The density of humid air is calculated by determining the sum of the densities of dry air and water vapor, weighted by their respective partial pressures.
Here's a breakdown of the process:
1. Understanding the Components
Humid air is a mixture of two gases:
- Dry Air: Composed primarily of nitrogen, oxygen, argon, and trace gases.
- Water Vapor: Gaseous form of water.
2. Key Concepts
- Partial Pressure: The pressure exerted by each individual gas in a mixture. In humid air, we consider the partial pressure of dry air (Pd) and the partial pressure of water vapor (Pv).
- Ideal Gas Law: PV = nRT, where:
- P = Pressure
- V = Volume
- n = Number of moles
- R = Ideal gas constant (8.314 J/(mol·K))
- T = Temperature (in Kelvin)
3. Calculation Steps
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a. Determine the Partial Pressures: You'll need to know the total atmospheric pressure (P), relative humidity (RH), and temperature (T).
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Calculate the saturation vapor pressure (Ps) at the given temperature. Several empirical formulas can be used for this, such as the Antoine equation or approximations. Ps represents the maximum amount of water vapor the air can hold at that temperature.
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Calculate the partial pressure of water vapor: Pv = RH * Ps
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Calculate the partial pressure of dry air: Pd = P - Pv
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b. Calculate the Densities: Use a modified version of the Ideal Gas Law to find the density of each component. Density (ρ) = (PM)/(RT), where:
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ρ = Density
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P = Partial Pressure (Pd for dry air, Pv for water vapor)
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M = Molar Mass (approximately 28.97 g/mol for dry air, 18.015 g/mol for water vapor)
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R = Ideal Gas Constant (8.314 J/(mol·K) or 8.314 kg⋅m2/(s2⋅mol⋅K))
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T = Temperature (in Kelvin)
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Density of dry air (ρd) = (Pd Md) / (R T)
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Density of water vapor (ρv) = (Pv Mv) / (R T)
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c. Calculate the Density of Humid Air: Sum the densities of dry air and water vapor.
- Density of humid air (ρh) = ρd + ρv
4. Example
Let's say we have air at:
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Total Pressure (P) = 101325 Pa (1 atm)
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Temperature (T) = 25°C (298.15 K)
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Relative Humidity (RH) = 60%
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Saturation Vapor Pressure (Ps): Assume we've calculated Ps to be 3169 Pa using a suitable equation.
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Partial Pressure of Water Vapor (Pv): Pv = 0.60 * 3169 Pa = 1901.4 Pa
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Partial Pressure of Dry Air (Pd): Pd = 101325 Pa - 1901.4 Pa = 99423.6 Pa
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Density of Dry Air (ρd): ρd = (99423.6 Pa 0.02897 kg/mol) / (8.314 J/(mol·K) 298.15 K) ≈ 1.16 kg/m³
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Density of Water Vapor (ρv): ρv = (1901.4 Pa 0.018015 kg/mol) / (8.314 J/(mol·K) 298.15 K) ≈ 0.0138 kg/m³
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Density of Humid Air (ρh): ρh = 1.16 kg/m³ + 0.0138 kg/m³ ≈ 1.1738 kg/m³
Important Considerations:
- Units: Ensure consistent units throughout the calculation. Convert temperatures to Kelvin (K = °C + 273.15) and use appropriate units for pressure, molar mass, and the gas constant.
- Accuracy: The accuracy of the result depends on the accuracy of the input parameters (temperature, pressure, humidity) and the equation used to calculate saturation vapor pressure.
- Molar Mass of Dry Air: The molar mass of dry air is an approximation. The actual value varies slightly depending on the composition of the air.
In summary, the density of humid air is determined by calculating the densities of dry air and water vapor separately using their partial pressures and then summing these densities. This method accurately reflects the contribution of each component to the overall density of the mixture.
