# 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 {:}

$$N_A$$

$$\frac{1}{mol}$$

$$\rho_{x}$$

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 {:}

$$N_A$$

$$\frac{1}{mol}$$

$$\rho$$

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.

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