In order to compare two air parcels at the same temperatures and different pressures, the concept of potential temperature is useful.

The relationship between temperature and pressure can be derived using the first law of thermodynamics as follows:

Consider adiabatic flow

dH = 0

0 = Cp dT - a dp

0 = Cp dT - (RT/p)*dp

Cp dT = (RT/p)*dp

dT/T = (R/Cp)*(dp/p)

d (ln T) = (R/Cp)*d(ln p)

Integrate from ( T, p ) to ( To, po )

Define potential temperature  as the temperature corresponding to the pressure po = 1000 mb,