. This fundamental concept is pivotal for analyzing situations where a liquid is in contact with its vapor, as it sets the conditions for equilibrium concentrations or pressures.

Vapor Pressure and Relative Humidity

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid (or solid) phase at a given temperature. Relative humidity is the ratio of the current vapor pressure to the equilibrium vapor pressure (i.e., the saturated vapor pressure) expressed as a percentage. When relative humidity is 100%, the air is saturated with water vapor, and the vapor pressure equals the equilibrium vapor pressure. This concept is crucial in determining the mole fraction of water vapor in the gas phase under equilibrium conditions.

Mole Fraction in Gas Mixtures

The mole fraction is a way of expressing the concentration of a component in a mixture by taking the ratio of the number of moles of that component to the total number of moles in the mixture. In gas mixtures, the mole fraction of a component can be determined by dividing its partial pressure by the total pressure. This concept allows one to relate the physical pressure measurements to the composition of the gas mixture.

Gas Solubility in Liquids

Gas solubility in liquids, often described by Henry's Law, relates the concentration of a dissolved gas in a liquid to its partial pressure in the gas phase. In the context of air in water, this concept explains how gases such as nitrogen and oxygen dissolve in water. The mole fraction of the dissolved air in water is determined by the equilibrium established between the gas phase and the liquid phase, reflecting the limited solubility of gases in liquids.

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