# molar mass derivations

1. molar mass of total air from density and number density

symbol

description

unit

variable name

$$M_{air}$$

molar mass of total air

$$\frac{g}{mol}$$

molar_mass {:}

$$n$$

number density

$$\frac{molec}{m^3}$$

number_density {:}

$$N_A$$

Avogadro constant

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

$$\rho$$

mass density

$$\frac{kg}{m^3}$$

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.

$M_{air} = 10^{3}\frac{\rho N_{A}}{n}$
2. molar mass of total air from H2O mass mixing ratio

symbol

description

unit

variable name

$$M_{air}$$

molar mass of total air

$$\frac{g}{mol}$$

molar_mass {:}

$$M_{dry\_air}$$

molar mass of dry air

$$\frac{g}{mol}$$

$$M_{H_{2}O}$$

molar mass of H2O

$$\frac{g}{mol}$$

$$q_{H_{2}O}$$

mass mixing ratio of H2O

$$\frac{kg}{kg}$$

H2O_mass_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.

$M_{air} = \frac{M_{H_{2}O}M_{dry\_air}}{\left(1-q_{H_{2}O}\right)M_{H_{2}O} + q_{H_{2}O}M_{dry\_air}}$
3. molar mass of total air from H2O volume mixing ratio

symbol

description

unit

variable name

$$M_{air}$$

molar mass of total air

$$\frac{g}{mol}$$

molar_mass {:}

$$M_{dry\_air}$$

molar mass of dry air

$$\frac{g}{mol}$$

$$M_{H_{2}O}$$

molar mass of H2O

$$\frac{g}{mol}$$

$$\nu_{H_{2}O}$$

mass mixing ratio of H2O

$$ppv$$

H2O_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.

$M_{air} = M_{dry\_air}\left(1 - \nu_{H_{2}O}\right) + M_{H_{2}O}\nu_{H_{2}O}$