Virtual Temperature - Derivation

Derivation

Consider an air parcel containing masses and of water vapor in a given volume . The density is given by:

where and are the densities of dry air and water vapor would respectively have when occupying the volume of the air parcel. Rearranging the standard ideal gas equation with these variables gives:

and

Solving for the densities in each equation and combining with the law of partial pressures yields:

Then, solving for and using is approximately 0.622 in Earth's atmosphere:

where the virtual temperature is:

We now have a non-linear scalar for temperature dependent purely on the unitless value allowing for varying amounts of water vapor in an air parcel. This virtual temperature in units of Kelvin can be used seamlessly in any thermodynamic equation necessitating it.