# mass density derivations

1. mass density of air component from number 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.

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

2. mass density of total air from number 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.

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

3. mass density of air component from column mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}
   \(\rho_{x}\) | mass density of air component x (e.g. \(\rho_{O_{3}}\)) | \(\frac{kg}{m^3}\) | <species>_density {:}
   \(\sigma_{x}\) | column mass density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{kg}{m^2}\) | <species>_column_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.

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

4. mass density of total air from dry air mass density and H2O mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\rho\) | mass density | \(\frac{kg}{m^3}\) | density {:}
   \(\rho_{dry\_air}\) | mass density of dry air | \(\frac{kg}{m^3}\) | dry_air_density {:}
   \(\rho_{H_{2}O}\) | mass density of H2O | \(\frac{kg}{m^3}\) | H2O_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.

   \[\rho = \rho_{dry\_air} + \rho_{H_{2}O}\]

5. mass density of dry air from total air mass density and H2O mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\rho\) | mass density | \(\frac{kg}{m^3}\) | density {:}
   \(\rho_{dry\_air}\) | mass density of dry air | \(\frac{kg}{m^3}\) | dry_air_density {:}
   \(\rho_{H_{2}O}\) | mass density of H2O | \(\frac{kg}{m^3}\) | H2O_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.

   \[\rho_{dry\_air} = \rho - \rho_{H_{2}O}\]

6. mass density of H2O from total air mass density and dry air mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\rho\) | mass density | \(\frac{kg}{m^3}\) | density {:}
   \(\rho_{dry\_air}\) | mass density of dry air | \(\frac{kg}{m^3}\) | dry_air_density {:}
   \(\rho_{H_{2}O}\) | mass density of H2O | \(\frac{kg}{m^3}\) | H2O_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.

   \[\rho_{H_{2}O} = \rho - \rho_{dry\_air}\]

7. mass density of total air from column mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}
   \(\rho\) | mass density of total air | \(\frac{kg}{m^3}\) | density {:}
   \(\sigma\) | column mass density of total air | \(\frac{kg}{m^2}\) | column_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.

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