Question

Consider two rooms that are identical except that one is maintained at \(25^{\circ} \mathrm{C}\) and 40 percent relative humidity while the other is maintained at \(20^{\circ} \mathrm{C}\) and 55 percent relative humidity. Noting that the amount of moisture is proportional to the vapor pressure, determine which room contains more moisture.

Short answer

Solution: Step 1: Calculate the saturation vapor pressure for each room. Room 1: \(T_1 = 298.15\:K \) Room 2: \(T_2 = 293.15\:K \) Calculate the saturation vapor pressures for both rooms: Room 1: \(P_{sat1} = 611 \text{exp}\left[\frac{2.501 \times 10^6}{461.5}\left(\frac{1}{273.15} - \frac{1}{298.15}\right)\right] \) Room 2: \(P_{sat2} = 611 \text{exp}\left[\frac{2.501 \times 10^6}{461.5}\left(\frac{1}{273.15} - \frac{1}{293.15}\right)\right] \) Step 2: Calculate the vapor pressure in each room. Calculate the vapor pressure in each room: Room 1: \(P_{actual1} = P_{sat1} \times 0.40\) Room 2: \(P_{actual2} = P_{sat2} \times 0.55\) Step 3: Compare the vapor pressure in both rooms. Calculate and compare \(P_{actual1}\) and \(P_{actual2}\) to determine which room has more moisture.

Step-by-step solution

Calculate the saturation vapor pressure for each room.

We'll use the Clausius-Clapeyron equation to find the saturation vapor pressure for each room. The equation is: \(P_{sat} = P_{ref} \text{exp}\left[\frac{L_v}{R_v}\left(\frac{1}{T_{ref}} - \frac{1}{T}\right)\right]\) Where: - \(P_{sat}\) is the saturation vapor pressure - \(P_{ref}\) is a reference vapor pressure with saturated point (\(611\: \text{Pa}\)) - \(L_v\) is the latent heat of vaporization of water (\(2.501 \times 10^6 \: \text{J/kg}\)) - \(R_v\) is the specific gas constant for water vapor (\(461.5\: \text{J/kg⋅K}\)) - \(T_{ref}\) is the reference temperature in Kelvin \((273.15\:K)\) - \(T\) is the temperature in Kelvin First, convert temperatures from Celsius to Kelvin: Room 1: \(T_1 = 25^{\circ}C + 273.15 = 298.15\:K \) Room 2: \(T_2 = 20^{\circ}C + 273.15 = 293.15\:K \) Next, calculate the saturation vapor pressures for both rooms: Room 1: \(P_{sat1} = 611 \text{exp}\left[\frac{2.501 \times 10^6}{461.5}\left(\frac{1}{273.15} - \frac{1}{298.15}\right)\right]\) Room 2: \(P_{sat2} = 611 \text{exp}\left[\frac{2.501 \times 10^6}{461.5}\left(\frac{1}{273.15} - \frac{1}{293.15}\right)\right]\)

Calculate the vapor pressure in each room.

The vapor pressure in each room is calculated by multiplying the saturation vapor pressure by the corresponding relative humidity: \(P_{actual} = P_{sat} \times RH\) Where \(P_{actual}\) is the actual vapor pressure and \(RH\) is the relative humidity. For each room, calculate the vapor pressure: Room 1: \(P_{actual1} = P_{sat1} \times 0.40\) Room 2: \(P_{actual2} = P_{sat2} \times 0.55\)

Compare the vapor pressure in both rooms.

Determine which room has a higher vapor pressure, indicating more moisture in the air: If \(P_{actual1} > P_{actual2}\), then Room 1 contains more moisture. If \(P_{actual1} < P_{actual2}\), then Room 2 contains more moisture. Calculate and compare \(P_{actual1}\) and \(P_{actual2}\) to determine which room has more moisture.

Key concepts

Relative Humidity

Relative humidity (RH) is a measure of the amount of water vapor that air contains compared to the maximum amount it could hold at that temperature. This concept is central to understanding moisture levels in different environments. It is expressed as a percentage. For instance, a relative humidity of 40% means the air is holding 40% of the maximum moisture it can contain at a given temperature.

The RH affects how we perceive temperature; higher humidity can make temperatures feel warmer due to the reduced effectiveness of sweating. When solving problems that involve relative humidity, remember that the actual vapor pressure (the current amount of moisture in the air) is a portion of the saturation vapor pressure (the maximum amount of moisture air can hold at that temperature), which can be calculated using the RH percentage.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a pivotal tool in thermodynamics that establishes a relationship between vapor pressure and temperature. Specifically, it shows how the saturation vapor pressure of a substance changes in response to temperature. The equation is especially useful in meteorology for predicting the saturation vapor pressure for water in the air.

The equation considers the latent heat of vaporization of water, the specific gas constant for water vapor, and two temperatures: the reference temperature and the temperature for which we're solving. By using this equation, we can understand how changes in temperature impact the amount of moisture air can potentially hold. When working with changes in temperature and humidity, as shown in the provided exercise, understanding this relationship ensures a comprehensive grasp of how moisture content in air shifts in different conditions.

Saturation Vapor Pressure

Saturation vapor pressure (SVP) is the maximum pressure that water vapor can exert when the air is saturated with moisture at a given temperature. When SVP is reached, the air cannot hold any more water vapor without condensation occurring – this state is commonly known as 100% relative humidity.

The Clausius-Clapeyron equation helps us calculate SVP, which is crucial for determining the potential maximum moisture content in the air. It is temperature-dependent: as the temperature increases, so does the SVP. A higher SVP means the air can hold more water vapor, which is essential in identifying the moisture content at different temperatures, as explored in the original exercise.

Moisture Content in Air

Understanding moisture content in air is vital for various applications, including weather forecasting, HVAC systems design, and understanding human comfort levels. The moisture content refers to the actual amount of water vapor present in the air. It can be quantified using actual vapor pressure, which we obtain by applying the relative humidity to the saturation vapor pressure.

As seen in the exercise, by comparing the actual vapor pressures of two rooms with different temperatures and relative humidities, we can assess which room contains more moisture. This comparison helps control indoor environments for comfort and prevents issues like mold growth, which are influenced by the amount of moisture in the air.