# number density derivations

1. number density of air component from mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(M_{x}\) | molar mass of air component x | \(\frac{g}{mol}\) |
   \(n_{x}\) | number density of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
   \(N_A\) | Avogadro constant | \(\frac{1}{mol}\) |
   \(\rho_{x}\) | mass density of air component x (e.g. \(\rho_{O_{3}}\)) | \(\frac{kg}{m^3}\) | <species>_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \frac{\rho_{x}N_{A}}{10^{-3}M_{x}}\]

2. number density of total air from mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(M_{air}\) | molar mass of total air | \(\frac{g}{mol}\) | molar_mass {:}
   \(n\) | number density of total air | \(\frac{molec}{m^3}\) | number_density {:}
   \(N_A\) | Avogadro constant | \(\frac{1}{mol}\) |
   \(\rho\) | mass density of total air | \(\frac{kg}{m^3}\) | density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n = \frac{\rho N_{A}}{10^{-3}M_{air}}\]

3. number density of total air from pressure and temperature

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(k\) | Boltzmann constant | \(\frac{kg m^2}{K s^2}\) |
   \(n\) | number density | \(\frac{molec}{m^3}\) | number_density {:}
   \(p\) | pressure | \(Pa\) | pressure {:}
   \(T\) | temperature | \(K\) | temperature {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n = \frac{p}{kT}\]

4. number density from volume mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(n\) | number density of total air | \(\frac{molec}{m^3}\) | number_density {:}
   \(n_{x}\) | number density of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
   \(\nu_{x}\) | volume mixing ratio of air component x (e.g. \(n_{O_{3}}\)) | \(ppv\) | <species>_volume_mixing_ratio {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \nu_{x}n\]

5. number density from volume mixing ratio dry air

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(n_{dry\_air}\) | number density of dry air | \(\frac{molec}{m^3}\) | dry_air_number_density {:}
   \(n_{x}\) | number density of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
   \(\bar{\nu}_{x}\) | volume mixing ratio of air component x (e.g. \(n_{O_{3}}\)) | \(ppv\) | <species>_volume_mixing_ratio_dry_air {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \bar{\nu}_{x}n_{dry\_air}\]

6. number density of air component from column number density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c_{x}\) | column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
   \(n_{x}\) | number density of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
   \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \frac{c_{x}}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]

7. number density of total air from dry air number density and H2O number density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(n\) | number density | \(\frac{molec}{m^3}\) | number_density {:}
   \(n_{dry\_air}\) | number density of dry air | \(\frac{molec}{m^3}\) | dry_air_number_density {:}
   \(n_{H_{2}O}\) | number density of H2O | \(\frac{molec}{m^3}\) | H2O_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n = n_{dry\_air} + n_{H_{2}O}\]

8. number density of dry air from total air number density and H2O number density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(n\) | number density | \(\frac{molec}{m^3}\) | number_density {:}
   \(n_{dry\_air}\) | number density of dry air | \(\frac{molec}{m^3}\) | dry_air_number_density {:}
   \(n_{H_{2}O}\) | number density of H2O | \(\frac{molec}{m^3}\) | H2O_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{dry\_air} = n - n_{H_{2}O}\]

9. number density of H2O from total air number density and dry air number density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(n\) | number density | \(\frac{molec}{m^3}\) | number_density {:}
   \(n_{dry\_air}\) | number density of dry air | \(\frac{molec}{m^3}\) | dry_air_number_density {:}
   \(n_{H_{2}O}\) | number density of H2O | \(\frac{molec}{m^3}\) | H2O_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{H_{2}O} = n - n_{dry\_air}\]

10. number density of total air from column number density

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(c\) | column number density | \(\frac{molec}{m^2}\) | column_number_density {:}
    \(n\) | number density | \(\frac{molec}{m^3}\) | number_density {:}
    \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[n = \frac{c}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]

11. number density of air component from partial pressure and temperature

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(k\) | Boltzmann constant | \(\frac{kg m^2}{K s^2}\) |
    \(n_{x}\) | number density of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
    \(p_{x}\) | partial pressure of air component x (e.g. \(p_{O_{3}}\)) | \(Pa\) | <species>_partial_pressure {:}
    \(T\) | temperature | \(K\) | temperature {:}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[n_{x} = \frac{p_{x}}{kT}\]

12. surface number density of total air from surface pressure and surface temperature

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(k\) | Boltzmann constant | \(\frac{kg m^2}{K s^2}\) |
    \(n_{surf}\) | surface number density | \(\frac{molec}{m^3}\) | surface_number_density {:}
    \(p_{surf}\) | surface pressure | \(Pa\) | surface_pressure {:}
    \(T_{surf}\) | surface temperature | \(K\) | surface_temperature {:}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[n_{surf} = \frac{p_{surf}}{kT_{surf}}\]