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Density of air

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Mass per unit volume of the Earth's atmosphere

Thedensity of air oratmospheric density, denotedρ,^{[note 1]} is themass per unitvolume ofEarth's atmosphere at a given point and time. Air density, like air pressure, decreases with increasing altitude. It also changes with variations in atmospheric pressure, temperature, andhumidity. According to theISO International Standard Atmosphere (ISA), the standard sea level density of air at 101.325kPa (abs) and 15 °C (59 °F) is 1.2250 kg/m³ (0.07647 lb/cu ft).^[1] This is about1⁄800 that ofwater, which has a density of about 1,000 kg/m³ (62 lb/cu ft).

Air density is a property used in many branches of science, engineering, and industry, includingaeronautics;^[2]^[3]^[4]gravimetric analysis;^[5] the air-conditioning industry;^[6]atmospheric research andmeteorology;^[7]^[8]^[9] agricultural engineering (modeling and tracking of Soil-Vegetation-Atmosphere-Transfer (SVAT) models);^[10]^[11]^[12] and the engineering community that deals with compressed air.^[13]

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

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Other things being equal (most notably the pressure and humidity), hotter air is less dense than cooler air and will thus rise while cooler air tends to fall due tobuoyancy. This can be seen by using theideal gas law as an approximation.

Dry air

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The density of dry air can be calculated using theideal gas law, expressed as a function oftemperature and pressure:^{[citation needed]} ${\begin{aligned}\rho &={\frac {p}{R_{\text{specific}}T}}\\R_{\text{specific}}&={\frac {R}{M}}={\frac {k_{\rm {B}}}{m}}\\\rho &={\frac {pM}{RT}}={\frac {pm}{k_{\rm {B}}T}}\\\end{aligned}}$

where:

$\rho$ , air density (kg/m³)^{[note 2]}
$p {\displaystyle p}$ , absolutepressure (Pa)^{[note 2]}
$T {\displaystyle T}$ ,absolute temperature (K)^{[note 2]}
$R {\displaystyle R}$ is thegas constant,8.31446261815324 inJ⋅K⁻¹⋅mol⁻¹^{[note 2]}
$M {\displaystyle M}$ is themolar mass of dry air, approximately0.0289652 inkg⋅mol⁻¹.^{[note 2]}
$k_{\rm {B}}$ is theBoltzmann constant,1.380649×10⁻²³ inJ⋅K⁻¹^{[note 2]}
$m {\displaystyle m}$ is themolecular mass of dry air, approximately4.81×10⁻²⁶ inkg.^{[note 2]}
$R_{\text{specific}}$ , thespecific gas constant for dry air, which using the values presented above would be approximately287.0500676 in J⋅kg⁻¹⋅K⁻¹.^{[note 2]}

Therefore:

AtIUPAC standard temperature and pressure (0 °C and 100 kPa), dry air has a density of approximately 1.2754 kg/m³.
At 20 °C and 101.325 kPa, dry air has a density of 1.2041 kg/m³.
At 70 °F and 14.696 psi, dry air has a density of 0.074887 lb/ft³.

The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa:^{[citation needed]}

Effect of temperature on properties of air
Celsius temperature θ [°C]	Speed of sound c [m/s]	Density of air ρ [kg/m³]	Characteristic specific acoustic impedance z₀ [Pa⋅s/m]
35	351.88	1.1455	403.2
30	349.02	1.1644	406.5
25	346.13	1.1839	409.4
20	343.21	1.2041	413.3
15	340.27	1.2250	416.9
10	337.31	1.2466	420.5
5	334.32	1.2690	424.3
0	331.30	1.2922	428.0
−5	328.25	1.3163	432.1
−10	325.18	1.3413	436.1
−15	322.07	1.3673	440.3
−20	318.94	1.3943	444.6
−25	315.77	1.4224	449.1

Humid air

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Further information:Humidity

Effect of temperature and relative humidity on air density

The addition ofwater vapor to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. This occurs because themolar mass of water vapor (18 g/mol) is less than the molar mass of dry air^{[note 3]} (around 29 g/mol). For any ideal gas, at a given temperature and pressure, the number of molecules is constant for a particular volume (seeAvogadro'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 from increasing or temperature from decreasing. 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 ofideal gases. In this case, thepartial pressure ofwater vapor is known as thevapor 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:^[14] $\rho _{\text{humid air}}={\frac {p_{\text{d}}}{R_{\text{d}}T}}+{\frac {p_{\text{v}}}{R_{\text{v}}T}}={\frac {p_{\text{d}}M_{\text{d}}+p_{\text{v}}M_{\text{v}}}{RT}}$

where:

$\rho _{\text{humid air}}$ , density of the humid air (kg/m³)
$p_{\text{d}}$ , partial pressure of dry air (Pa)
$R_{\text{d}}$ , specific gas constant for dry air, 287.058 J/(kg·K)
$T {\displaystyle T}$ , temperature (K)
$p_{\text{v}}$ , pressure of water vapor (Pa)
$R_{\text{v}}$ , specific gas constant for water vapor, 461.495 J/(kg·K)
$M_{\text{d}}$ , molar mass of dry air, 0.0289652 kg/mol
$M_{\text{v}}$ , molar mass of water vapor, 0.018016 kg/mol
$R {\displaystyle R}$ ,universal gas constant, 8.31446 J/(K·mol)

The vapor pressure of water may be calculated from thesaturation vapor pressure andrelative humidity. It is found by: $p_{\text{v}}=\phi p_{\text{sat}}$

where:

$p_{\text{v}}$ , vapor pressure of water
$\phi$ , relative humidity (0.0–1.0)
$p_{\text{sat}}$ , saturation vapor pressure

The saturationvapor pressure of water at any given temperature is the vapor pressure when relative humidity is 100%. One formula isTetens' equation from^[15] used to find the saturation vapor pressure is: $p_{\text{sat}}=0.61078\exp \left({\frac {17.27(T-273.15)}{T-35.85}}\right)$ where:

$p_{\text{sat}}$ , saturation vapor pressure (kPa)
$T {\displaystyle T}$ , temperature (K)

Seevapor pressure of water for other equations.

The partial pressure of dry air $p_{\text{d}}$ is found consideringpartial pressure, resulting in: $p_{\text{d}}=p-p_{\text{v}}$ where $p {\displaystyle p}$ simply denotes the observedabsolute pressure.

Variation with altitude

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Further information:Barometric formula § Density equations

Standard atmosphere:p₀ = 101.325 kPa,T₀ = 288.15 K,ρ₀ = 1.225 kg/m³

Troposphere

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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 theInternational Standard Atmosphere, using for calculation theuniversal gas constant instead of the air specific constant:

$p_{0}$ , sea level standard atmospheric pressure, 101325 Pa
$T_{0}$ , sea level standard temperature, 288.15 K
$g {\displaystyle g}$ , earth-surface gravitational acceleration, 9.80665 m/s²
$L {\displaystyle L}$ ,temperature lapse rate, 0.0065 K/m
$R {\displaystyle R}$ , ideal (universal) gas constant, 8.31446 J/(mol·K)
$M {\displaystyle M}$ ,molar mass of dry air, 0.0289652 kg/mol

Temperature at altitude $h {\displaystyle h}$ meters above sea level is approximated by the following formula (only valid inside thetroposphere, no more than ~18 km above Earth's surface (and lower away from Equator)): $T=T_{0}-Lh$

The pressure at altitude $h {\displaystyle h}$ is given by: $p=p_{0}\left(1-{\frac {Lh}{T_{0}}}\right)^{\frac {gM}{RL}}$

Density can then be calculated according to a molar form of theideal gas law: $\rho ={\frac {pM}{RT}}={\frac {pM}{RT_{0}\left(1-{\frac {Lh}{T_{0}}}\right)}}={\frac {p_{0}M}{RT_{0}}}\left(1-{\frac {Lh}{T_{0}}}\right)^{{\frac {gM}{RL}}-1}$

where:

$M {\displaystyle M}$ ,molar mass
$R {\displaystyle R}$ ,ideal gas constant
$T {\displaystyle T}$ ,absolute temperature
$p {\displaystyle p}$ ,absolute pressure

Note that the density close to the ground is ${\textstyle \rho _{0}={\frac {p_{0}M}{RT_{0}}}}$

It can be easily verified that thehydrostatic equation holds: ${\frac {dp}{dh}}=-g\rho .$

Exponential approximation

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As the temperature varies with height inside the troposphere by less than 25%, ${\textstyle {\frac {Lh}{T_{0}}}<0.25}$ and one may approximate: $\rho =\rho _{0}e^{\left({\frac {gM}{RL}}-1\right)\ln \left(1-{\frac {Lh}{T_{0}}}\right)}\approx \rho _{0}e^{-\left({\frac {gM}{RL}}-1\right){\frac {Lh}{T_{0}}}}=\rho _{0}e^{-\left({\frac {gMh}{RT_{0}}}-{\frac {Lh}{T_{0}}}\right)}$

Thus: $\rho \approx \rho _{0}e^{-h/H_{n}}$

Which is identical to theisothermal solution, except thatH_n, the height scale of the exponential fall for density (as well as fornumber density n), is not equal toRT₀/gM as one would expect for an isothermal atmosphere, but rather: ${\frac {1}{H_{n}}}={\frac {gM}{RT_{0}}}-{\frac {L}{T_{0}}}$

Which givesH_n = 10.4 km.

Note that for different gasses, the value ofH_n differs, according to the molar massM: It is 10.9 for nitrogen, 9.2 for oxygen and 6.3 forcarbon 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: $p=p_{0}e^{{\frac {gM}{RL}}\ln \left(1-{\frac {Lh}{T_{0}}}\right)}\approx p_{0}e^{-{\frac {gM}{RL}}{\frac {Lh}{T_{0}}}}=p_{0}e^{-{\frac {gMh}{RT_{0}}}}$

Which is identical to theisothermal solution, with the same height scaleH_p =RT₀/gM. Note that the hydrostatic equation no longer holds for the exponential approximation (unlessL is neglected).

H_p is 8.4 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

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Further note that sinceg, Earth'sgravitational acceleration, is approximately constant with altitude in the atmosphere, the pressure at heighth is proportional to the integral of the density in the column aboveh, and therefore to the mass in the atmosphere above heighth. Therefore, the mass fraction of the troposphere out of all the atmosphere is given using the approximated formula forp: $1-{\frac {p(h=11{\text{ km}})}{p_{0}}}=1-\left({\frac {T(11{\text{ km}})}{T_{0}}}\right)^{\frac {gM}{RL}}\approx 76\%$

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

Tropopause

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Higher than the troposphere, at thetropopause, the temperature is approximately constant with altitude (up to ~20 km) and is 220 K. This means that at this layerL = 0 andT = 220 K, so that the exponential drop is faster, withH_TP = 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 asU:

${\begin{aligned}p&=p(U)e^{-{\frac {h-U}{H_{\text{TP}}}}}=p_{0}\left(1-{\frac {LU}{T_{0}}}\right)^{\frac {gM}{RL}}e^{-{\frac {h-U}{H_{\text{TP}}}}}\\\rho &=\rho (U)e^{-{\frac {h-U}{H_{\text{TP}}}}}=\rho _{0}\left(1-{\frac {LU}{T_{0}}}\right)^{{\frac {gM}{RL}}-1}e^{-{\frac {h-U}{H_{\text{TP}}}}}\end{aligned}}$

Composition

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Composition of dry atmosphere, by volume^{[▽ note 1]}^{[▽ note 2]}
Gas(and others)		Various^[16]		CIPM-2007^[17]		ASHRAE^[18]		Schlatter^[19]		ICAO^[20]		US StdAtm76^[21]		▼ Tap to expand or collapse table ▲
Gas(and others)		ppmv^{[▽ note 3]}	percentage	ppmv	percentage	ppmv	percentage	ppmv	percentage	ppmv	percentage	ppmv	percentage
Nitrogen	N₂	780,800	78.080%	780,848	78.0848%	780,818	78.0818%	780,840	78.084%	780,840	78.084%	780,840	78.084%
Oxygen	O₂	209,500	20.950%	209,390	20.9390%	209,435	20.9435%	209,460	20.946%	209,476	20.9476%	209,476	20.9476%
Argon	Ar	9,340	0.9340%	9,332	0.9332%	9,332	0.9332%	9,340	0.9340%	9,340	0.9340%	9,340	0.9340%
Carbon dioxide	CO₂	397.8	0.03978%	400	0.0400%	385	0.0385%	384	0.0384%	314	0.0314%	314	0.0314%
Neon	Ne	18.18	0.001818%	18.2	0.00182%	18.2	0.00182%	18.18	0.001818%	18.18	0.001818%	18.18	0.001818%
Helium	He	5.24	0.000524%	5.2	0.00052%	5.2	0.00052%	5.24	0.000524%	5.24	0.000524%	5.24	0.000524%
Methane	CH₄	1.81	0.000181%	1.5	0.00015%	1.5	0.00015%	1.774	0.0001774%	2	0.0002%	2	0.0002%
Krypton	Kr	1.14	0.000114%	1.1	0.00011%	1.1	0.00011%	1.14	0.000114%	1.14	0.000114%	1.14	0.000114%
Hydrogen	H₂	0.55	0.000055%	0.5	0.00005%	0.5	0.00005%	0.56	0.000056%	0.5	0.00005%	0.5	0.00005%
Nitrous oxide	N₂O	0.325	0.0000325%	0.3	0.00003%	0.3	0.00003%	0.320	0.0000320%	0.5	0.00005%	-	-
Carbon monoxide	CO	0.1	0.00001%	0.2	0.00002%	0.2	0.00002%	-	-	-	-	-	-
Xenon	Xe	0.09	0.000009%	0.1	0.00001%	0.1	0.00001%	0.09	0.000009%	0.087	0.0000087%	0.087	0.0000087%
Nitrogen dioxide	NO₂	0.02	0.000002%	-	-	-	-	-	-	Up to 0.02	Up to 0.000002%	-	-
Iodine	I₂	0.01	0.000001%	-	-	-	-	-	-	Up to 0.01	Up to 0.000001%	-	-
Ammonia	NH₃	trace	trace	-	-	-	-	-	-			-	-
Sulfur dioxide	SO₂	trace	trace	-	-	-	-	-	-	Up to 1.00	Up to 0.0001%	-	-
Ozone	O₃	0.02 to 0.07	2 to 7×10⁻⁶%	-	-	-	-	0.01 to 0.10	1 to 10×10⁻⁶%	Up to 0.02 to 0.07^{[▽ note 4]}	Up to 2 to 7×10⁻⁶%^{[▽ note 4]}	-	-
Trace to 30 ppm^{[▽ note 5]}	—	-	-	-	-	2.9	0.00029%	-	-	-	-	-	-
Dry air total	air	1,000,000	100.00%	1,000,000	100.00%	1,000,000	100.00%	1,000,000	100.00%	1,000,000	100.00%	1,000,080	100.00%
Not included in above dry atmosphere
Water vapor	H₂O	~0.25% by mass over full atmosphere, locally 0.001–5% by volume.^[22]						~0.25% by mass over full atmosphere, locally 0.001–5% by volume.^[22]
▽ notes ^Concentration pertains to the troposphere ^Total values may not add up to exactly 100% due to roundoff and uncertainty. ^ppmv:parts per million by volume.Volume fraction is equal tomole fraction for ideal gas only, seevolume (thermodynamics). ^^a ^bO₃ concentration up to 0.07 ppmv (7×10⁻⁶%) in summer and up to 0.02 ppmv (2×10⁻⁶%) in winter. ^Volumetric composition value adjustment factor (sum of all trace gases, below the CO₂, and adjusts for 30 ppmv)

References

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Notes

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^Rho is widely used as a generic symbol for density
^^a ^b ^c ^d ^e ^f ^g ^hIn the SI unit system. However, other units can be used.
^as dry air is a mixture of gases, its molar mass is the weighted average of the molar masses of its components

Citations

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^Torenbeek, Egbert (2013). "Appendix B: International Standard Atmosphere".Advanced Aircraft Design. John Wiley & Sons, Ltd.ISBN 978-1-118-56810-1. Retrieved2025-04-27.
^Olson, Wayne M. (2000) AFFTC-TIH-99-01, Aircraft Performance Flight
^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.
^Grigorie, T.L., Dinca, L., Corcau J-I. and Grigorie, O. (2010) Aircraft's Altitude Measurement Using Pressure Information:Barometric Altitude and Density Altitude
^Picard et al. 2008.
^Herrmann, Kretzschmar & Gatley 2009.
^F.R. Martins, R.A. Guarnieri e E.B. Pereira, (2007) O aproveitamento da energia eólica (The wind energy resource).
^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
^Marshall, John and Plumb, R. Alan (2008), Atmosphere, ocean, and climate dynamics: an introductory textISBN 978-0-12-558691-7.
^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.
^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.
^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.
^Perry, R.H. and Chilton, C.H., eds., Chemical Engineers' Handbook, 5th ed., McGraw-Hill, 1973.
^Shelquist, R (2009) Equations - Air Density and Density Altitude
^Shelquist, R (2009) Algorithms - Schlatter and Baker
^Partial sources for figures: Base constituents,Nasa earth factsheet, (updated 2014-03). Carbon dioxide,NOAA Earth System Research Laboratory, (updated 2014-03). Methane and Nitrous Oxide, The NOAA Annual greenhouse gas index(AGGI)Greenhouse gas-Figure 2, (updated 2014-03).
^A., Picard, R.S., Davis, M., Gläser and K., Fujii (2008), Revised formula for the density of moist air (CIPM-2007), Metrologia 45 (2008) 149–155 doi:10.1088/0026-1394/45/2/004, pg 151 Table 1
^S. Herrmann, H.-J. Kretzschmar, and D.P. Gatley (2009), ASHRAE RP-1485 Final Report Thermodynamic Properties of Real Moist Air,Dry Air, Steam, Water, and Ice pg 16 Table 2.1 and 2.2
^Thomas W. Schlatter (2009), Atmospheric Composition and Vertical Structure pg 15 Table 2
^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. pg E-x Table B
^U.S. Committee on Extension to the Standard Atmosphere (COESA) (1976) U.S. Standard Atmosphere, 1976 pg 03 Table 3
^^a ^bWallace, John M. and Peter V. Hobbs. Atmospheric Science;An Introductory Survey. Elsevier. Second Edition, 2006.ISBN 978-0-12-732951-2. Chapter 1

Sources

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Herrmann, Sebastian; Kretzschmar, Hans-Joachim; Gatley, Donald P. (2009). "Thermodynamic Properties of Real Moist Air, Dry Air, Steam, Water, and Ice (RP-1485)".HVAC&R Research.15 (5).doi:10.1080/10789669.2009.10390874.
Picard, A.; Davis, R.S.; Gläser, M.; Fujii, K. (2008). "Revised formula for the density of moist air".Metrologia.45 (2).doi:10.1088/0026-1394/45/2/004.