Potential Temperature - Derivation

Derivation

The enthalpy form of the first law of thermodynamics can be written as:

where denotes the enthalpy change, the temperature, the change in entropy, the specific volume, and the pressure.

For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to:

For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, can be substituted into the 1st law yielding, after some rearrangement:

where the was used and both terms were divided by the product

Integrating yields:

and solving for, the temperature a parcel would acquire if moved adiabatically to the pressure level, you get: