mass density derivations

1. mass density for air component from number density:

   symbol	description	unit	variable name
   \(M_{x}\)	molar mass for air component x	\(\frac{g}{mol}\)
   \(n_{x}\)	number density for 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 for 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 for total air from number density:

   symbol	description	unit	variable name
   \(M_{air}\)	molar mass for total air	\(\frac{g}{mol}\)	molar_mass {:}
   \(n\)	number density for total air	\(\frac{molec}{m^3}\)	number_density {:}
   \(N_A\)	Avogadro constant	\(\frac{1}{mol}\)
   \(\rho\)	mass density for 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 for 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 for air component x (e.g. \(\rho_{O_{3}}\))	\(\frac{kg}{m^3}\)	<species>_density {:}
   \(\sigma_{x}\)	column mass density for 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 for 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 for 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 for 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 for 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 for 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 for 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 for 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 for total air	\(\frac{kg}{m^3}\)	density {:}
   \(\sigma\)	column mass density for 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.

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