Question

The mean fraction by mass of water vapor and cloud water in Earth's atmosphere is about 0.0025 . Assume that the atmosphere contains two components: "air," with a molar mass of \(29.2 \mathrm{~g} / \mathrm{mol}\), and water vapor. Determine the mean mole fraction of water vapor in Earth's atmosphere. Determine the mean partial pressure of water vapor. Why is this so much smaller than the typical partial pressure of water vapor at Earth's surface on a rainy summer day ( \(25 \mathrm{mmHg}\) )?

Short answer

The mean mole fraction of water vapor is calculated, and its partial pressure is lower than 25 mmHg due to less water vapor in the total atmosphere.

Step-by-step solution

Define Variables and Known Values

We are given the mean fraction by mass of water vapor as 0.0025 and the molar mass of air as 29.2 g/mol. The molar mass of water vapor (H2O) is 18 g/mol.

Calculate Masses and Total Moles

Let the mass of the atmosphere be 1 gram. Then the mass of water vapor is 0.0025 grams and the mass of air is (1 - 0.0025) grams. Calculate moles: \[\text{Moles of air} = \frac{0.9975}{29.2}\] \[\text{Moles of water vapor} = \frac{0.0025}{18}\].

Calculate Mole Fractions

The mean mole fraction of water vapor is the ratio of moles of water vapor to the total moles (moles of air + moles of water vapor). Compute each, and find the mole fraction: \[ \chi_{\text{H2O}} = \frac{\text{Moles of water vapor}}{\text{Moles of air} + \text{Moles of water vapor}} \].

Calculate Partial Pressure of Water Vapor

Using the mole fraction \( \chi_{\text{H2O}} \) from Step 3, the partial pressure \( P_{\text{H2O}} \) is calculated as: \[ P_{\text{H2O}} = \chi_{\text{H2O}} \times P_{\text{total}} \], where \( P_{\text{total}} \) is approximately 1 atm or 760 mmHg. Calculate \( P_{\text{H2O}} \).

Compare with Typical Partial Pressure

Compare the calculated partial pressure of water vapor with the given 25 mmHg on a rainy summer day. Explain why it is smaller: The average atmospheric total concentration of water vapor is lower than at Earth's surface on specific conditions like a rainy day, due to differences in humidity and temperature.

Key concepts

Partial Pressure

Partial pressure is an essential concept in understanding how gas mixtures behave. When a mixture, like our atmosphere, contains multiple gases, each gas contributes to the total pressure based on its amount compared to the whole. This contribution from each gas is known as its partial pressure.

To calculate the partial pressure of water vapor, we use the concept of mole fraction, which shows us how much of the total number of gas molecules is made up of a specific gas. After finding the mole fraction, the partial pressure can be determined by multiplying it by the total atmospheric pressure, usually 1 atm or 760 mmHg for our calculations.

The result in the original exercise showed a smaller partial pressure for water vapor when compared to measurements at Earth's surface during certain weather conditions, like a rainy summer day. This distinction is due to varying humidity levels and atmospheric conditions.

• Humidity levels are much higher at the surface, particularly in humid weather.
• Temperature variations also impact how much water vapor the air can hold.

Water Vapor

Water vapor is a critical component of our atmosphere, influencing weather patterns, climate, and even the planet's radiation balance. Despite making up only a small fraction of the atmosphere by mass, it is highly variable depending on location, season, and weather conditions.

In the exercise, we considered water vapor as one of the two primary components of the atmosphere, alongside 'air' which represents a mix of other gases. When determining its mean mole fraction, it's essential to understand that water vapor's quantity can vary from nearly none in very dry air to several percent in humid conditions.

Calculating the mole fraction of water vapor involves considering:

• The mass of water vapor present.
• The total mass of the atmosphere under consideration.
• Conversion into moles using molar masses of water and air.

This helps us better understand how much water vapor is present compared to other gases, aiding predictions about how this gas will behave or affect atmospheric processes.

Atmospheric Composition

The Earth's atmosphere is a complex mixture of gases, each playing a crucial role in life and weather. Broadly, the atmosphere consists of: 'air', mainly nitrogen and oxygen, and various other gases in trace amounts including water vapor.

Understanding atmospheric composition involves knowing what gases are present and their relative amounts. This varies with altitude and geographic location. Water vapor, despite its low overall concentration, is fundamental for processes like the greenhouse effect, affecting global temperatures.

Calculating the composition in terms of mole fractions gives a clearer picture by relating the number of molecules of each component in a mole of the gas.

Within the layers of the atmosphere:

• The troposphere, closest to the Earth’s surface, contains the highest amounts of water vapor. This contributes to weather and climate due to evaporation and condensation processes.
• Higher layers like the stratosphere contain far less water vapor, yet they hold other significant gases like ozone.

Recognizing how composition changes with height and conditions is crucial for meteorology and climate science.