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 for air component x (e.g. \(n_{O_{3}}\))	\(\frac{molec}{m^3}\)	<species>_number_density {:}
   \(p_{x}\)	partial pressure for 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 for air component x (e.g. \(p_{O_{3}}\))	\(Pa\)	<species>_partial_pressure {:}
   \(\nu_{x}\)	volume mixing ratio for 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 for air component x (e.g. \(p_{O_{3}}\))	\(Pa\)	<species>_partial_pressure {:}
   \(\bar{\nu}_{x}\)	volume mixing ratio for 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.

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