Density of air


The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variation in atmospheric pressure, temperature and humidity. At 1013.25 hPa and 15°C, air has a density of approximately 1.225 kg/m³ according to ISA.
Air density is a property used in many branches of science, engineering, and industry, including aeronautics; gravimetric analysis; the air-conditioning industry; atmospheric research and meteorology; agricultural engineering ; and the engineering community that deals with compressed air.
Depending on the measuring instruments used, different sets of equations for the calculation of the density of air can be applied. Air is a mixture of gases and the calculations always simplify, to a greater or lesser extent, the properties of the mixture.

Dry air

The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure:
where:
The specific gas constant for dry air is 287.058 J/ in SI units, and 53.35 / in United States customary and Imperial units. This quantity may vary slightly depending on the molecular composition of air at a particular location.
Therefore:
The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa:

Humid air

The addition of water vapor to air reduces the density of the air, which may at first appear counter-intuitive.
This occurs because the molar mass of water is less than the molar mass of dry air. For any ideal gas, at a given temperature and pressure, the number of molecules is constant for a particular volume. So when water molecules are added to a given volume of air, the dry air molecules must decrease by the same number, to keep the pressure or temperature from increasing. Hence the mass per unit volume of the gas decreases.
The density of humid air may be calculated by treating it 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:
where:
The vapor pressure of water may be calculated from the saturation vapor pressure and relative humidity. It is found by:
where:
The saturation vapor pressure of water at any given temperature is the vapor pressure when relative humidity is 100%.
One formula used to find the saturation vapor pressure is:
where is in degrees C. See vapor pressure of water for other equations.
The partial pressure of dry air is found considering partial pressure, resulting in:
Where simply denotes the observed absolute pressure.

Variation with altitude

Troposphere

To calculate the density of air as a function of altitude, one requires additional parameters. For the troposphere, the lowest part of the atmosphere, they are listed below, along with their values according to the International Standard Atmosphere, using for calculation the universal gas constant instead of the air specific constant:
Temperature at altitude meters above sea level is approximated by the following formula :
The pressure at altitude is given by:
Density can then be calculated according to a molar form of the ideal gas law:
where:
Note that the density close to the ground is
It can be easily verified that the hydrostatic equation holds:

Exponential approximation

As the temperature varies with height inside the troposphere by less than 25%, and one may approximate:
Thus:
Which is identical to the isothermal solution, except that Hn, the height scale of the exponential fall for density, is not equal to RT0/g M as one would expect for an isothermal atmosphere, but rather:
Which gives Hn = 10.4 km.
Note that for different gasses, the value of Hn differs, according to the molar mass M: It is 10.9 for nitrogen, 9.2 for oxygen and 6.3 for carbon dioxide. The theoretical value for water vapor is 19.6, but due to vapor condensation the water vapor density dependence is highly variable and is not well approximated by this formula.
The pressure can be approximated by another exponent:
Which is identical to the isothermal solution, with the same height scale Hp = RT0/gM. Note that the hydrostatic equation no longer holds for the exponential approximation.
Hp is 8.4 km, but for different gasses, it is again different and depends upon molar mass, giving 8.7 for nitrogen, 7.6 for oxygen and 5.6 for carbon dioxide.

Total content

Further note that since g, Earth's gravitational acceleration, is approximately constant with altitude in the atmosphere, the pressure at height h is proportional to the integral of the density in the column above h, and therefore to the mass in the atmosphere above height h.
Therefore the mass fraction of the troposphere out of all the atmosphere is given using the approximated formula for p:
For nitrogen, it is 75%, while for oxygen this is 79%, and for carbon dioxide - 88%.

Tropopause

Higher than the troposphere, at the tropopause, the temperature is approximately constant with altitude and is 220 K. This means that at this layer L = 0 and T= 220K, so that the exponential drop is faster, with HTP = 6.3 km for air. Both the pressure and density obey this law, so, denoting the height of the border between the troposphere and the tropopause as U:

Composition