View source for Physics:Density of air

{{Short description|Mass per unit volume of earths atmosphere}}
The '''density of air''' or '''atmospheric density''', denoted ''ρ'', is the [[Physics:Mass|mass]] per unit [[Physics:Volume|volume]] of [[Earth:Atmosphere of Earth|Earth's atmosphere]]. Air density, like air pressure, decreases with increasing altitude. It also changes with variations in atmospheric pressure, temperature and [[Physics:Humidity|humidity]]. At 101.325 kPa (abs) and 20 °C (68 °F), air has a density of approximately {{Cvt|1.204|kg/m3|lb/ft3|abbr=on}}, according to the [[International Standard Atmosphere]] (ISA). At 101.325{{nbsp}}kPa (abs) and {{cvt|15|C|F}}, air has a density of approximately {{cvt|1.225|kg/m3|lb/ft3|lk=on}}, which is about {{fract|1|800}} that of [[Chemistry:Water|water]], according to the [[International Standard Atmosphere]] (ISA).{{Citation needed|date=January 2019}} Pure liquid water is {{cvt|1000|kg/m3|lb/ft3}}.

Air density is a property used in many branches of science, engineering, and industry, including [[Earth:Aeronautics|aeronautics]];<ref name="aero_01en">Olson, Wayne M. (2000) AFFTC-TIH-99-01, Aircraft Performance Flight</ref><ref name="aero_02en">ICAO, Manual of the ICAO Standard Atmosphere (extended to 80 kilometres (262 500 feet)), Doc 7488-CD, Third Edition, 1993, {{ISBN|92-9194-004-6}}.</ref><ref name="aero_03en">Grigorie, T.L., Dinca, L., Corcau J-I. and Grigorie, O. (2010) Aircrafts'{{sic}} Altitude Measurement Using Pressure Information:Barometric Altitude and Density Altitude</ref> [[Chemistry:Gravimetric analysis|gravimetric analysis]];<ref name="grav_01en">A., Picard, R.S., Davis, M., Gläser and K., Fujii (CIPM-2007) Revised formula for the density of moist air</ref> the air-conditioning<ref name="hvac_01en">S. Herrmann, H.-J. Kretzschmar, and D.P. Gatley (2009), ASHRAE RP-1485 Final Report</ref> industry; [[Physics:Atmospheric sciences|atmospheric research]] and [[Physics:Meteorology|meteorology]];<ref name="mete_01en">F.R. Martins, R.A. Guarnieri e E.B. Pereira, (2007) O aproveitamento da energia eólica (The wind energy resource).</ref><ref name="mete_02en">Andrade, R.G., Sediyama, G.C., Batistella, M., Victoria, D.C., da Paz, A.R., Lima, E.P., Nogueira, S.F. (2009) Mapeamento de parâmetros biofísicos e da evapotranspiração no Pantanal usando técnicas de sensoriamento remoto</ref><ref name="mete_03en">Marshall, John and Plumb, R. Alan (2008), Atmosphere, ocean, and climate dynamics: an introductory text {{ISBN|978-0-12-558691-7}}.</ref> agricultural engineering (modeling and tracking of Soil-Vegetation-Atmosphere-Transfer (SVAT) models);<ref name="agri_01en">Pollacco, J. A., and B. P. Mohanty (2012), Uncertainties of Water Fluxes in Soil-Vegetation-Atmosphere Transfer Models: Inverting Surface Soil Moisture and Evapotranspiration Retrieved from Remote Sensing, Vadose Zone Journal, 11(3), {{doi|10.2136/vzj2011.0167}}.</ref><ref name="agri_02en">Shin, Y., B. P. Mohanty, and A.V.M. Ines (2013), Estimating Effective Soil Hydraulic Properties Using Spatially Distributed Soil Moisture and Evapotranspiration, Vadose Zone Journal, 12(3), {{doi|10.2136/vzj2012.0094}}.</ref><ref name="agri_03en">Saito, H., J. Simunek, and B. P. Mohanty (2006), Numerical Analysis of Coupled Water, Vapor, and Heat Transport in the Vadose Zone, Vadose Zone J. 5: 784-800.</ref> and the engineering community that deals with compressed air.<ref name="eng_01en">Perry, R.H. and Chilton, C.H., eds., Chemical Engineers' Handbook, 5th ed., McGraw-Hill, 1973.</ref>

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.

==Temperature==
Other things being equal, hotter air is less dense than cooler air and will thus rise through cooler air. This can be seen by using the [[Physics:Ideal gas law|ideal gas law]] as an approximation.

==Dry air==
The density of dry air can be calculated using the [[Physics:Ideal gas law|ideal gas law]], expressed as a function of [[Physics:Thermodynamic temperature|temperature]] and pressure:{{citation needed|date=November 2021}}
<math display=block>\begin{align}
\rho &= \frac{p}{R_\text{specific} T}\\
R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\
\rho &= \frac{pM}{RT} = \frac{pm}{k_{\rm B}T}\\
\end{align}</math>


*<math>\rho</math>, air density (kg/m<sup>3</sup>)<ref group="note" name="SInote01">In the SI unit system. However, other units can be used.</ref>
*<math>p</math>, absolute [[Physics:Pressure|pressure]] (Pa)<ref group="note" name="SInote01"/>
*<math>T</math>, [[Physics:Absolute temperature|absolute temperature]] (K)<ref group="note" name="SInote01"/>
*<math>R</math> is the [[Physics:Gas constant|gas constant]], {{val|8.31446261815324}} in [[Joule|J]]⋅[[Kelvin|K]]<sup>−1</sup>⋅[[Mole (unit)|mol]]<sup>−1</sup> <ref group="note" name="SInote01"/>
*<math>M</math> is the [[Physics:Molar mass|molar mass]] of dry air, approximately {{val|0.0289652}} in [[Kilogram|kg]]⋅[[Mole (unit)|mol]]<sup>−1</sup>.<ref group="note" name="SInote01"/>
*<math>k_{\rm B}</math> is the [[Physics:Boltzmann constant|Boltzmann constant]], {{val|1.380649||e=-23}} in [[Joule|J]]⋅[[Kelvin|K]]<sup>−1</sup><ref group="note" name="SInote01"/>
*<math>m</math> is the [[Physics:Molecular mass|molecular mass]] of dry air, approximately {{val|4.81|e=-26}} in [[Kilogram|kg]].<ref group="note" name="SInote01"/>
*<math>R_\text{specific}</math>, the specific gas constant for dry air, which using the values presented above would be approximately {{val|287.0500676}} in J⋅kg<sup>−1</sup>⋅K<sup>−1</sup><ref group="note" name="SInote01"/>.

Therefore:
* At [[Organization:International Union of Pure and Applied Chemistry|IUPAC]] [[Standard temperature and pressure|standard temperature and pressure]] (0{{nbsp}}[[Celsius|°C]] and 100{{nbsp}}k[[Pascal (unit)|Pa]]), dry air has a density of approximately 1.2754{{nbsp}}[[Kilogram|kg]]/m<sup>3</sup>.{{citation needed|date=November 2021}}

The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa:{{citation needed|date=November 2021}}
{{Temperature effect}}

==Humid air==
{{further|Physics:Humidity}}
[[File:Air density dependence on temperature and relative humidity.svg|thumb|right|400px|Effect of temperature and relative humidity on air density]]
The addition of [[Physics:Water vapor|water vapor]] to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. This occurs because the [[Physics:Molar mass|molar mass]] of water vapor (18{{nbsp}}g/mol) is less than the molar mass of dry air<ref group="note">as dry air is a mixture of gases, its molar mass is the weighted average of the molar masses of its components</ref> (around 29{{nbsp}}g/mol).  For any ideal gas, at a given temperature and pressure, the number of molecules is constant for a particular volume (see Avogadro's Law).  So when water molecules (water vapor) 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 (its density) decreases.

The density of humid air may be calculated by treating it as a mixture of [[Physics:Ideal gas|ideal gas]]es. In this case, the [[Physics:Partial pressure|partial pressure]] of [[Physics:Water vapor|water vapor]] is known as the [[Physics:Vapor pressure|vapor pressure]]. Using this method, error in the density calculation is less than 0.2% in the range of −10&nbsp;°C to 50&nbsp;°C. The density of humid air is found by:<ref name=wahiduddin_01>[http://wahiduddin.net/calc/density_altitude.htm Shelquist, R (2009) Equations - Air Density and Density Altitude]</ref>
<math display="block">
\rho_\text{humid air} = \frac{p_\text{d}}{R_\text{d} T} + \frac{p_\text{v}}{R_\text{v} T} = \frac{p_\text{d}M_\text{d} + p_\text{v}M_\text{v}}{R T}
</math>

where:
*<math>\rho_\text{humid air}</math>, density of the humid air (kg/m<sup>3</sup>)
*<math>p_\text{d}</math>, partial pressure of dry air (Pa)
*<math>R_\text{d}</math>, specific gas constant for dry air, 287.058{{nbsp}}J/(kg·K)
*<math>T</math>, temperature ([[Kelvin (unit)|K]])
*<math>p_\text{v}</math>, pressure of water vapor (Pa)
*<math>R_\text{v}</math>, specific gas constant for water vapor, 461.495{{nbsp}}J/(kg·K)
*<math>M_\text{d}</math>, molar mass of dry air, 0.0289652{{nbsp}}kg/mol
*<math>M_\text{v}</math>, molar mass of water vapor, 0.018016{{nbsp}}kg/mol
*<math>R</math>, [[Physics:Gas constant|universal gas constant]], 8.31446{{nbsp}}J/(K·mol)

The vapor pressure of water may be calculated from the saturation vapor pressure and [[Physics:Relative humidity|relative humidity]]. It is found by:
<math display="block">p_\text{v} = \phi p_\text{sat}</math>

where:
*<math>p_\text{v}</math>, vapor pressure of water
*<math>\phi</math>, relative humidity (0.0–1.0)
*<math>p_\text{sat}</math>, saturation vapor pressure

The saturation [[Physics:Vapour pressure of water|vapor pressure of water]] at any given temperature is the vapor pressure when relative humidity is 100%. One formula is [[Physics:Tetens equation|Tetens' equation]] from<ref name=wahiduddin_02>[http://wahiduddin.net/calc/density_algorithms.htm Shelquist, R (2009) Algorithms - Schlatter and Baker<!-- Bot generated title -->]</ref> used to find the saturation vapor pressure is:
<math display="block">p_\text{sat} = 6.1078 \times 10^{\frac{7.5 T}{T + 237.3}} </math>
where:
*<math>p_\text{sat}</math>, saturation vapor pressure (hPa)
*<math>T</math>, temperature (°[[Celsius|C]])

See [[Physics:Vapour pressure of water|vapor pressure of water]] for other equations.

The partial pressure of dry air <math>p_\text{d}</math> is found considering [[Physics:Partial pressure|partial pressure]], resulting in:
<math display="block">p_\text{d} = p - p_\text{v}</math>
where <math>p</math> simply denotes the observed absolute pressure.

==Variation with altitude==
[[Image:StandardAtmosphere.png|thumb|upright=2.0|Standard atmosphere: {{nowrap|1='''''p'''''<sub>0</sub> = 101.325 kPa}}, {{nowrap|1='''''T'''''<sub>0</sub> = 288.15 K}}, {{nowrap|1='''ρ'''<sub>0</sub> = 1.225 kg/m<sup>3</sup>}}]]

===Troposphere===
To calculate the density of air as a function of altitude, one requires additional parameters. For the troposphere, the lowest part (~10 km) of the atmosphere, they are listed below, along with their values according to the [[International Standard Atmosphere]], using for calculation the [[Physics:Gas constant|universal gas constant]] instead of the air specific constant:

*<math>p_0</math>, sea level standard atmospheric pressure, 101325{{nbsp}}[[Pascal (unit)|Pa]]
*<math>T_0</math>, sea level standard temperature, 288.15{{nbsp}}[[Kelvin (unit)|K]]
*<math>g</math>, earth-surface gravitational acceleration, 9.80665{{nbsp}}m/s<sup>2</sup>
*<math>L</math>, temperature lapse rate, 0.0065{{nbsp}}K/m
*<math>R</math>, ideal (universal) gas constant, 8.31446{{nbsp}}J/([[Mole (unit)|mol]]·K)
*<math>M</math>, [[Physics:Molar mass|molar mass]] of dry air, 0.0289652{{nbsp}}kg/mol

Temperature at altitude <math>h</math> meters above sea level is approximated by the following formula (only valid inside the [[Earth:Troposphere|troposphere]], no more than ~18{{nbsp}}km above Earth's surface (and lower away from Equator)):
<math display="block">T = T_0 - L h</math>

The pressure at altitude <math>h</math> is given by:
<math display="block">p = p_0 \left(1 - \frac{L h}{T_0}\right)^\frac{g M}{R L}</math>

Density can then be calculated according to a molar form of the [[Physics:Ideal gas law|ideal gas law]]:
<math display="block">
  \rho = \frac{p M}{R T}
       = \frac{p M}{R T_0 \left(1 - \frac{Lh}{T_0}\right)}
       = \frac{p_0 M}{R T_0} \left(1 - \frac{L h}{T_0} \right)^{\frac{g M}{R L} - 1}
</math>

where:
*<math>M</math>, [[Physics:Molar mass|molar mass]]
*<math>R</math>, [[Physics:Ideal gas constant|ideal gas constant]]
*<math>T</math>, [[Physics:Absolute temperature|absolute temperature]]
*<math>p</math>, absolute pressure

Note that the density close to the ground is <math display="inline">\rho_0 = \frac{p_0 M}{R T_0}</math>

It can be easily verified that the [[Physics:Hydrostatics#Hydrostatic pressure|hydrostatic equation]] holds:
<math display="block">\frac{dp}{dh} = -g\rho .</math>

====Exponential approximation====
As the temperature varies with height inside the troposphere by less than 25%, <math display="inline">\frac{Lh}{T_0} < 0.25</math> and one may approximate:
<math display="block">
   \rho = \rho_0 e^{\left(\frac{g M}{R L} - 1\right) \ln \left(1 - \frac{L h}{T_0}\right)}
  \approx \rho_0 e^{-\left(\frac{g M}{R L} - 1\right)\frac{L h}{T_0}}
        = \rho_0 e^{-\left(\frac{g M h}{R T_0} - \frac{L h}{T_0}\right)}
</math>

Thus:
<math display="block">\rho \approx \rho_0 e^{-h/H_n}</math>

Which is identical to the [[Physics:Isothermal|isothermal]] solution, except that ''H''<sub>''n''</sub>, the height scale of the exponential fall for density (as well as for [[Physics:Number density|number density]] n), is not equal to ''RT''<sub>0</sub>/''gM'' as one would expect for an isothermal atmosphere, but rather:
<math display="block">
\frac{1}{H_n} = \frac{g M}{R T_0} - \frac{L}{T_0}
</math>

Which gives ''H''<sub>''n''</sub> = 10.4{{nbsp}}km.

Note that for different gasses, the value of ''H''<sub>''n''</sub> differs, according to the molar mass ''M'': It is 10.9 for nitrogen, 9.2 for oxygen and 6.3 for [[Chemistry:Carbon dioxide|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:
<math display="block">
      p = p_0 e^{\frac{g M}{R L} \ln \left(1 - \frac{L h}{T_0}\right)}
  \approx p_0 e^{-\frac{g M}{R L}\frac{L h}{T_0}}
        = p_0 e^{-\frac{g M h}{R T_0}}
</math>

Which is identical to the [[Physics:Isothermal|isothermal]] solution, with the same height scale {{nowrap|''H''<sub>''p''</sub> {{=}} ''RT''<sub>0</sub>/''gM''}}. Note that the hydrostatic equation no longer holds for the exponential approximation (unless ''L'' is neglected).

''H''<sub>''p''</sub> is 8.4{{nbsp}}km, but for different gasses (measuring their partial pressure), 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 [[Physics:Gravitational acceleration|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'':
<math display="block">1 - \frac{p(h = 11\text{ km})}{p_0}  = 1 - \left(\frac{T(11\text{ km})}{T_0} \right)^\frac{g M}{R L} \approx 76\%</math>

For nitrogen, it is 75%, while for oxygen this is 79%, and for carbon dioxide, 88%.

===Tropopause===
Higher than the troposphere, at the [[Earth:Tropopause|tropopause]], the temperature is approximately constant with altitude (up to ~20{{nbsp}}km) and is 220{{nbsp}}K. This means that at this layer {{nowrap|''L'' {{=}} 0}} and {{nowrap|''T'' {{=}} 220 K}}, so that the exponential drop is faster, with {{nowrap|''H''<sub>TP</sub> {{=}} 6.3 km}} for air (6.5 for nitrogen, 5.7 for oxygen and 4.2 for carbon dioxide). Both the pressure and density obey this law, so, denoting the height of the border between the troposphere and the tropopause as ''U'':

<math display="block">\begin{align}
     p &=    p(U) e^{-\frac{h - U}{H_\text{TP}}} =    p_0 \left(1 - \frac{L U}{T_0}\right)^\frac{g M}{R L} e^{-\frac{h - U}{H_\text{TP}}} \\
  \rho &= \rho(U) e^{-\frac{h - U}{H_\text{TP}}} = \rho_0 \left(1 - \frac{L U}{T_0}\right)^{\frac{g M}{R L} - 1} e^{-\frac{h - U}{H_\text{TP}}}
\end{align}</math>

==Composition==


==See also==
*[[Earth:Atmosphere of Earth|Air]]
*Atmospheric drag
*[[Physics:Lighter than air|Lighter than air]]
*[[Physics:Density|Density]]
*[[Earth:Atmosphere of Earth|Atmosphere of Earth]]
*[[International Standard Atmosphere]]
*[[Earth:U.S. Standard Atmosphere|U.S. Standard Atmosphere]]
*[[Earth:NRLMSISE-00|NRLMSISE-00]]

==Notes==
{{Reflist|group=note}}

==References==
{{Reflist}}

==External links==
*[http://www.sengpielaudio.com/ConvDensi.htm Conversions of density units ρ by Sengpielaudio]
*[http://wahiduddin.net/calc/density_altitude.htm Air density and density altitude calculations and by Richard Shelquist]
*[http://www.sengpielaudio.com/calculator-airpressure.htm Air density calculations by Sengpielaudio (section under Speed of sound in humid air)]
*[http://www.enggcyclopedia.com/calculators/physical-properties/air-density-calculator/ Air density calculator by Engineering design encyclopedia]
*[https://web.archive.org/web/20140222174637/http://www.wolfdynamics.com/tools/atmospheric-pressure-calculator.html/ Atmospheric pressure calculator by wolfdynamics]
*[https://itunes.apple.com/en/app/air-itools/id598469643?mt=8 Air iTools - Air density calculator for mobile by JSyA]
*[https://www.nist.gov/system/files/documents/calibrations/CIPM-2007.pdf Revised formula for the density of moist air (CIPM-2007) by NIST]

[[Category:Atmospheric thermodynamics]]
[[Category:Density|A]]

{{Sourceattribution|Density of air|1}}