density of air

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 to buoyancy. This can be seen by using the ideal gas law as an approximation.

The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure:{{citation needed|date=November 2021}}

$$\backslash begin\{align\}$$

\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}

where:

• $\backslash rho$, air density (kg/m³)In the SI unit system. However, other units can be used.
• $p$, absolute pressure (Pa)
• $T$, absolute temperature (K)
• $R$ is the gas constant, {{val|8.31446261815324}} in J⋅K⁻¹⋅mol⁻¹
• $M$ is the molar mass of dry air, approximately {{val|0.0289652}} in kg⋅mol⁻¹.
• $k\_\{\backslash rm\; B\}$ is the Boltzmann constant, {{val|1.380649||e=-23}} in J⋅K⁻¹
• $m$ is the molecular mass of dry air, approximately {{val|4.81|e=-26}} in kg.
• $R\_\backslash text\{specific\}$, the specific gas constant for dry air, which using the values presented above would be approximately {{val|287.0500676}} in J⋅kg⁻¹⋅K⁻¹.

Therefore:

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

The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa:

{{Temperature effect}}

{{further|Humidity}}

File:Air density dependence on temperature and relative humidity.svg

The addition of 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 molar mass of water vapor (18{{nbsp}}g/mol) is less than the molar mass of dry airas dry air is a mixture of gases, its molar mass is the weighted average of the molar masses of its components (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 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 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:[http://wahiduddin.net/calc/density_altitude.htm Shelquist, R (2009) Equations - Air Density and Density Altitude]

\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}

where:

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

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

$$p\_\backslash text\{v\}\; =\; \backslash phi\; p\_\backslash text\{sat\}$$

where:

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

The saturation vapor pressure of water at any given temperature is the vapor pressure when relative humidity is 100%. One formula is Tetens' equation from[http://wahiduddin.net/calc/density_algorithms.htm Shelquist, R (2009) Algorithms - Schlatter and Baker] used to find the saturation vapor pressure is:

$$p\_\backslash text\{sat\}\; =0.61078\; \backslash exp\backslash left(\backslash frac\{17.27\; (T-273.15)\}\{T-35.85\}\backslash right)$$

where:

• $p\_\backslash text\{sat\}$, saturation vapor pressure (kPa)
• $T$, temperature (K)

See vapor pressure of water for other equations.

The partial pressure of dry air $p\_\backslash text\{d\}$ is found considering partial pressure, resulting in:

$$p\_\backslash text\{d\}\; =\; p\; -\; p\_\backslash text\{v\}$$

where $p$ simply denotes the observed absolute pressure.

{{further|Barometric formula#Density equations}}

Image:StandardAtmosphere.png

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 universal gas constant instead of the air specific constant:

• $p\_0$, sea level standard atmospheric pressure, 101325{{nbsp}}Pa
• $T\_0$, sea level standard temperature, 288.15{{nbsp}}K
• $g$, earth-surface gravitational acceleration, 9.80665{{nbsp}}m/s²
• $L$, temperature lapse rate, 0.0065{{nbsp}}K/m
• $R$, ideal (universal) gas constant, 8.31446{{nbsp}}J/(mol·K)
• $M$, molar mass of dry air, 0.0289652{{nbsp}}kg/mol

Temperature at altitude $h$ meters above sea level is approximated by the following formula (only valid inside the troposphere, no more than ~18{{nbsp}}km above Earth's surface (and lower away from Equator)):

$$T\; =\; T\_0\; -\; L\; h$$

The pressure at altitude $h$ is given by:

$$p\; =\; p\_0\; \backslash left(1\; -\; \backslash frac\{L\; h\}\{T\_0\}\backslash right)^\backslash frac\{g\; M\}\{R\; L\}$$

Density can then be calculated according to a molar form of the ideal gas law:

\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}

where:

• $M$, molar mass
• $R$, ideal gas constant
• $T$, absolute temperature
• $p$, absolute pressure

Note that the density close to the ground is $\backslash rho\_0\; =\; \backslash frac\{p\_0\; M\}\{R\; T\_0\}$

It can be easily verified that the hydrostatic equation holds:

$$\backslash frac\{dp\}\{dh\}\; =\; -g\backslash rho\; .$$

As the temperature varies with height inside the troposphere by less than 25%, and one may approximate:

\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)}

Thus:

$$\backslash rho\; \backslash approx\; \backslash rho\_0\; e^\{-h/H\_n\}$$

Which is identical to the isothermal solution, except that H_n, the height scale of the exponential fall for density (as well as for number density n), is not equal to RT₀/gM as one would expect for an isothermal atmosphere, but rather:

\frac{1}{H_n} = \frac{g M}{R T_0} - \frac{L}{T_0}

Which gives H_n = 10.4{{nbsp}}km.

Note that for different gasses, the value of H_n 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:

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}}

Which is identical to the isothermal solution, with the same height scale {{nowrap|H_p {{=}} RT₀/gM}}. Note that the hydrostatic equation no longer holds for the exponential approximation (unless L is neglected).

H_p 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.

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:

$$1\; -\; \backslash frac\{p(h\; =\; 11\backslash text\{\; km\})\}\{p\_0\}\; =\; 1\; -\; \backslash left(\backslash frac\{T(11\backslash text\{\; km\})\}\{T\_0\}\; \backslash right)^\backslash frac\{g\; M\}\{R\; L\}\; \backslash approx\; 76\backslash \%$$

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

Higher than the troposphere, at the 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_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 as U:

$$\backslash begin\{align\}$$

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}

{{Table composition of dry atmosphere}}

• Air
• Atmospheric drag
• Lighter than air
• Density
• Atmosphere of Earth
• International Standard Atmosphere
• U.S. Standard Atmosphere
• NRLMSISE-00

{{Reflist|group=note}}

{{Reflist}}

• [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] {{Webarchive|url=https://web.archive.org/web/20211218111827/http://www.enggcyclopedia.com/calculators/physical-properties/air-density-calculator/ |date=2021-12-18 }}
• [https://web.archive.org/web/20140222174637/http://www.wolfdynamics.com/tools/atmospheric-pressure-calculator.html/ Atmospheric pressure calculator by wolfdynamics]
• [https://web.archive.org/web/20160306233150/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

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Category:Meteorological quantities

Category:Vertical distributions