Using properties of saturated water, explain how you would determine the mole fraction of water vapor at the surface of a lake when the temperature of the lake surface and the atmospheric pressure are specified.

When looking at the surface of a lake, it's important to understand how the water interacts with the air to create water vapor. This process is closely related to the properties of saturated water, which is water that is in equilibrium with its vapor at a given temperature and pressure. At this point, the water cannot evaporate or condense without a change in the system (like a temperature increase or pressure decrease).

For our lake's surface, the saturated water properties are dictated by temperature. These properties include saturated vapor pressure, which is the pressure exerted by the water vapor when it is in this equilibrium state. This value can be found in steam tables referencing the temperature of the lake surface, or through calculations such as the Antoine equation. Understanding these properties is essential to determine the mole fraction of water vapor.

Vapor pressure is a term that refers to the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The vapor pressure of water increases with an increase in temperature, meaning that warmer water has a higher potential to become water vapor than colder water.

Vapor pressure is critical when calculating the mole fraction of water vapor because it represents the maximum pressure that water vapor can exert at a given temperature, i.e., when the water is saturated. In the context of a lake, the vapor pressure would be the pressure exerted by water vapor at the surface of the lake, assuming the air above it is saturated with moisture. When the atmosphere's actual pressure equals the vapor pressure, the atmosphere is holding all the water vapor it can without condensation occurring.

This scientific principle, named after John Dalton, tells us how to approach the study of mixtures of gases, like the combination of water vapor and dry air over our lake. Dalton's Law of Partial Pressures states that in a mixture of non-reacting gases, the total pressure exerted by the mixture is equal to the sum of the partial pressures of individual gases.

Therefore, to find the mole fraction of water vapor at the lake surface, you first need to understand the partial pressures involved. These include the pressure from the water vapor and the pressure from the other gases, predominantly nitrogen and oxygen, that make up dry air. By applying Dalton's Law, we can isolate the partial pressure of the water vapor when we know the total atmospheric pressure and the pressure from the dry air components.

Mole fraction calculation is a dimensionless number that represents the ratio of the number of moles of a component to the total number of moles of all components in the mixture. To find the mole fraction of water vapor (Y_w) in the air at the lake's surface, we divide the partial pressure of water vapor (P_w) by the total pressure, which includes both the water vapor and the dry air (P_w + P_dry_air).

It's worth noting that the mole fraction is an important concept because it is independent of the overall amount of the mixture or the presence of other gases. It allows us to measure composition in a way that can be easily compared between different systems. By calculating the mole fraction of water vapor, you gain valuable insight into the humidity of the air, which has practical implications for weather prediction, HVAC systems, and understanding local climate behavior around bodies of water like lakes.