# partial pressure derivations

1. partial pressure from number density 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.

   \[p_{x} = n_{x}kT\]

2. partial pressure from volume mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(p\) | pressure | \(Pa\) | pressure {:}
   \(p_{x}\) | partial pressure of air component x (e.g. \(p_{O_{3}}\)) | \(Pa\) | <species>_partial_pressure {:}
   \(\nu_{x}\) | volume mixing ratio of air component x (e.g. \(\nu_{O_{3}}\)) | \(ppv\) | <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.

   \[p_{x} = \nu_{x}p\]

3. partial pressure from volume mixing ratio dry air

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(p_{x}\) | partial pressure of air component x (e.g. \(p_{O_{3}}\)) | \(Pa\) | <species>_partial_pressure {:}
   \(\bar{\nu}_{x}\) | volume mixing ratio of air component x (e.g. \(\nu_{O_{3}}\)) | \(ppv\) | <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.

   \[p_{x} = \bar{\nu}_{x}p_{dry\_air}\]