Problem 88 - University Physics with Modern Physics

Question

The vapor pressure is the pressure of the vapor phase of a substance when it is in equilibrium with the solid or liquid phase of the substance. The relative humidity is the partial pressure of water vapor in the air divided by the vapor pressure of water at that same terperature, expressed as a percentage. The air is saturated when the humidity is 100\(\%\) . (a) The vapor pressure of water at \(20.0^{\circ} \mathrm{C}\) is \(2.34 \times 10^{3} \mathrm{Pa}\) . If the air temperature is \(20.0^{\circ} \mathrm{C}\) and the relative humidity is 60\(\%\) what is the partial pressure of water vapor in the atmosphere (that is, the pressure due to water vapor alone)? (b) Under the conditions of part (a), what is the mass of water in 1.00 \(\mathrm{m}^{3}\) of air? (The molar mass of water is 18.0 \(\mathrm{g} / \mathrm{mol}\) . Assume that water vapor can be treated as an ideal gas.)

Step-by-Step Solution

Answer

(a) 1404 Pa; (b) 10.38 g/m³.

Step 1: Calculate the Partial Pressure of Water Vapor

Since the relative humidity (RH) is given as 60% when the vapor pressure at 20.0°C is 2340 Pa, the partial pressure of water vapor can be found using the formula: \( P_{w} = \text{RH} \times P_0 \), where:- \( P_{w} \) is the partial pressure of water vapor,- \( \text{RH} = 0.60 \), - \( P_0 = 2340 \text{ Pa} \).Therefore, \( P_{w} = 0.60 \times 2340 = 1404 \text{ Pa} \).

Step 2: Use Ideal Gas Law to Find Density of Water Vapor

To determine the mass of water in 1.00 m³ of air, use the ideal gas law equation: \( PV = nRT \). Rearrange to find \( n \), the amount of substance (in moles): \( n = \frac{PV}{RT} \).Given:- Pressure \( P_{w} = 1404 \text{ Pa} \),- Volume \( V = 1.00 \text{ m}^3 \),- Ideal gas constant \( R = 8.314 \text{ J/mol·K} \),- Temperature \( T = 293.15 \text{ K} \) (which is 20°C in absolute temperature).So, \( n = \frac{1404 \times 1}{8.314 \times 293.15} = 0.5767 \text{ moles} \).

Step 3: Calculate the Mass of Water in the Air

To convert the amount of substance in moles to mass, use the formula: \( m = n \times M \), where:- \( m \) is the mass,- \( n = 0.5767 \text{ moles} \),- Molar Mass \( M = 18.0 \text{ g/mol} \).Thus, \( m = 0.5767 \times 18.0 = 10.38 \text{ g} \).

Key Concepts

Relative Humidity

Relative humidity is an important measurement in meteorology and atmospheric science. It refers to the amount of moisture present in the air compared to the maximum amount of water vapor that air could hold at a given temperature. This value is expressed as a percentage. A relative humidity of 50% means the air holds half of the water vapor needed for it to be fully saturated at that temperature.

Understanding relative humidity helps us predict weather patterns and human comfort levels. High humidity can make temperatures feel hotter, while low humidity can make the air feel cooler. Relative humidity is calculated using the formula:
\( \text{Relative Humidity} = \frac{\text{Partial Pressure of Water Vapor}}{\text{Vapor Pressure of Water}} \times 100 \% \).

In the exercise, when the relative humidity is 60%, it means the air contains 60% of the water vapor it could hold at that temperature. Calculating relative humidity involves knowing both the partial pressure of water vapor in the air and the vapor pressure at that particular temperature.

Ideal Gas Law

The Ideal Gas Law is a useful equation in chemistry and physics that relates the pressure, volume, temperature, and number of moles of a gas. It is usually expressed as:
\( PV = nRT \), where:

• \(P\) is the pressure of the gas,
• \(V\) is the volume the gas occupies,
• \(n\) is the number of moles of the gas,
• \(R\) is the ideal gas constant, and
• \(T\) is the temperature in Kelvin.

This equation assumes gases behave ideally, which works well under a range of conditions but can deviate at high pressures or low temperatures. For water vapor, assuming ideal behavior simplifies calculations, like in the exercise.

By rearranging the Ideal Gas Law, you can solve for any variable, such as determining the number of moles. This makes it a versatile tool in many scientific calculations, such as finding the mass of water vapor in air.

Partial Pressure

Partial pressure is the pressure exerted by a single type of gas in a mixture of gases. It's essential in understanding how different gases behave independently within a mix, such as atmospheric air, which contains nitrogen, oxygen, water vapor, and other gases.

The concept of partial pressures is part of Dalton's Law of Partial Pressures, which states that the total pressure exerted by a mixture is the sum of the partial pressures of the individual gases present.
\( P_{\text{total}} = P_{1} + P_{2} + P_{3} + ... \)

In the exercise, we calculated the partial pressure of water vapor using the given relative humidity. This value represents the contribution of water vapor to the total atmospheric pressure. Knowing the partial pressure is crucial for determining humidity levels and for further calculations using the Ideal Gas Law.

Phase Equilibrium

Phase equilibrium occurs when a substance is in a state where its various phases—vapor, liquid, or solid—exist together without any net change over time. At equilibrium, the rate of evaporation equals the rate of condensation in a closed system, like the point where vapor pressure is measured.

This concept is key to understanding vapor pressure itself. Vapor pressure reflects the tendency of molecules to escape from a liquid or solid into the gaseous phase. When these phases are in dynamic balance, the vapor pressure of the system is constant if the temperature remains stable.

For instance, the vapor pressure of water at \(20.0^{\circ} \text{C}\) given in the problem is a characteristic value indicating phase equilibrium at this temperature. Understanding this equilibrium helps in predicting if condensation or evaporation will occur under changing environmental conditions.