Density of air

From Wikipedia, the free encyclopedia

The density of air, ρ (Greek: rho) (air density), is the mass per unit volume of Earth's atmosphere, and is a useful value in aeronautics. As does air pressure, air density decreases with increasing altitude and temperature. At sea level and at 20 °C, dry air has a density of approximately 1.2 kg/m3.

The density of water, which is about 1000 kg/m3 (1 g/cm³), is about 800 times more than the density of air.

Contents

[edit] Effects of temperature and pressure

The formula for the density of dry air is given by:

\rho = \frac{p}{R \cdot T}

where ρ is the air density in kilograms per cubic meter, p is pressure in pascals, R is the specific gas constant, and T is temperature in kelvins.

The specific gas constant for dry air is:

R_\mathrm{dry\,air} = 287.05 \frac{\mbox{J}}{\mbox{kg} \cdot \mbox{K}}

Therefore:

[edit] Effect of water vapor

The addition of water vapor to air (making the air humid) reduces the density of the air, which may at first appear contrary to logic.

This occurs because the molecular mass of water (18) is less than the molecular mass of air (around 29). For any gas, at a given temperature and pressure, the number of molecules present is constant for a particular volume. So when water molecules (vapor) are introduced to the air, the number of air molecules must reduce by the same number in a given volume, without the pressure or temperature increasing. Hence the mass per unit volume of the gas (its density) decreases.

The density of humid air may be calculated as a mixture of ideal gases. In this case, the partial pressure of water vapor is known as the vapor pressure. Using this method, error in the density calculation is less than 0.2% in the range of −10 °C to 50 °C. The density of humid air is found by:

\rho~_{_{humid~air}} = \frac{p_{d}}{R_{d} \cdot T} + \frac{p_{v}}{R_{v} \cdot T} [1]

Where:

\rho~_{_{humid~air}} = Density of the humid air (kg/m³)
pd = Partial pressure of dry air (Pa)
Rd = Specific gas constant for dry air, 287.05 J/(kg·K)
T = Temperature (K)
pv = Pressure of water vapor (Pa)
Rv = Specific gas constant for water vapor, 461.495 J/(kg·K)

The vapor pressure of water may be calculated from the saturation vapor pressure and relative humidity. It is found by:

p_{v} = \phi~ \cdot p_{sat}

Where:

pv = Vapor pressure of water
\phi~ = Relative humidity
psat = Saturation vapor pressure

The saturation vapor pressure of water at any given temperature is the vapor pressure when relative humidity is 100%. A simplification of the regression [1] used to find this, can be formulated as:

p(mb)_{sat} = 6.1078 \cdot 10^{\frac{7.5 \cdot T-2048.625}{T-35.85}}

IMPORTANT:

  • This will give a result in mb (millibar), 1 mb=100 Pa
pd = p − pv

Where p simply notes the absolute pressure in the observed system.

[edit] Effects of altitude

To calculate the density of air as a function of altitude, one requires additional parameters. They are listed below, along with their values according to the International Standard Atmosphere, using the universal gas constant instead of the specific one:

  • sea level standard atmospheric pressure p0 = 101325 Pa
  • sea level standard temperature T0 = 288.15 K
  • Earth-surface gravitational acceleration g = 9.80665 m/s2.
  • temperature lapse rate L = −.0065 °K/m
  • universal gas constant R = 8.31447 J/(mol·K)
  • molar mass of dry air M = 0.0289644 kg/mol = (28.9644 g/mol)

Temperature at altitude h meters above sea level is given by the following formula (only valid inside the troposphere):

T = T_0 + L \cdot h

The pressure at altitude h is given by:

p = p_0 \cdot \left(1 + \frac{L \cdot h}{T_0} \right)^\frac{g \cdot M}{-R \cdot L}

Density can then be calculated according to a molar form of the original formula:

\rho = \frac{p \cdot M}{R \cdot T}

[edit] Importance of temperature

The below table demonstrates that the properties of air change significantly with temperature.

Table — speed of sound in air c, density of air ρ, acoustic impedance Z vs. temperature °C

Effect of temperature
°C c in m/s ρ in kg/m³ Z in Pa·s/m
−10 325.2 1.342 436.1
−5 328.3 1.317 432.0
0 331.3 1.292 428.4
+5 334.3 1.269 424.3
+10 337.3 1.247 420.6
+15 340.3 1.225 416.8
+20 343.2 1.204 413.2
+25 346.1 1.184 409.8
+30 349.0 1.165 406.3

[edit] See also

[edit] References

  1. ^ a b Equations - Air Density and Density Altitude

[edit] External links