Editing Density of air (section)

===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 [[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>, [[adiabatic lapse rate|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>, [[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 [[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 [[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>, [[molar mass]]
*<math>R</math>, [[ideal gas constant]]
*<math>T</math>, [[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 [[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 [[isothermal]] solution, except that ''H''<sub>''n''</sub>, the height scale of the exponential fall for density (as well as for [[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 [[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 [[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 [[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%.