Which is denser at the same temperature and pressure, dry air or air saturated with water vapor? Explain.

The ideal gas law is a fundamental principle in chemistry and physics, which describes the relationship between the pressure, volume, temperature, and amount of an ideal gas. It is expressed by the equation

\[ PV = nRT \]
where \(P\) is the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the amount of substance in moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

In the context of comparing the density of air, whether dry or moist, the ideal gas law explains that for a given temperature and pressure, the volume is determined by the number of moles and does not change regardless of the gas's composition. However, because different gases have different molar masses, those with lower molar masses will occupy the same volume at a lighter weight, resulting in a lower density.

Molecular mass, often referred to as molar mass, is the mass of one mole of a substance, usually measured in grams per mole (g/mol). This value is particularly important when discussing the ideal gas law and density as it directly influences the mass of a gas occupying a specific volume.

For example, the molecular mass of water vapor (\(H_2O\)) is approximately 18 g/mol, and the average molecular mass of dry air — which consists primarily of \(N_2\), \(O_2\), and \(Ar\) — is around 28.97 g/mol. Consequently, dry air has a higher molecular mass compared to moist air, as water vapor replaces some of the nitrogen and oxygen, resulting in a lower average molecular mass for moist air. To put it simply, where dry air might weigh around 28.97 g per mole, moist air might be lighter due to the inclusion of water vapor.

The composition of air plays a vital role in determining its properties, including its density. Dry air is a mixture of several different gases, predominantly nitrogen (about 78%), oxygen (about 21%), and trace amounts of argon, carbon dioxide, and other gases. In contrast, moist air contains the same gases in addition to varying quantities of water vapor.

Since water vapor has a lower molecular mass than the other primary gases in air, moist air has a lower average molecular mass than dry air. It is this difference in molecular mass that provides the key to understanding why moist air is less dense, as demonstrated by the ideal gas law and its usage in our exercise solution.

In a real-world scenario, the presence of water vapor (humidity) in the air is constantly changing, which means the density of the air shifts as well. This is why humid air can feel 'heavier' or 'thicker' to breathe, but in actuality, it is less dense than dry air, a fact that may seem counterintuitive.