# 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.

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