Question

Explain how vapor pressure of the ambient air is determined when the temperature, total pressure, and relative humidity of the air are given.

Short answer

Answer: To determine the vapor pressure of ambient air, apply the following steps: 1. Understand the concept of saturation vapor pressure, which is the pressure of water vapor in the air when it is saturated and depends on temperature. 2. Use the formula for relative humidity: Relative Humidity (RH) = (Vapor Pressure) / (Saturation Vapor Pressure) * 100 3. Calculate saturation vapor pressure using the August-Roche-Magnus equation: Saturation Vapor Pressure = 0.61094 * e^((17.625 * T) / (T + 243.04)), where T is the temperature in degrees Celsius. 4. Rearrange the RH formula to solve for vapor pressure: Vapor Pressure = (Relative Humidity / 100) * Saturation Vapor Pressure 5. Plug in the given temperature and relative humidity values into the formulas to calculate the vapor pressure of the ambient air.

Step-by-step solution

Understanding Saturation Vapor Pressure

Saturation vapor pressure is the pressure of water vapor in the air when the air is saturated, which means it can no longer hold more water vapor. The saturation vapor pressure depends on the temperature of the air, and it increases as the temperature increases.

Introduce the formula for Relative Humidity

Relative humidity (RH) is the ratio of the current amount of water vapor in the air (vapor pressure) to the maximum amount of water vapor the air can hold (saturation vapor pressure). The formula for relative humidity can be written as: Relative Humidity (RH) = (Vapor Pressure) / (Saturation Vapor Pressure) * 100

Calculate Saturation Vapor Pressure

To determine saturation vapor pressure, we can use the Clausius-Clapeyron equation or a simpler approximation known as the August-Roche-Magnus equation. We'll use the latter for this example: Saturation Vapor Pressure = 0.61094 * e^((17.625 * T) / (T + 243.04)) Where T is the temperature in degrees Celsius, and e is the base of the natural logarithm (approximately 2.718).

Calculate Vapor Pressure from Relative Humidity

Now that we have the formula for saturation vapor pressure, we can rearrange the RH formula to solve for vapor pressure: Vapor Pressure = (Relative Humidity / 100) * Saturation Vapor Pressure

Determine Vapor Pressure of Ambient Air

Plug the given temperature and relative humidity values into the formulas from Steps 3 and 4 to calculate the vapor pressure of the ambient air. For example, if the temperature is 25°C and the relative humidity is 50%, the vapor pressure would be: 1. Calculate the saturation vapor pressure: Saturation Vapor Pressure = 0.61094 * e^((17.625 * 25) / (25 + 243.04)) Saturation Vapor Pressure ≈ 3.169 kPa 2. Calculate the vapor pressure: Vapor Pressure = (50 / 100) * 3.169 Vapor Pressure ≈ 1.584 kPa Thus, the vapor pressure of the ambient air is approximately 1.584 kPa.

Key concepts

Relative Humidity

Relative humidity is a measure of how much water vapor is present in the air relative to the maximum amount it can hold at a given temperature. It is expressed as a percentage, indicating how close the air is to being saturated. When the air is fully saturated, its relative humidity is 100%. At 0%, the air contains no moisture.
To understand relative humidity, consider it as a ratio of two pressures:

• The actual vapor pressure which is the amount of water vapor currently in the air.
• The saturation vapor pressure which is the maximum amount of water vapor the air can hold at the same temperature.

As temperature increases, the air can hold more moisture, so the saturation vapor pressure rises, affecting relative humidity. If the actual vapor pressure remains constant but the saturation vapor pressure increases, the relative humidity decreases.
Calculating relative humidity uses the simple formula:\[\text{Relative Humidity (RH)} = \left(\frac{\text{Vapor Pressure}}{\text{Saturation Vapor Pressure}}\right) \times 100\]This formula tells us that relative humidity increases if the vapor pressure increases or if saturation vapor pressure decreases, assuming the other factor remains constant.

Saturation Vapor Pressure

Saturation vapor pressure is a critical concept in understanding humidity and weather patterns. It represents the pressure exerted by water vapor in a completely saturated atmosphere. When the air reaches its saturation vapor pressure, it cannot hold any more water vapor without condensation occurring.
The saturation vapor pressure is highly dependent on temperature. As temperature rises, the saturation vapor pressure also increases due to the enhanced activity of water molecules, which escape more readily from liquid to vapor state. This increase occurs because warmer air holds more energy, allowing it to carry more moisture.
To calculate saturation vapor pressure, a common equation used is the August-Roche-Magnus equation:\[\text{Saturation Vapor Pressure} = 0.61094 \times e^{\left(\frac{17.625 \times T}{T + 243.04}\right)}\]Here, \(T\) is the temperature in degrees Celsius, and \(e\) is the base of the natural logarithm, approximately 2.718. This calculation helps predict how much moisture air can hold before water vapor turns into liquid, which is crucial for weather forecasting and understanding humidity levels.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation describes how vapor pressure changes with temperature. It is fundamental in thermodynamics, playing a pivotal role in understanding phase transitions, such as evaporation and condensation, especially in meteorology.
This equation is derived from the principles of energy conservation and the properties of an ideal gas. It expresses the relationship between temperature and pressure for a phase change at equilibrium. Essentially, it helps quantify the increase in vapor pressure needed for a substance to remain in a vapor state as the temperature rises.
The Clausius-Clapeyron equation is given by:\[\frac{dP}{dT} = \frac{L}{T(V_v - V_l)}\]Here,

• \(dP/dT\) is the rate of change of pressure with temperature.
• \(L\) is the latent heat of the phase transition.
• \(T\) is the absolute temperature.
• \(V_v\) and \(V_l\) are the molar volumes of the vapor and liquid phases, respectively.

By simplifying these relationships under certain assumptions, like constant latent heat and ideal gas behavior, the equation becomes practical for estimating vapor pressures and is a tool used in atmospheric science to understand how temperature changes affect humidity and atmospheric pressure.