What is the adiabatic lapse rate of air?

That’s the question posed by Scottish Sceptic here.

When explaining the greenhouse warming effect, I’ve avoided going into the cause of the adiabatic temperature change of the atmosphere as we get higher and instead used a hand waving argument that expanding air is cooler which has been enough to explain the necessary temperature gradient up through the atmosphere (link).

However, this isn’t really the mechanism behind the adiabatic lapse rate.