Atmospheric Density
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The density of air or atmospheric density, denoted '' ρ'', is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
per unit
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of
Earth's atmosphere The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing fo ...
. Air density, like air pressure, decreases with increasing altitude. It also changes with variation in atmospheric pressure, temperature and
humidity Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation, dew, or fog to be present. Humidity depe ...
. At 101.325 kPa (abs) and 20 °C (68 °F), air has a density of approximately , according to the
International Standard Atmosphere The International Standard Atmosphere (ISA) is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations. It has been established to provide a ...
(ISA). At 101.325kPa (abs) and , air has a density of approximately , which is about that of
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
, according to the
International Standard Atmosphere The International Standard Atmosphere (ISA) is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations. It has been established to provide a ...
(ISA). Pure liquid water is . Air density is a property used in many branches of science, engineering, and industry, including
aeronautics Aeronautics is the science or art involved with the study, design, and manufacturing of air flight–capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society identifies ...
;Olson, Wayne M. (2000) AFFTC-TIH-99-01, Aircraft Performance FlightICAO, Manual of the ICAO Standard Atmosphere (extended to 80 kilometres (262 500 feet)), Doc 7488-CD, Third Edition, 1993, .Grigorie, T.L., Dinca, L., Corcau J-I. and Grigorie, O. (2010) Aircrafts' Altitude Measurement Using Pressure Information:Barometric Altitude and Density Altitude
gravimetric analysis Gravimetric analysis describes a set of methods used in analytical chemistry for the quantitative determination of an analyte (the ion being analyzed) based on its mass. The principle of this type of analysis is that once an ion's mass has been ...
;A., Picard, R.S., Davis, M., Gläser and K., Fujii (CIPM-2007) Revised formula for the density of moist air the air-conditioningS. Herrmann, H.-J. Kretzschmar, and D.P. Gatley (2009), ASHRAE RP-1485 Final Report industry; atmospheric research and
meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
;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 remotoMarshall, John and Plumb, R. Alan (2008), Atmosphere, ocean, and climate dynamics: an introductory text . agricultural engineering (modeling and tracking of Soil-Vegetation-Atmosphere-Transfer (SVAT) models);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), .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), .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. and the engineering community that deals with compressed air.Perry, R.H. and Chilton, C.H., eds., Chemical Engineers' Handbook, 5th ed., McGraw-Hill, 1973. 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
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
as an approximation.


Dry air

The density of dry air can be calculated using the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, expressed as a function of
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
and pressure: \begin \rho &= \frac\\ R_\text &= \frac = \frac\\ \rho &= \frac = \frac\\ \end where: :\rho, air density (kg/m3)In the SI unit system. However, other units can be used. :p, absolute
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
(Pa) :T,
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
(K) :R is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, in JK−1mol−1 :M is the
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
of dry air, approximately in kgmol−1. :k_ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, in JK−1 :m is the
molecular mass The molecular mass (''m'') is the mass of a given molecule: it is measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quanti ...
of dry air, approximately in kg. :R_\text, the
specific gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. Therefore: * At
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
standard temperature and pressure Standard temperature and pressure (STP) are standard sets of conditions for experimental measurements to be established to allow comparisons to be made between different sets of data. The most used standards are those of the International Union o ...
(0
°C The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The ...
and 100k Pa), dry air has a density of approximately 1.2754 kg/m3. * At 20°C and 101.325kPa, dry air has a density of 1.2041 kg/m3. * At 70 °F and 14.696
psi Psi, PSI or Ψ may refer to: Alphabetic letters * Psi (Greek) (Ψ, ψ), the 23rd letter of the Greek alphabet * Psi (Cyrillic) (Ѱ, ѱ), letter of the early Cyrillic alphabet, adopted from Greek Arts and entertainment * "Psi" as an abbreviatio ...
, 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:


Humid air

The addition of
water vapor (99.9839 °C) , - , Boiling point , , - , specific gas constant , 461.5 J/( kg·K) , - , Heat of vaporization , 2.27 MJ/kg , - , Heat capacity , 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous pha ...
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 In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
of water vapor (18g/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 29g/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 Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) or Avogadro-Ampère's hypothesis is an experimental gas law relating the volume of a gas to the amount of substance of gas present. The law is a specific c ...
). 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
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
es. In this case, the
partial pressure In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas ...
of
water vapor (99.9839 °C) , - , Boiling point , , - , specific gas constant , 461.5 J/( kg·K) , - , Heat of vaporization , 2.27 MJ/kg , - , Heat capacity , 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous pha ...
is known as the
vapor pressure Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
. 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_\text = \frac + \frac = \frac   Shelquist, R (2009) Equations - Air Density and Density Altitude
/ref> where: :\rho_\text, density of the humid air (kg/m3) :p_\text, partial pressure of dry air (Pa) :R_\text, specific gas constant for dry air, 287.058J/(kg·K) :T, temperature ( K) :p_\text, pressure of water vapor (Pa) :R_\text, specific gas constant for water vapor, 461.495J/(kg·K) :M_\text, molar mass of dry air, 0.0289652kg/mol :M_\text, molar mass of water vapor, 0.018016kg/mol :R,
universal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, 8.31446J/(K·mol) The vapor pressure of water may be calculated from the
saturation vapor pressure Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
and
relative humidity Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation, dew, or fog to be present. Humidity depe ...
. It is found by: :p_\text = \phi p_\text where: :p_\text, vapor pressure of water :\phi, relative humidity (0.0–1.0) :p_\text, saturation vapor pressure The saturation
vapor pressure of water The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The saturation vapour pressure is the pressure at which water vapour is in thermodynamic ...
at any given temperature is the vapor pressure when relative humidity is 100%. One formula is Tetens' equation fromShelquist, R (2009) Algorithms - Schlatter and Baker
/ref> used to find the saturation vapor pressure is: :p_\text = 6.1078 \times 10^ where: :p_\text, saturation vapor pressure (hPa) :T, temperature (° C) See
vapor pressure of water The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The saturation vapour pressure is the pressure at which water vapour is in thermodynamic ...
for other equations. The partial pressure of dry air p_\text is found considering
partial pressure In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas ...
, resulting in: :p_\text = p - p_\text Where p simply denotes the observed
absolute pressure Pressure measurement is the measurement of an applied force by a fluid (liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pressur ...
.


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 (~10 km) of the atmosphere, they are listed below, along with their values according to the
International Standard Atmosphere The International Standard Atmosphere (ISA) is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations. It has been established to provide a ...
, using for calculation the
universal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
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, earth-surface gravitational acceleration, 9.80665m/s2 :L,
temperature lapse rate The lapse rate is the rate at which an atmospheric variable, normally temperature in Earth's atmosphere, falls with altitude. ''Lapse rate'' arises from the word ''lapse'', in the sense of a gradual fall. In dry air, the adiabatic lapse rate is ...
, 0.0065K/m :R, ideal (universal) gas constant, 8.31446J/( mol·K) :M,
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
of dry air, 0.0289652kg/mol Temperature at altitude h meters above sea level is approximated by the following formula (only valid inside the
troposphere The troposphere is the first and lowest layer of the atmosphere of the Earth, and contains 75% of the total mass of the planetary atmosphere, 99% of the total mass of water vapour and aerosols, and is where most weather phenomena occur. From ...
, no more than ~18km 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 \left(1 - \frac\right)^\frac Density can then be calculated according to a molar form of the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
: : \rho = \frac = \frac = \frac \left(1 - \frac \right)^ where: :M,
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
:R,
ideal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
:T,
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
:p,
absolute pressure Pressure measurement is the measurement of an applied force by a fluid (liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pressur ...
Note that the density close to the ground is \rho_0 = \frac It can be easily verified that the
hydrostatic equation Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body " fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imm ...
holds: :\frac = -g\rho .


Exponential approximation

As the temperature varies with height inside the troposphere by less than 25%, \frac < 0.25 and one may approximate: : \rho = \rho_0 e^ \approx \rho_0 e^ = \rho_0 e^ Thus: :\rho \approx \rho_0 e^ Which is identical to the
isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a ...
solution, except that ''H''''n'', the height scale of the exponential fall for density (as well as for
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
n), is not equal to ''RT''0/''gM'' as one would expect for an isothermal atmosphere, but rather: : \frac = \frac - \frac Which gives ''H''''n'' = 10.4km. 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 Carbon dioxide (chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is transpar ...
. 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^ \approx p_0 e^ = p_0 e^ Which is identical to the
isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a ...
solution, with the same height scale . Note that the hydrostatic equation no longer holds for the exponential approximation (unless ''L'' is neglected). ''H''''p'' is 8.4km, 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
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
, 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 - \frac = 1 - \left(\frac \right)^\frac \approx 76\% 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 tropopause is the atmospheric boundary that demarcates the troposphere from the stratosphere; which are two of the five layers of the atmosphere of Earth. The tropopause is a thermodynamic gradient-stratification layer, that marks the end of ...
, the temperature is approximately constant with altitude (up to ~20km) and is 220K. This means that at this layer and , so that the exponential drop is faster, with 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'': :\begin p &= p(U) e^ = p_0 \left(1 - \frac\right)^\frac e^ \\ \rho &= \rho(U) e^ = \rho_0 \left(1 - \frac\right)^ e^ \end


Composition


See also

*
Air The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
*
Atmospheric drag In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
*
Lighter than air A lifting gas or lighter-than-air gas is a gas that has a density lower than normal atmospheric gases and rises above them as a result. It is required for aerostats to create buoyancy, particularly in lighter-than-air aircraft, which include free ...
*
Density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
*
Atmosphere of Earth The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...
*
International Standard Atmosphere The International Standard Atmosphere (ISA) is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations. It has been established to provide a ...
*
U.S. Standard Atmosphere The U.S. Standard Atmosphere is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations. The model, based on an existing international stand ...
*
NRLMSISE-00 NRLMSISE-00 is an empirical, global reference atmospheric model of the Earth from ground to space. It models the temperatures and densities of the atmosphere's components. A primary use of this model is to aid predictions of satellite orbital deca ...


Notes


References

{{Reflist, 30em


External links


Conversions of density units ρ by Sengpielaudio
* ttp://www.sengpielaudio.com/calculator-airpressure.htm Air density calculations by Sengpielaudio (section under Speed of sound in humid air)br>Air density calculator by Engineering design encyclopediaAir iTools - Air density calculator for mobile by JSyA
Atmospheric thermodynamics A