Problem 100 - Physics

Question

Concept Questions On a cool autumn morning, the relative humidity is \(100 \%\). (a) Does this mean that the partial pressure of water in the air equals atmospheric pressure? Why or why not? (b) Suppose the air warms up in the afternoon, but the partial pressure of water in the air does not change. Is the humidity still \(100 \% ?\) Provide a reason for your answer. Problem The vapor pressure of water at \(20^{\circ} \mathrm{C}\) is \(2500 \mathrm{~Pa}\). (a) What percentage of atmospheric pressure is this? (b) What percentage of the total air pressure at \(20^{\circ} \mathrm{C}\) is due to water vapor if the relative humidity is \(100 \% ?\) (c) The vapor pressure of water at \(35^{\circ} \mathrm{C}\) is \(5500 \mathrm{~Pa}\). What is the relative humidity at this temperature if the partial pressure of water in the air has not changed from that at \(20^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

(a) No, the partial pressure doesn't equal atmospheric pressure but matches the vapor pressure. (b) Relative humidity decreases if the temperature increases without changing water vapor pressure. Problem: (a) 2.47%, (b) 2.47%, (c) 45.45%.

Step 1: Understanding Relative Humidity

Relative humidity (RH) is the ratio of the current partial pressure of water vapor in the air to the equilibrium vapor pressure of water at the same temperature. If RH is 100%, it implies that the air is fully saturated with water vapor at that temperature. It does not mean the partial pressure equals atmospheric pressure.

Step 2: Effect of Temperature on Humidity

If the air warms up but the partial pressure of water vapor does not change, the relative humidity decreases. This is because warm air can hold more moisture before reaching saturation than cold air.

Step 3: Calculating Water Vapor Pressure Percentage

The vapor pressure of water at 20°C is 2500 Pa. To find what percentage this is of the standard atmospheric pressure (101325 Pa), use the formula: \( \text{percentage} = \left( \frac{2500}{101325} \right) \times 100 \).

Step 4: Calculating Water Vapor's Contribution to Air Pressure

At 20°C with 100% RH, the partial pressure of water vapor is equal to its vapor pressure, 2500 Pa. The percentage of the total air pressure due to water vapor is calculated similarly: \( \text{percentage} = \left( \frac{2500}{101325} \right) \times 100 \).

Step 5: Determining Relative Humidity at Higher Temperature

At 35°C, the vapor pressure of water is 5500 Pa. With the partial pressure of water vapor unchanged at 2500 Pa (from 20°C), the relative humidity is \( \text{RH} = \left( \frac{2500}{5500} \right) \times 100 \).

Key Concepts

Vapor Pressure

Vapor pressure is a key concept in understanding how water exists in the air as a gas. It's the pressure exerted by water vapor when it is in equilibrium with liquid water at a given temperature. When we say "equilibrium," we mean there's a balance between the water molecules evaporating from the liquid into the air and those condensing back into the liquid. This balance results in a specific pressure value, known as vapor pressure.

• The vapor pressure changes with temperature: as the temperature rises, more water molecules have enough energy to escape into the air, increasing the vapor pressure.
• Vapor pressure is an inherent property of water that tells us how much water vapor the air can hold at a particular temperature before it becomes saturated.

You'll often hear about vapor pressure when dealing with humidity. It helps predict whether water will condense at a given temperature and pressure. Remember that when the air is fully saturated with vapor, the vapor pressure equals the partial pressure of water in the air, corresponding to 100% relative humidity.

Temperature Effect on Humidity

The temperature of the air greatly influences its capacity to hold water vapor. This effect is crucial in understanding why the relative humidity (RH) changes throughout the day or in different conditions. When air temperature increases, the amount of water vapor it can hold also increases.

• Warm air can accommodate more moisture than cold air. For instance, at a higher temperature, the same amount of water vapor in the air results in a lower RH because the air's moisture-holding capacity has increased.
• If the air temperature rises from morning to afternoon, without adding more moisture, the relative humidity decreases, reflecting the increase in capacity to hold moisture.

The role of temperature is why warm and moist climates often feel more humid than cooler ones, even if the amount of water vapor in the air isn't significantly different. It's about how full the air is relative to its capacity.

Partial Pressure of Water

Partial pressure is a concept used to describe the pressure exerted by a single type of gas in a mixture, like water vapor in the air. Even though air is made up of many gases like nitrogen, oxygen, and water vapor, each gas exerts its own pressure as if it were the only gas present.

• The partial pressure of water is the portion of atmospheric pressure that comes from water vapor. This measurement is critical when calculating relative humidity.
• At 100% RH, the partial pressure equals the vapor pressure, indicating full saturation of the air with water vapor at that temperature.

Understanding partial pressure helps grasp why changes in temperature alter relative humidity, despite constant levels of water vapor. The air can support different levels of saturation depending on its temperature and the partial pressures of its constituent gases, particularly water vapor.