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.

   $${\rho }_x=\frac{{\sigma }_x}{|z^B(2)-z^B(1)|}$$

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 {:}
   ${\rho }_{dry\mathrm{\_}air}$	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.

   $$\rho ={\rho }_{dry\mathrm{\_}air}+{\rho }_{H_{2}O}$$

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 {:}
   ${\rho }_{dry\mathrm{\_}air}$	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.

   $${\rho }_{dry\mathrm{\_}air}=\rho -{\rho }_{H_{2}O}$$

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 {:}
   ${\rho }_{dry\mathrm{\_}air}$	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.

   $${\rho }_{H_{2}O}=\rho -{\rho }_{dry\mathrm{\_}air}$$

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.

   $$\rho =\frac{\sigma }{|z^B(2)-z^B(1)|}$$