# 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.

$n_{surf} = \frac{p_{surf}}{kT_{surf}}$